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Vale-Silva, Maria F¨alth Savitski, Jovan Tanevski, Julio Saez-Rodriguez +! +Abstract +Single-cell RNA sequencing (scRNA-seq) and spatially-resolved imaging/sequencing technologies have revolutionized biomedical +research. On one hand, scRNA-seq data provides for individual cells information about a large portion of the transcriptome, but does not +include the spatial context of the cells. On the other hand, spatially resolved measurements come with a trade-off between resolution, +throughput and gene coverage. Combining data from these two modalities can provide a spatially resolved picture with enhances +resolution and gene coverage. Several methods have been recently developed to integrate these modalities, but they use only the +expression of genes available in both modalities. They don’t incorporate other relevant and available features, especially the spatial +context. We propose DOT, a novel optimization framework for assigning cell types to tissue locations. Our model (i) incorporates ideas +from Optimal Transport theory to leverage not only joint but also distinct features, such as the spatial context, (ii) introduces scale- +invariant distance functions to account for differences in the sensitivity of different measurement technologies, and (iii) provides control +over the abundance of cells of different types in the tissue. We present a fast implementation based on the Frank-Wolfe algorithm and +we demonstrate the effectiveness of DOT on correctly assigning cell types or estimating the expression of missing genes in spatial data +coming from two areas of the brain, the developing heart, and breast cancer samples. +Index Terms +Optimal Transport, optimization, Frank-Wolfe, single-cell, biology, spatial, tissue, decomposition, deconvolution +Arezou Rahimi is with the Institute for Computational Biomedicine, Faculty of Medicine, Heidelberg University, Heidelberg University Hospital, and Cellzome +GmbH, GlaxoSmithKline, Heidelberg, Germany (e-mail: arezou.rahimi@uni-heidelberg.de). +Luis A. Vale-Silva is with Cellzome GmbH, GlaxoSmithKline, Heidelberg, Germany (e-mail: luis.a.valesilva@gsk.com) +Maria F¨alth Savitski is with Cellzome GmbH, GlaxoSmithKline, Heidelberg, Germany (e-mail: maria.x.faelth-savitski@gsk.com) +Jovan Tanevski is with the Institute for Computational Biomedicine, Faculty of Medicine, Heidelberg University and Heidelberg University Hospital, Heidelberg, +Germany, and Department of Knowledge Technologies, Joˇzef Stefan Institute, Ljubljana, Slovenia (e-mail: jovan.tanevski@uni-heidelberg.de). +Julio Saez-Rodriguez is with the Institute for Computational Biomedicine, Faculty of Medicine, Heidelberg University and Heidelberg University Hospital, +Heidelberg, Germany (e-mail: pub.saez@uni-heidelberg.de). +Jovan Tanevski and Julio Saez-Rodriguez cosupervised this work. +arXiv:2301.01682v1 [cs.CE] 4 Jan 2023 + +1 +DOT: Fast Cell Type Decomposition of Spatial +Omics by Optimal Transport +1 +INTRODUCTION +The organization of cells within human tissues, their molec- +ular programs and their response to perturbations are cen- +tral to better understand physiology, disease progression +and to eventual identification of targets for therapeutic in- +tervention [1], [2]. Cell types are distinct subpopulations of +cells with unique trasncriptional signatures, which are often +identified by known markers and/or by data-driven tech- +niques, most commonly clustering based on transcriptomic +profiles [3]. Single-cell RNA sequencing can profile the +entire transcriptome (mRNA expression of the full range of +genes) of large portions of individual (single) cells. This has +made scRNA-seq an essential tool for revealing distinct cell +types in complex tissues and has profoundly impacted our +understanding of biological processes and the underlying +mechanisms that control cellular functions [4], [5], [6], [7]. +However, scRNA-seq requires dissociation of the tissue [8], +losing the information about the spatial context and physical +relationship between cells. +To overcome these limitations, there has been recent ad- +vancements in spatially resolved transcriptomics methods +[9]. Spatial transcriptomics methods measure gene expres- +sion in locations, hereafter referred to as spots, coupled with +their two- or three-dimensional position. These methods +vary in two axes: spatial resolution and gene throughput. +On one hand, technologies such as Multiplexed Error- +Robust Fluorescence In-Situ Hybridization (MERFISH) and +In-Situ Sequencing (ISS), achieve cellular or even subcellular +resolution [10] through cell segmentation [11], [12], but +are limited to measuring up to a couple of hundred pre- +selected genes. On the other hand, spatially resolved RNA +sequencing, such as Spatial Transcriptomics [13], commer- +cially available as 10x’s Visium, and Slide-seq [14], enable +high-throughput gene profiling by capturing mRNAs in-situ +at the cost of spots with the size of tens of cells. Thus, there +is a trade-off between the resolution and the richness of the +data. +A strategy to overcome these limitations is to combine +scRNA-seq data with high resolution spatial data to map +dissociated cells to spatial locations or more generally to +combine it with low-resolution spatial data to estimate the +composition of cell types and expression in each spot. We +refer to this task as decomposition. Alternatively, we can +attempt to enrich high-resolution data by predicting the +expression of unmeasured genes. As the latter requires +extrapolation to various degrees, machine learning and opti- +mization methods are generally suited to the decomposition +task. We will show that our tailored Optimal Transport +formulation is capable of tackling both decomposition and +enrichment tasks in high- and low-resolution spatial data. +Since the initial efforts to bridge this gap [15], there has +been an increased interest in improvement and new method +development (see Section 2). However, so far the methods +rely on the genes that are captured both by scRNA-seq and +spatial data without using the remaining genes captured in +each modality, do not use the spatial relationships between +spots in the spatial data, and usually come with high com- +putation cost for large instances. Neglecting the spatial con- +text is equivalent to assuming random placement of spots in +the space, which is in contrast to the established structure- +function relationship of tissues. Considering only a subset +of genes limits the applicability of these methods to cases +where the two data sets share several informative genes, +which might not be the case when different technologies are +used for profiling, or when few genes are measured in the +spatial data (e.g., in MERFISH). +We address these limitations by incorporating ideas +from the Optimal Transport (OT) theory and adapting +a Gromov-Wasserstein (GW) distance [16], [17] between +scRNA-seq and spatial data. We present DOT (Fast Cell +Type Decomposition by Optimal Transport), a fast and +scalable optimization framework to integrate scRNA-seq +and spatial data for cell type localization by solving a multi- +criteria probabilistic matching problem. We summarize the +main contributions of our work as follows: +(i) We propose a novel formulation for mapping cell types +from scRNA-seq to spots in spatial data by casting +this problem to a multi-objective probabilistic matching +problem. Our model is applicable to both high- and +low-resolution spatial data, in the form of inferring +membership probabilities for the former and relative +abundance of cell types in the latter, and is capable of +estimating the expression of genes that are missing in +the spatial data but present in the scRNA-seq data. +(ii) We adapt a generalization of OT with a Gromov- +Wasserstein objective to leverage spatial information +and to go beyond the use of genes common to the two +modalities. +(iii) We introduce a scale-invariant metric based on cosine- +similarity to account for differences in the scale of gene +expressions in different technologies. +(iv) We present a very fast implementation for our model +based on the Frank-Wolfe algorithm, ensuring scalabil- +ity and efficient solvability in large-scale datasets. +2 +RELATED WORK +Cell type decomposition. Several decomposition methods +(also known as deconvolution methods) have been pro- +posed in recent years. As cell type decomposition, partic- +ularly in the high-resolution spatial data, is inherently a +multiclass classification task, classification methods, such as +Random Forests [18], can be used for tackling this problem. + +2 +However, because of the domain-specific properties of this +problem, including differences in gene coverage, resolution, +measurement sensitivity, and modality-specific characteris- +tics, tailored approaches are needed. +While most of these models are designed specifically for +low-resolution spatial data, some are also applicable to high- +resolution spatial data. [19] proposed SPOTlight, which es- +timates relative abundance of cell types in spots using non- +negative matrix factorization regression and non-negative +least squares. Robust cell type decomposition (RCTD) [20] +fits a statistical model by maximum-likelihood estimation, +assuming a Poisson distribution for the expression of each +gene at each spot. cell2location assumes a two-step Bayesian +model for inferring cell type composition of spots [21]. +Tangram [22] proposes a deep learning model to find the +best placement of single cells in spots using a designed +loss function and can thus carry cell type information as a +byproduct. Seurat V3 workflow [23] is a widely-used toolkit +for analyzing scRNA-seq data, which offers an “anchoring” +technique based on mutual nearest neighbours classifier for +aligning two modalities in the PCA space. +Optimal Transport. Optimal Transport (OT) [24] is a way +to match, with minimal cost, data points/histograms be- +tween two domains embedded in possibly different spaces +using different variants of the Wasserstein distance [25], [26], +[27]. Over the past years, OT has been applied to various +machine learning problems in a wide variety of contexts, +including but not limited to generative modeling [28], [29], +Wasserstein auto-encoders [30], feature aggregation [31], +generalization error prediction [32], dataset denoising [33], +graph matching/classification [34], and domain adaptation +[35], [36], [37]. +Recently, OT has been employed in biology, in particular +to analyse single-cell data. For example, [38] model cellu- +lar dynamics as an unbalanced dynamic transport, with +the goal of transporting entities from one cross sectional +measurement to the next. [39] use OT for studying devel- +opmental time courses and understanding the molecular +programs that guide differentiation during development +by incorporating temporal information and modeling cell +growth over time. Similarly, [40] employ graphical models +and OT to reconstruct developmental trajectories from time +courses with snapshots of cell states and lineages. +3 +MODEL +3.1 +Preliminaries +Given a reference scRNAseq data (R for short), which is +a collection of single cells each annotated with a cell type +c ∈ C, and a target spatially resolved transcriptomics data +(S for short), which consists of a set I of spots without cell +type annotations, the goal of decomposition is to determine +the composition of cell types in spots of S. Note that the +term “spot” can refer to one or a group of cells in certain +spatial contexts. We denote by ni the given size (number of +cells) of spot i ∈ I. When such information is not available, +or when spots are at single-cell resolution, we set ni = 1 to +compute the proportion or probability of cell types in each +spot rather than computing the number of cells of each type. +Let XR +c,g denote the mean expression of gene g ∈ GR in +cell type c ∈ C, where GR is the set of genes measured in R. +Each spot i ∈ I of S consists of spatial coordinates xi ∈ R2 or +R3 and gene expressions XS +i,g for g ∈ GS, where GS is the set +of genes that are measured in S. Further, if prior information +about the expected abundance of cell types in S is available +(e.g., estimated from a neighboring single-cell level tissue), +we denote the expected abundance of cell type c ∈ C in S +by rc. Note that r is scaled such that � +i∈I ni = � +c∈C rc. +For convenience, we also define G = GR ∩ GS as the set of +genes that are common between R and S. In the following, +unless otherwise mentioned, vectors of gene expressions are +assumed to be in the space of common genes. +To assess dissimilarity between expression vectors a and +b, we also introduce the distance function +dcos(a, b) := +� +1 − cos (a, b), +(1) +where cos (a, b) = +1 +∥a∥∥b∥⟨a, b⟩. Note that dcos is convex for +positive vectors a and b, and is scale-invariant, in the sense +that it is indifferent to the magnitudes of the vectors. This +is by design, since we want to assess dissimilarity between +expression vectors regardless of the measurement sensitiv- +ities of different technologies. We also note the following +important property of dcos (proofs given in Appendix C). +Proposition 1. Unlike cosine dissimilarity (i.e., 1 − cos(·, ·)), +dcos is a metric distance function. +3.2 +High-Level Model +Our model relies on determining a “many-to-many” map- +ping Y of cell types in R to spots in S, with Yc,i denoting the +proportion (or probability when ni = 1) of spot i ∈ I that is +of cell type c ∈ C. A high-quality mapping should naturally +match the expression of common genes across R and S. We +ensure this by considering the following genomic criteria: +(i) Expression of genes in each spot of S should match +expression of genes mapped to that spot via Y . +(ii) Centroid of each cell type in R should match the cen- +troid of that cell type in S as determined via Y . +(iii) Distribution of expression of each gene across spots of S +should be similar to the distribution of that gene across +spots as mapped from R to S via Y . +Additionally, we may incorporate prior knowledge in the +form of spatial location of spots as well as expected abun- +dance of cell types using the following auxiliary criteria: +(iv) Spots that are both adjacent in space and have similar +expression profiles should attain similar cell type pro- +files. +(v) If prior information about abundance of cell types in S +is available (e.g., when R and S correspond to adjacent +tissues), abundance of cell types mapped to S should +match with the given abundances. +The genomic objectives naturally take precedence over +the auxiliary objectives, especially when a large number +of genes are common between R and S, but the auxiliary +objectives are useful when the common genes are limited. +Note that objective (v) is meant to provide additional control +over the abundance of cell types in the spatial data, but can +be ignored if prior information about the abundance of cell +types is not available. We elaborate on these objectives in +the following. + +3 +3.3 +Formulation +Objective (i) ensures that the vector of gene expressions in +spot i ∈ I (i.e., XS +i,:) is most similar to the vector of gene ex- +pressions mapped to spot i through Y (i.e., � +c∈C Yc,iXR +c,:). +To achieve this objective, we minimize dissimilarity between +these vectors by using +di(Y ) := dcos(XS +i,:, +� +c∈C Yc,iXR +c,:). +(2) +Objective (ii) is in nature similar to objective (i). Here, +we would like to minimize dissimilarity between centroid +of cell type c ∈ C in R (i.e., XR +c,:) and centroid of cell type +c in S as determined via Y (i.e., +1 +ρc +� +i∈I Yc,iXS +i,:). Given +the scale-invariance property of dcos, we can drop 1/ρc and +measure the dissimilarity between these centroids using the +following distance function +dc(Y ) := dcos(XR +c,:, ρ−1 +c +� +i∈I Yc,iXS +i,:) += dcos(XR +c,:, +� +i∈I Yc,iXS +i,:). +(3) +Our goal in objective (iii) is to match distribution of +expression of gene g ∈ G in S (i.e., XS +:,g) with the one +mapped to S through Y (i.e., � +c∈C Yc,:XR +c,g). Hence, we +minimize dissimilarity between these vectors by using +dg(Y ) := dcos(XS +:,g, +� +c∈C Yc,:XR +c,g). +(4) +To achieve objective (iv), we borrow ideas from Optimal +Transport theory and the Gromov-Wasserstein metric. Let +M R and M S be metrics in R and S, respectively, in that +M R +c,k defines distance between cell types c and k, while +M S +i,j defines distance between spots i and j. Note that these +distances are defined for each dataset independently; hence, +we can use the entire features in each set: the entire genome +in R, including the genes not measured in S, and the un- +common/common genes as well as the spatial coordinates +in S (see Section 4 for how these matrices are computed). +The 2-Gromov-Wasserstein distance [16] between R and S +for given mapping Y , denoted dGW(Y ), is defined in (5). +Minimizing dGW(Y ) ensures that similar pair of spots in +S (with respect to their locations and expressions) are not +assigned to dissimilar pair of cell types in R, and vice versa. +dGW(Y ) := +� +� +� +� +� +i∈I +� +j∈I +� +c∈C +� +k∈C +� +M R +c,k − M S +i,j +�2 +Yc,iYk,j +(5) +Let ρc := � +i∈I Yc,i denote the abundance of cell type c in +S as determined by mapping Y . As noted by [41], we may +simplify (5) as stated in Proposition 2 below. +Proposition 2. Define parameter +¯mi +:= +� +j∈I(M S +i,j)2nj +and auxiliary variables ¯mc := � +k∈C(M R +c,k)2ρk and Z := +M RY M S. GW distance function in (5) is equivalent to +dGW(Y ) = +�� +c∈C +� +i∈I Yc,i( ¯mc + ¯mi − 2Zc,i). +(6) +Objective (v) provides optional control over abundance +of cell types mapped to S, when prior information about +expected abundance of cell types is available. We employ +Jensen-Shannon divergence between ρ and r to measure +their dissimilarity +dA(Y ) := 1 +2DKL +� +ρ +���� +ρ + r +2 +� ++ 1 +2DKL +� +r +���� +ρ + r +2 +� +, +(7) +where DKL (p∥q) = � +j pj log(pj/qj) denotes the Kull- +back–Leibler divergence [42]. In addition, to avoid overfit- +ting, we may require that all cell types are at least minimally +represented in the mapping. To achieve this goal, we define +dR(Y ) := − +� +c∈C log(ρc) = DKL (¯r∥ρ) , +(8) +where ¯rc = 1 for all c ∈ C. Equation (8) is in fact a +Nash fairness [43] objective and its logarithmic form ensures +presence of all cell types (i.e., ρc > 0). +We treat these criteria as objectives in a multi-objective +optimization problem and to achieve them simultaneously +(i.e., produce a Pareto-optimal solution), we optimize Y +against a linear combination of these objectives as formu- +lated below, hereafter referred to DOT model: +min +� +i∈I +nidi(Y ) + λC +� +c∈C +ρcdc(Y ) + λG +� +g∈G +dg(Y ) ++ λGWdGW(Y ) + λAdA(Y ) + λRdR(Y ) +(9) +w.r.t. +Y ∈ R|C|×|I| ++ +, ρ ∈ R|C| +(10) +s.t. +� +c∈C Yc,i = ni +∀i ∈ I, +(11) +� +i∈I Yc,i = ρc +∀c ∈ C, +(12) +where λC, λG, λGW, λA and λR are the user-defined penalty +weights, and coefficients ni and ρc in (9) balance the scales +of deviations in spots and cell types, respectively. +Remark 1. Unlike the conventional OT formulations, DOT does +not require the cell type abundances in S (i.e., ρ) to be strictly +equal to their expected abundances (i.e., r), and rather penalizes +their deviation in the objective function. +4 +ALGORITHM +We propose a solution to the DOT model based on the +Frank-Wolfe (FW) algorithm [44], [45], which is a first-order +method for solving non-linear optimization problems of the +form minx∈X f(x), where f : Rn → R is a (potentially +non-convex) continuously differentiable function over the +convex and compact set X. FW operates by replacing the +non-linear objective function f with its linear approximation +˜f(x) = f(x(0))+∇xf(x(0))⊤(x−x(0)) at a trial point x(0) ∈ +X, and solving a simpler problem ˆx = arg minx∈X ˜f(x) to +produce an “atom” solution ˆx. The algorithm then iterates +by taking a convex combination of x(0) and ˆx to produce +the next trial point x(1), which remains feasible thanks to +convexity of X. The FW algorithm is described in Algorithm +1, in which f(Y ) is the objective function in (9). +4.1 +Distance Matrices +Distance matrices M R and M S incorporate the features that +are not shared across R and S. To compute M R +c,k, we calculate +the dissimilarity between the centroids of cell types c and k +considering all genes in R (i.e., XR +c,: = (XR +c,g)g∈GR for each +c ∈ C) +M R +c,k = dcos(XR +c,:, XR +k,:). + +4 +Algorithm 1: Frank-Wolfe algorithm for DOT +1 Initialization: Setup distance matrices M R and M S. +2 Set t = 0 and find an initial map Y (0) (see Appendix +A.1). +3 while not converged do +4 +Compute gradient ∆(t) = ∇Y f(Y (t)) (see +Appendix A.2) +5 +for each spot i ∈ I do +6 +Find current best cell type +ˆc = arg minc∈C{∆(t) +c,i} +7 +Compute atom solution ˆY (t) +ˆc,i = ni and +ˆY (t) +c,i = 0 for c ̸= ˆc +8 +Update Y (t+1) = Y (t) + +2 +2+t( ˆY (t) − Y (t)) +9 +t ← t + 1 +Remark 2. M R is a metric in the domain of R since dcos is a +metric. +The matrix M S captures the dissimilarity of S spots +in terms of their locations and expressions. Let D1 +i,j and +D2 +i,j represent distance of spots (i, j) with respect to their +locations and expressions, respectively, as defined below: +D1 +i,j = 1condition +�∥xi − xj∥ > ¯d +� +D2 +i,j = dcos +� +XS +i,:, XS +j,: +� +, +where ¯d is a given distance threshold, and D2 +i,j is computed +with respect to all genes in S (i.e., GS). Finally, we take M S +to be the average of D1 and D2: +M S = (D1 + D2)/2 +(13) +Remark 3. M S is a metric in the domain of S, since both D1 +and D2 are metrics. +To see why this definition of M S makes sense, we first +note that cell types, by definition, are distinct subpopula- +tions in the scRNA-seq data. Therefore, it is reasonable to +assume that their centroids are dissimilar (i.e., Mc,k ≈ 1 for +c ̸= k). This yields the following result. +Proposition 3. Let α = � +i∈I +� +j∈I(1−M S +i,j)2ninj. Assuming +that cell types are relatively distinct, so that M R +c,k ≈ 1, for c, k ∈ +C, c ̸= k, then +dGW(Y ) ≈ +� +α + +� +i∈I +� +j∈I +� +2M S +i,j − 1 +� +⟨Y:,i, Y:,j⟩ +Remark 4. Observe that ⟨Y:,i, Y:,j⟩ measures similarity between +cell type profiles of spots i and j. Therefore, dGW (i) rewards +⟨Y:,i, Y:,j⟩ when 2M S +i,j −1 ≈ +1 (i.e., encourages adjacent spots +to attain similar cell types if their expressions are similar) and +(ii) penalizes ⟨Y:,i, Y:,j⟩ when 2M S +i,j − 1 ≈ −1 (i.e., prevents +distant spots from attaining similar cell types if their expressions +are different). Moreover, (iii) dGW is indifferent to pair (i, j) +when 2M S +i,j − 1 ≈ 0 (i.e., if i and j are distant or different +in expressions, but not both). +4.2 +Producing an Atom Solution +While the DOT model is not separable, its linear approxi- +mation can be decomposed to |I| independent subproblems, +one for each spot i ∈ I. This is because, unlike conventional +OT formulations, we do not require the distribution of cell +types (i.e., ρ) to be equal to their expected distribution +(i.e., r), but have penalized their deviations in the objective +function using dA (7). The subproblem i then becomes +min +� +⟨Y:,i, ∆(t) +:,i ⟩ : Y:,i ∈ R|C| ++ , +� +c∈C Yc,i = ni +� +which, in turn, is a simple sorting problem. This property +of Algorithm 1 enables it to efficiently tackle problems with +large number of spots in the spatial data. +4.3 +Convergence +Under suitable conditions, FW converges to an optimal so- +lution in linear rate when optimizing a convex function over +a polytope domain [46]. Given the non-convex objective +function in (9), Algorithm 1 instead obtains a first-order sta- +tionary point at a rate of O(1/ +√ +t) [47], [48]. We numerically +assess the convergence of Algorithm 1 at iteration t using +the so-called “FW-gap” [45] +δ(t) := +� +i∈I +� +c∈C(Y (t) +c,i − ˆY (t) +c,i )∆(t) +c,i. +We also implemented acceleration techniques such as av- +eraging gradients [49] and away steps [46], [50], but did +not observe practical gains compared to the vanilla FW. +Moreover, while it is common practice to use entropic or +other strongly-convex regularizations in OT to facilitate +producing the atom solutions, we did not incorporate such +regularizations because an atom solution can be produced +easily in our formulation. +5 +PRACTICAL ENHANCEMENTS +In this section, we introduce practical enhancements to +incorporate the domain-specific properties of the problems. +5.1 +Cell Heterogeneity +While cell types are distinct subpopulations of cells, signif- +icant variations may naturally exist within each cell type. +This means, a single vector XR +c,: may not properly represent +the distribution of cells within this cell type. Consequently, +mapping cell types solely based on the centroids of cell types +can be error-prone. To capture the intrinsic heterogeneity +of cell types, we cluster each cell type into predefined κ +smaller groups using an unsupervised learning method, +and produce a total of κ|C| centroids to replace the original +|C| centroids. With this definition of centroids, we treat all +terms except dA and dR as before. For dA and dR, since +prior information about cell types (and not sub-clusters) +are available, we keep ρ to represent the abundance of +original cell types by setting ρc = � +k∈Kc +� +i∈I Yk,i, where +Kc denotes the set of sub-clusters of cell type c. Finally, once +Y is obtained, � +k∈Kc Yk,i determines probability that spot +i is of cell type c. + +5 +5.2 +Sparse Mapping +As previously discussed, spatial data are either high- +resolution (single-cell level) or low-resolution (multicell +level). In the case of high-resolution spatial data, given that +each spot corresponds to an individual cell (i.e., ni = 1), +it is desirable to produce sparse allocations, in the sense +that we prefer Yc,i close to 0 or 1. In general, assuming that +Yc,i ∈ {0, ni}, then (11) implies that Yc,i = ni for exactly one +cell type c and is zero for all other cell types. Consequently, +for binary Y we obtain +dcos +� +XS +i,:, +� +c∈C +Yc,iXR +c,: +� += 1 +ni +� +c∈C +Yc,idcos +� +XS +i,:, XR +c,: +� +, +which is linear in Y . As linear objectives promote sparse (or +corner point) solutions, we may control the level of sparsity +of the mapping by introducing a parameter θ ∈ [0, 1] and +redefining di(Y ) as +di(Y ) =(1 − θ)dcos +� +XS +i,:, +� +c∈C Yc,iXR +c,: +� ++ θ +ni +� +c∈C Yc,idcos +� +XS +i,:XR +c,: +� +. +(14) +Note that a higher value for θ yields a sparser solution. +Indeed, with θ = 1 and zero weights assigned to other +objectives, the optimal mapping will be completely binary. +6 +RESULTS +We compared the performance of our method, abbreviated +DOT, against five state of the art models in the litera- +ture: SPOTlight [19], RCTD [20], cell2location [21], +Tangram [22], and Seurat [23]. We designed three exper- +iments to evaluate the performance of DOT from different +perspectives. Briefly, in Section 6.2, we evaluate the per- +formance of models in predicting the cell type of single- +cell level spots in high-resolution spatial data, followed by +cell type decomposition in multicell spots in low-resolution +spatial data in Section 6.3. Finally, in Section 6.4, we evaluate +capability of DOT in estimating the expression of genes that +are missing in the spatial data but present in the reference +single-cell data. +We performed experiments on data coming from (i) the +primary motor cortex of the mouse brain, (ii) the primary +somatosensory cortex of the mouse brain, (iii) the develop- +ing human heart, and (iv) the human breast cancer, specifics +of which are presented in Appendix B. +6.1 +Experimental Setup +6.1.1 +Parameter Setting +For DOT, we set penalty weights λC += 1 and λG += +|n|/|G| to balance the scales of different objectives, where +|n| := � +i∈I ni. This is because both � +i∈I nidi(Y ) and +� +c∈C rcdc(Y ) are in the range of 0 and |n|, while 0 ≤ +� +g∈G dg(Y ) ≤ |G|. For the GW objective, it is not difficult +to verify that 0 ≤ dGW(Y ) ≤ |n|. However, although +spatial information contributes to the accuracy of cell type +mapping, meaning that λGW > 0 is desirable, a large value +for λGW may dominate the genomic objectives di(Y ), dc(Y ) +and dg(Y ), thus reduce accuracy. A middle-ground is to +set a small positive value for λGW. In our computations, we +found that λGW = 0.1 works best in most cases. Whenever +prior information about expected abundance of cell types +is available, we set λA = 1 and λR = 1. We computed +ρc, the expected abundance of cell type c, based on the +observed fraction of cell type c in the reference scRNA- +seq data multiplied by |n|. We set the sparsity parameter +θ = 1 for high resolution spatial data, and set θ = 0 for +low resolution spatial data. To capture heterogeneity of cell +types, we clustered each cell type into κ = 10 clusters. +The distance threshold ¯d is computed as follows. For each +spot we computed its Euclidean distance to 8 closest spots +in space1, yielding 8|I| values. We then took ¯d as the 99th +percentile of these values. +For RCTD, SPOTlight, Tangram, and C2L we used the +default parameters suggested by the authors with the fol- +lowing exceptions. For RCTD we set the parameter UMI_min +to 50 to prevent the model from removing too many cells +from the data. Given the large number of cell types in +the mouse MOp datasets, for SPOTlight we reduced the +number of cells per cell type to 100 to enhance the computa- +tion time. Similarly, as Tangram was not able to produce +results in a reasonable time for the MOp instances, we +randomly selected 500 cells per cell type to reduce the com- +putation time. For C2L, we used 20000 epochs to balance +computation performance and accuracy. For Seurat and +SingleR, we followed the package documentations, with +functions used with default parameters. For RF we used +the implementation provided in the R package ranger [51] +with all parameters set at their default values. +6.1.2 +Performance Metrics +We compared the predictive performance of DOT against +the other methods using three metrics. Accuracy in the +context of high-resolution spatial data (i.e., when each spot +corresponds to an individual cell) is the proportion of +correctly classified spots (i.e., sum of the main diagonal in +the confusion matrix) over all spots. To assess the accuracy +of membership probabilities produced by each model, we +compared the models using Brier Score, also known as mean +squared error: +Brier Score = |I|−1 � +i∈I +� +c∈C(Yc,i − Pc,i)2, +where Pc,i = 1 if spot i is of cell type c and Pc,i = 0 +otherwise, and Yc,i is the predicted probability that spot i is +of cell type c. As Brier Score is a strictly proper scoring rule +for measuring the accuracy of probabilistic predictions [52], +a model with lower Brier Score produces better-calibrated +probabilities. +Besides the cell type that each spot is annotated with, we +can produce a cell type probability distribution for each spot +by considering the cell type of its neighboring spots, using +a Gaussian smoothing kernel of the form +˜Pc,i =( +� +j∈I Ki,j)−1 � +j∈I Ki,jPc,j, +where Ki,j = exp +�−∥xi − xj∥2/2σ2� +and σ is the kernel +width parameter which we set to 0.5 ¯d. Note that as spot j +becomes closer to spot i, its label contributes more to the +1. We used 8 closest neighbors to mimic the number of adjacent tiles +in a 2D regular grid. + +6 +probability distribution at spot i. Using these probabilities, +we also introduce the Spatial Jensen-Shannon (SJS) divergence +to compare the probability distributions assigned to spots +(i.e., Y ) with the smoothed probabilities (i.e., ˜P ) +SJS = |I|−1 � +i∈I JS(Y:,i, ˜P:,i), +where JS(Y:,i, ˜P:,i) is the Jensen-Shannon divergence be- +tween probability distributions Y:,i and ˜P:,i with base 2 +logarithm [42], also defined in (7). +6.2 +Experiment +1: +Cell +Type +Prediction +in +High- +Resolution Spatial Data +Our goal with our first set of experiments is to evaluate +the performance of different models in determining the +probability distribution of cell types at each spot. Since the +identity of the cell type represented by the spot is known in +our high resolution spatial data, we can use this information +as ground-truth when evaluating the performance of the +different models. In addition to deconvolution methods, +we used SingleR [53], a method to define cell type from +single-cell resolution data. Given the multiclass classifica- +tion nature of this task, we also used RF [18] as a multiclass +classifier baseline. +We use the high-resolution MERFISH spatial data of the +primary motor cortex region (MOp) of the mouse brain [54], +which contains the spatial information of 280,186 cells across +75 samples (Appendix B.1). With each sample, we created a +reference scRNA-seq data using all the 280,186 cells, except +the cells contained in the sample, and the 254 genes to +estimate the centroids of the 99 reference cell types. We +further created 15 high resolution spatial datasets for each +sample (i.e., a total of 1125 spatial datasets) as follows. To +simulate the effect of number of shared features between the +spatial and scRNA-seq data, we assumed that only a subset +of the 254 genes are available in the spatial data by selecting +the first |G| genes, where |G| ∈ {50, 75, 100, 125, 150} (i.e., +20%, 30%, 40%, 50%, 60% of genes). Moreover, to simulate +the effect of differences in measurement sensitivities of +different technologies, we introduced random noise in the +spatial data by multiplying the expression of gene g in spot +i by 1+βi,g, where βi,g ∼ U(−ϕ, ϕ) with ϕ ∈ {0, 0.25, 0.5}. +We compare the predictive performance of DOT to +Seurat, RCTD, Tangram, SingleR and RF in Fig. 1. We +removed SPOTlight and C2L from these plots due to their +clear under-performance in the high resolution spatial data. +We observe that not only does DOT dominate the three alter- +natives in assigning correct cell types to the spots (Fig. 1a), +but also produces well-calibrated probabilities (Fig. 1b) and +better captures the relationships between cell types in space +(Fig. 1c), owing to its capacity to incorporate the spatial +information in dGW through the distance matrices. We also +observe that even with very few genes in common between +the spatial data and the reference scRNA-seq data (e.g., +|G| ≤ 75), DOT is able to reliably determine the cell type +of spots in the space with high accuracy. In contrast, RCTD +fails to produce results due to lack of shared information, +and Seurat and Tangram produce results with low accu- +racy. Additionally, we observe that DOT is more immune to +fluctuations in expressions in the spatial data, implying the +effectiveness of our dcos distance function in accounting for +ϕ = 0 +ϕ = 0.25 +ϕ = 0.5 +50 +75 +100 +125 +150 +50 +75 +100 +125 +150 +50 +75 +100 +125 +150 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Accuracy +(a) +ϕ = 0 +ϕ = 0.25 +ϕ = 0.5 +50 +75 +100 +125 +150 +50 +75 +100 +125 +150 +50 +75 +100 +125 +150 +0.4 +0.6 +0.8 +Brier Score +(b) +ϕ = 0 +ϕ = 0.25 +ϕ = 0.5 +50 +75 +100 +125 +150 +50 +75 +100 +125 +150 +50 +75 +100 +125 +150 +0.5 +0.6 +0.7 +SJS +(c) +DOT +Seurat +RCTD +Tangram +RF +SingleR +Fig. 1: Predictive performance of the algorithms in the high- +resolution spatial data across different number of genes +in the spatial data (x-axis) and different noise factors (ϕ). +Points represent the median of 75 values, and the shaded ar- +eas correspond to their interquartile interval. Note: SingeR +does not produce probabilities and is compared based on +Accuracy only. +differences in measurement scales of different technologies. +In terms of algorithmic performance (Table 1), DOT takes +on average 441 seconds to solve each instance, which is an +order of magnitude faster than RCTD, Tangram, and RF, and +is comparable to Seurat and SingleR. +6.3 +Experiment 2: Cell Type Decomposition in Low- +Resolution Spatial Data +We evaluated the performance of models on low-resolution +spatial data. For these experiments, since there is no ground +truth for real multicell spatial data such as Visium and Slide- +seq, we resort to producing ground truth low-resolution +spatial data by pooling the adjacent cells in the high res- +olution spatial data of primary motor cortex of the mouse +brain (MOp), primary somatosensory cortex of the mouse +brain (SSp), and the developing human heart. Fig. A1 in +Appendix B.1 illustrates a sample low-resolution spatial +data obtained from a MERFISH MOp tissue. Unlike the +high-resolution spatial data, the ground truth Pc,i now +corresponds to relative abundance of cell type c in spot +i. We can therefore assess the performance of each model +by comparing the probability distributions P:,i and the +estimated probabilities (i.e., Y:,i) using Brier Score or Jensen- +Shannon metrics. +6.3.1 +Experiments on the Mouse MOp +To produce ground truth for MOp, using the common +subclass annotations between MERFISH MOp and scRNA- +seq MOp [55] (see Appendix B.1), for each of the 75 MER- +FISH MOp samples, we randomly assigned each cell in the +MERFISH MOp data to a cell in the scRNA-seq MOp data +of the same subclass. Next, we lowered the resolution of +spatial data by splitting each sample into regular grids of + +7 +Experiment +Resolution +Instances +DOT +Seurat +RCTD +Tangram +SPOTlight +C2L +SingleR +RF +MOp +High +1125 +441 +380 +4748 +10141 +7884 +3310 +303 +7427 +MOp +Low +75 +457 +1086 +4705 +8250 +52825 +6119 +— +— +SSp +Low +1 +4 +21 +117 +248 +705 +364 +— +— +Heart +Low +1 +8 +11 +185 +88 +316 +398 +— +— +TABLE 1: Average computation times (in seconds) of different models across different experiments. +0.1 +0.2 +0.3 +0.4 +0.2 +0.4 +0.6 +0.8 +Jensen−Shannon +Brier Score +DOT +Seurat +RCTD +SPOTlight +C2L +Tangram +Fig. 2: Predictive performance of the algorithms in the low- +resolution spatial data across 75 samples of MOp. Each point +in the scatter plots denoting the average performance across +all spots in the sample. +length 100µm to mimic the size and inter-distance of spots +in low-resolution spatial transcriptomics, such as Visium. +Finally, we aggregated the expression profiles of cells within +each tile as the expression profile of the respective spots. +Fig. 2 compares the performance of DOT against RCTD, +SPOTlight, cell2location (C2L), Tangram and Seurat in +determining the cell type composition of the multicell spots. +We observe that DOT outperforms other models with respect +to both metrics. As presented in Table 1, DOT took on +average 457 seconds to solve an instance, which proved to +be more than twice faster than Seurat, and orders of mag- +nitude faster than RCTD, SPOTlight, C2L and Tangram, +further highlighting the superiority of DOT in terms of both +accuracy and computational efficiency. +6.3.2 +Experiments on the Mouse SSp and the Developing +Human Heart +We also carried out experiments on data from the SSp +region of mouse brain as well as the developing human +heart to evaluate the performance of models on tissues of +different structures. We employed single-cell level spatial +data coming from osmFISH technology [56] to produce +multicell data for SSp (Appendix B.2). For the developing +human heart, we used subcellular spatial data generated +by the ISS platform [57] (Appendix B.3). We tested the +performance of the six deconvolution methods on these +two samples, results of which are illustrated in Fig. 3. Each +subplot shows the distribution of prediction error based on +the Jensen-Shannon divergence at each spot in the spatial +data, with the average value over all spots given on top +of each plot. DOT outperforms other models in the human +heart sample and is among the best-performing models in +the mouse SSp sample. Moreover, performance of DOT is not +sensitive to different regions/cell type of the tissue (compare +to Tangram and Seurat in SSp and RCTD in human heart). +These results further highlight the competitive performance +of DOT and its robustness in identifying the cell type com- +position of spots across different tissues. +6.4 +Experiment 3: Gene Expression Estimation in High- +Resolution Spatial Data +Recall that, in high-resolution spatial data, only a few genes +are measured. Hence, as a result of mapping scRNA-seq to +spatial data, we can estimate the expression of genes that +were not measured in the spatial data. Therefore, in our +final set of experiments, we evaluate the performance of +DOT in estimating the expression of missing genes in the +high resolution spatial data. For this experiment, we use +the spatial data from breast cancer tumor microenvironment +produced by the 10X Xenium In Situ technology [58]. The +dataset is unique in that it contains both high-resolution +(Xenium) and low-resolution (Visium) spatial data of serial +sections from the same tissue. The high-resolution data con- +tains spatial information of 313 genes across 167,782 single- +cell spots, while the low-resolution data contains the spatial +information of around 18,000 genes across 4,992 multicell +spots. The dataset also contains the dissociated scRNA-seq +data coming from a tissue adjacent to the tissues used for +high- and low-resolution spatial data, which contains the +measurements of 18,000 genes across 30,365 cells. Therefore, +we can use Visium as a proxy for ground truth to validate +the distribution of genes that are not measured in Xenium, +and are mapped by DOT from scRNA-seq. +For this experiment, we matched the common capture +areas of high- and low-resolution spatial data using the +Hematoxylin-Eosin (H&E) images accompanying these spa- +tial data (Fig. A2), which corresponded to 134,664 cells in +the high-resolution and 3,928 spots in the low-resolution +spatial data. Given that the task here is to complete the +expression of missing genes in the high-resolution spatial +data, we first performed community detection on the graph +of shared nearest neighbors of cells in scRNA-seq using the +Leiden implementation in [23], which is common practice +in single-cell analysis and is used as a first step towards +cell type identification. This resulted in 218 clusters, and +we then mapped the centroids of these clusters to the high- +resolution spatial data. (We also tried as high as 1000 fine- +grained clusters but got essentially the same results.) We +also turned off the cell-type-related objectives since we are +not mapping cell types. Although the datasets are extremely + +8 +DOT (0.308) +SPOTlight (0.456) +Tangram (0.484) +C2L (0.294) +RCTD (0.298) +Seurat (0.404) +0.00 +0.25 +0.50 +0.75 +1.00 +JS +DOT (0.392) +SPOTlight (0.548) +Tangram (0.408) +C2L (0.541) +RCTD (0.607) +Seurat (0.468) +0.25 +0.50 +0.75 +1.00 +JS +Mouse SSp +Human Heart +Fig. 3: Distribution of performance of models on each individual spot in the low-resolution spatial data of Mouse SSp (top) +and developing human heart (bottom). +large, DOT was able to perform the mapping in less than +two hours. +We start by evaluating the performance of DOT on genes +that are present in the high-resolution spatial data as a true +ground truth. The qualitative comparisons of gene maps of +eight genes associated with breast cancer [59] produced by +DOT with those of high-resolution (ground truth) and low- +resolution data (approximate ground truth) can be seen in +Fig. 4. As can be observed, the expression maps produced by +DOT match almost perfectly with the ground truth expres- +sion maps. Both DOT and the ground truth high-resolution +spatial data also match the low resolution gene expression +maps almost perfectly, which further validate the quality +of mapping produced by DOT. Note that due to the single- +cell resolution of the high-resolution spatial data, expression +levels are higher at high resolution, thus the brighter colors. +Nonetheless, the spatial patterns match between all three +rows. +Next, we compare the expression map of genes that +are not present in the high-resolution spatial data but are +estimated by DOT. Fig. 5a illustrates the expression maps +of five genes associated with breast cancer that are not +measured in the high-resolution spatial data. For a quanti- +tative comparison of expression maps, given that there is no +one-to-one mapping between single-cell spots in the high- +resolution and multicell spots in the low-resolution spatial +data, we split the tissue into a 10 by 10 grid, and aggregated +the expression of each gene within each tile. Consequently, +we obtained two 100 by 18,000 matrices, one for the low- +resolution spatial data and another for DOT, with rows +corresponding to tiles and columns corresponding to genes. +Fig. 5b compares the column-wise cosine similarities across +different genes. These results further confirm the ability of +DOT in reliably estimating the expression of missing genes +in high-resolution spatial data. +7 +CONCLUSION +Single-cell +RNA-seq +and +spatially-resolved +imag- +ing/sequencing technologies, the cutting edge technologies +in transcriptomic data generation, each provide a partial +picture in understanding the organization of complex +tissues. To obtain a full picture, computational methods aim +at combining data from these two modalities. We present +DOT, a fast and scalable optimization framework based +on Optimal Transport theory, for assigning cell types to +tissue locations by leveraging the spatial information as +well as both joint and distinct genes across scRNA-seq and +spatial data. Using experiments on data from mouse brain +and human heart, we show that DOT predicts the cell type +composition of spots in spatial data with high accuracy, +outperforming the state of the art methods both in terms of +predictive performance and computation time. +APPENDIX A +IMPLEMENTATION DETAILS OF THE FW ALGORITHM +A.1 +Initial Solution +A good quality initial solution plays a critical role in fast +convergence of FW. Given the multi-objective nature of our +model, we produce an initial solution as convex combina- +tion of two solutions. In the first solution, for each spot +i we first find cell type ˆc = arg minc∈C{dcos +� +XS +i,:, XR +c,: +� +} +and set Yc,i = ni if c = ˆc and Yc,i = 0 otherwise. Note +that this solution is optimal for the sparse case when di is +the only objective. In the second solution, we simply set +Yc,i = niρc/|I| for each i and c. Note that this solution +is optimal for dA. We then set the initial solution as the +convex combination of these two solutions, with 0.9 weight +assigned to the first solution. +A.2 +Derivatives +To find the derivatives of di(Y ) and dc(Y ), defined in (2) +and (3), we introduce auxiliary quantities ¯ +XS := Y ⊤XR +and ¯ +XR := Y XS to denote the expressions mapped through +Y to spots and cell types, respectively. Derivatives for di(Y ) +and dc(Y ) can then be calculated as: +∂di +∂Yc,i += +1 +∥XS +i,:∥⟨XR +c,:, T S +i,:⟩, +∂dc +∂Yc,i += +1 +∥XRc,:∥⟨XS +i,:, T R +c,:⟩, + +9 +CEACAM6 (Xenium) POSTN (Xenium) +ITGAX (Xenium) +VWF (Xenium) +KRT15 (Xenium) +FOXA1 (Xenium) +GATA3 (Xenium) +TACSTD2 (Xenium) +CEACAM6 (DOT) +POSTN (DOT) +ITGAX (DOT) +VWF (DOT) +KRT15 (DOT) +FOXA1 (DOT) +GATA3 (DOT) +TACSTD2 (DOT) +CEACAM6 (Visium) +POSTN (Visium) +ITGAX (Visium) +VWF (Visium) +KRT15 (Visium) +FOXA1 (Visium) +GATA3 (Visium) +TACSTD2 (Visium) +Fig. 4: Expression map of eight breast cancer markers measured in both Xenium (ground truth; top) and Visium (low- +resolution proxy; bottom), and as mapped from scRNA-seq to Xenium using DOT (estimated; middle). Brighter means +higher expression. +SCGB2A2 (DOT) +KRT17 (DOT) +CDH2 (DOT) +SFRP2 (DOT) +MT−ND1 (DOT) +SCGB2A2 (Visium) +KRT17 (Visium) +CDH2 (Visium) +SFRP2 (Visium) +MT−ND1 (Visium) +(a) +0% +10% +20% +30% +40% +50% +0.00 +0.25 +0.50 +0.75 +1.00 +Expression map similarity +% Genes +(b) +Fig. 5: (a) Expression map of five breast cancer markers that are measured in Visium (bottom) but are missing in Xenium +and are mapped from scRNA-seq using DOT (top). (b) Cosine similarity between expression maps of Visium and DOT for +the genes that are not measured in Xenium. +where +T S +i,g = +−1 +2di(Y ) +� +XS +i,g +∥ ¯ +XS +i,:∥ − +¯XS +i,g +∥ ¯ +XS +i,:∥3 ⟨XS +i,:, ¯ +XS +i,:⟩ +� +, +T R +c,g = +−1 +2dc(Y ) +� +XR +c,g +∥ ¯ +XRc,:∥ − +¯XR +c,g +∥ ¯ +XRc,:∥3 ⟨XR +c,:, ¯ +XR +c,:⟩ +� +. +Derivative of ρcdc(Y ) then can be computed using the +product rule. Similarly, we may derive the derivative for +dg(Y ) via +∂dg +∂Yc,i += +−1 +2dg(Y ) +XR +c,g +∥XS:,g∥ +� +XS +i,g +∥ ¯ +XS:,g∥ − +Yc,i +∥ ¯ +XS:,g∥3 ⟨XS +:,g, ¯ +XS +:,g⟩ +� +Taking into account the simplification from Proposi- +tion 2, noting that ¯mc and Zc,i are functions of Y while +¯mi is constant, we can show that +∂dGW +∂Yc,i += +1 +2dGW(Y )(2 ¯mc + ¯mi − 4Zc,i). +Finally, the derivatives for dA and dR, defined in (7) and +(8) respectively, can be calculated as: +∂dA +∂Yc,i += 1 +2 log +� +2ρc +ρc + rc +� +, +∂dR +∂Yc,i += − 1 +ρc +. +APPENDIX B +DATASETS +B.1 +Mouse Primary Motor Cortex (MOp) +MERFISH. For high-resolution spatial transcriptomics, we +used the spatially resolved cell atlas of the MOp recently +generated using multiplexed error-robust fluorescence in +situ hybridization (MERFISH) technology and made pub- +licly available by [54]. The processed dataset contains nor- +malized RNA counts of 254 genes and coordinates of the +boundaries of a total of 280,186 segmented cells across 75 +samples in the MOp of two adult mice, with the number of +cells within each sample ranging from 1000 to 7500 cells. We +computed the (x, y) coordinates of the center of each cell by +taking the average of the coordinates of its boundary. The +study also identifies 99 trasncriptionally distinct cell types + +10 +0 +500 +1000 +1500 +2000 +2500 +0 +500 +1000 +1500 +Fig. A1: Synthetic multicell spatial data from MERFISH. +Dots show individual cells and tiles represent multicell +spots (darker tile means denser spot). +by community detection applied on a cell similarity graph. +The clustering resulted in 39 excitatory neuronal cell types +(clusters), 42 inhibitory neuronal cell types, 14 non-neuronal +cell types, and four other cell types. +scRNA-seq. The corresponding scRNA-seq data comes +from a cell atlas of the MOp [55], which contains the +mRNA expression of the full range of genes for more than +500,000 individual cells across several omics layers. We used +the scRNA-seq dataset scRNA_10x_v2_A generated at the +Allen Institute, which contains 145,748 cells and 100 cell +types. After removing the unannotated cells and low quality +cell types (as categorized in the study), we retrieved 124,330 +cells and 90 distinct cell types. For computational efficiency, +we also selected the top 5,000 variable genes according to +their means and variances [23]. +Fig. A1 illustrates a sample low-resolution spatial data +obtained from a MERFISH MOp tissue. +B.2 +Mouse Primary Somatosensory Cortex (SSp) +Similar to MOp, another well-studied tissue area is the pri- +mary somatosensory cortex area (SSp). Here, we used high- +resolution spatial data coming from the osmFISH platform +[56], which contains measurements of 33 genes across 4,837 +cells, as well as annotations based on 11 major cell types. +For reference scRNA-seq data with matched cell types, we +used the annotations independently generated by [60] using +5,392 single cells in the same SSp region. +B.3 +Developing Human Heart +For the developing human heart, we used subcellular spatial +data generated by the ISS platform [57], which contains +tissue sections from human embryonic cardiac samples +collected at different times. We selected the PCW6.5 slide +which contains measurements of 69 genes across 17,454 cells +as well as annotations of 12 major cell types. The same +study also provides scRNA-seq data for similar slide, which +contains matched cell types for 3,253 cells. +Fig. A2: Common region (cyan) in the capture areas of +Visium (dashed blue lines) and Xenium (dark orange). The +pink region is the H&E image accompanying Visium. +B.4 +Human Breast Cancer +Breast cancer is a complex disease with significant cel- +lular and molecular heterogeneity. The breast cancer tu- +mor microvenvironment dataset generated in [58] contains +both single-cell and multicell data. The single-cell reso- +lution spatial dataset is produced by the recent 10X Xe- +nium In Situ technology, and contains two replications. +We used Xenium_FFPE_Human_Breast_Cancer_Rep1, +which contains the spatial information of 313 genes for +167,782 cells. The low-resolution multicell spatial dataset +is produced by the 10X Visium Spatial Transcriptomics +technology, which contains the spatial information of 18,000 +genes for 4,992 multicell spots. For the reference scRNA- +seq data, we used the Single Cell Gene Expression +Flex (FRP) data generated from a tissue section adjacent +to the tissue sections used for Visium and Xenium work- +flows, which contains expression of 18,000 genes across +30,365 cells. +APPENDIX C +C.1 +Proof of Proposition 1 +Proof: Note that +���� +a +∥a∥ − +b +∥b∥ +���� +2 += ∥a∥2 +∥a∥2 + ∥b∥2 +∥b∥2 − 2 ⟨a, b⟩ +∥a∥∥b∥ = 2 − 2 cos(a, b) +⇒ dcos(a, b) = +� +1 − cos(a, b) = +√ +2∥a/∥a∥ − b/∥b∥∥. +This shows that dcos is a metric since ∥ · ∥ is a metric. We +can easily show that cosine dissimilarity (i.e., 1 − cos(·, ·)) +is not a metric. For instance, consider a = (1, 0, 0), b = +(0, 1, 0) and c = (x, x, +√ +1 − 2x2) for arbitrary x ∈ ( 1 +2, +1 +√ +2), +and let f denote the cosine dissimilarity. Then f(a, b) = +1 − cos(a, b) = 1, and f(a, c) = f(c, b) = 1 − x, which +violates the triangular inequality since f(a, c) + f(c, b) = +2−2x < 1 = f(a, b). It is not difficult to see that dcos(a, c)+ +dcos(c, b) > 1. + +11 +C.2 +Proof of Proposition 2 +Proof: Rewrite g(Y ) as +g(Y ) = +� +i∈I +� +j∈I +� +c∈C +� +k∈C +� +M R +c,k − M S +i,j +�2 +Yc,iYk,j += +� +c∈C +� +i∈I +Yc,i +� +k∈C +� +j∈I +Yk,j +� +(M R +c,k)2 + (M S +i,j)2 − 2M R +c,kM S +i,j +� +. +Expanding the inner summations we obtain: +� +k∈C +� +j∈I +Yk,j(M R +c,k)2 = +� +k∈C +(M R +c,k)2 � +j∈I +Yk,j = ¯mc +� +k∈C +� +j∈I +Yk,j(M S +i,j)2 = +� +j∈I +(M S +i,j)2 � +k∈C +Yk,j = ¯mi +� +k∈C +� +j∈I +Yk,jM R +c,kM S +i,j = +� +M RY M S� +c,i = Zc,i, +where we have used � +j∈I +Yk,j = ρk and � +k∈C +Yk,j = nj. +Substituting these equations into g(Y ) gives the result. +C.3 +Proof of Proposition 3 +Proof: Provided that M R +c,k = 1 for c ̸= k and M R +c,c = 0, +we obtain +g(Y ) = +� +i∈I +� +j∈I +� +c∈C +� +M S +i,j +�2 +Yc,iYc,j ++ +� +i∈I +� +j∈I +� +c∈C +� +k∈C,k̸=c +� +1 − M S +i,j +�2 +Yc,iYk,j += +� +i∈I +� +j∈I +� +c∈C +�� +M S +i,j +�2 +− +� +1 − M S +i,j +�2� +Yc,iYc,j ++ +� +i∈I +� +j∈I +� +c∈C +� +k∈C +� +1 − M S +i,j +�2 +Yc,iYk,j += +� +i∈I +� +j∈I +� +2M S +i,j − 1 +� +⟨Y:,i, Y:,j⟩ + α, +where +α = +� +i∈I +� +j∈I +� +1 − M S +i,j +�2 � +c∈C +� +k∈C +Yc,iYk,j += +� +i∈I +� +j∈I +� +1 − M S +i,j +�2 +ninj +since � +c∈C +Yc,i = ni and � +k∈C +Yk,j = nj. +ETHIC STATEMENT +The human biological samples were sourced ethically and +their research use was in accord with the terms of the +informed consents under an IRB/EC approved protocol. +All animal studies were ethically reviewed and carried +out in accordance with European Directive 2010/63/EEC +and the GSK Policy on the Care, Welfare and Treatment of +Animals. +REFERENCES +[1] +C. 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Yao et al., “A taxonomy of transcriptomic cell types across the +isocortex and hippocampal formation,” Cell, vol. 184, no. 12, pp. +3222–3241, 2021. + diff --git a/09AzT4oBgHgl3EQftv3n/content/tmp_files/load_file.txt b/09AzT4oBgHgl3EQftv3n/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e9b72c22720def5a1833646d43eaedde72d075ce --- /dev/null +++ b/09AzT4oBgHgl3EQftv3n/content/tmp_files/load_file.txt @@ -0,0 +1,989 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf,len=988 +page_content='DOT: Fast Cell Type Decomposition of Spatial Omics by Optimal Transport Arezou Rahimi, Luis A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Vale-Silva, Maria F¨alth Savitski, Jovan Tanevski, Julio Saez-Rodriguez !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Abstract Single-cell RNA sequencing (scRNA-seq) and spatially-resolved imaging/sequencing technologies have revolutionized biomedical research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' On one hand, scRNA-seq data provides for individual cells information about a large portion of the transcriptome, but does not include the spatial context of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' On the other hand, spatially resolved measurements come with a trade-off between resolution, throughput and gene coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Combining data from these two modalities can provide a spatially resolved picture with enhances resolution and gene coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Several methods have been recently developed to integrate these modalities, but they use only the expression of genes available in both modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' They don’t incorporate other relevant and available features, especially the spatial context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We propose DOT, a novel optimization framework for assigning cell types to tissue locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Our model (i) incorporates ideas from Optimal Transport theory to leverage not only joint but also distinct features, such as the spatial context, (ii) introduces scale- invariant distance functions to account for differences in the sensitivity of different measurement technologies, and (iii) provides control over the abundance of cells of different types in the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We present a fast implementation based on the Frank-Wolfe algorithm and we demonstrate the effectiveness of DOT on correctly assigning cell types or estimating the expression of missing genes in spatial data coming from two areas of the brain, the developing heart, and breast cancer samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Index Terms Optimal Transport, optimization, Frank-Wolfe, single-cell, biology, spatial, tissue, decomposition, deconvolution Arezou Rahimi is with the Institute for Computational Biomedicine, Faculty of Medicine, Heidelberg University, Heidelberg University Hospital, and Cellzome GmbH, GlaxoSmithKline, Heidelberg, Germany (e-mail: arezou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='rahimi@uni-heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='de).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Luis A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Vale-Silva is with Cellzome GmbH, GlaxoSmithKline, Heidelberg, Germany (e-mail: luis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='valesilva@gsk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='com) Maria F¨alth Savitski is with Cellzome GmbH, GlaxoSmithKline, Heidelberg, Germany (e-mail: maria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='faelth-savitski@gsk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='com) Jovan Tanevski is with the Institute for Computational Biomedicine, Faculty of Medicine, Heidelberg University and Heidelberg University Hospital, Heidelberg, Germany, and Department of Knowledge Technologies, Joˇzef Stefan Institute, Ljubljana, Slovenia (e-mail: jovan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='tanevski@uni-heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='de).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Julio Saez-Rodriguez is with the Institute for Computational Biomedicine, Faculty of Medicine, Heidelberg University and Heidelberg University Hospital, Heidelberg, Germany (e-mail: pub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='saez@uni-heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='de).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Jovan Tanevski and Julio Saez-Rodriguez cosupervised this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='01682v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='CE] 4 Jan 2023 1 DOT: Fast Cell Type Decomposition of Spatial Omics by Optimal Transport 1 INTRODUCTION The organization of cells within human tissues, their molec- ular programs and their response to perturbations are cen- tral to better understand physiology, disease progression and to eventual identification of targets for therapeutic in- tervention [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Cell types are distinct subpopulations of cells with unique trasncriptional signatures, which are often identified by known markers and/or by data-driven tech- niques, most commonly clustering based on transcriptomic profiles [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Single-cell RNA sequencing can profile the entire transcriptome (mRNA expression of the full range of genes) of large portions of individual (single) cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' This has made scRNA-seq an essential tool for revealing distinct cell types in complex tissues and has profoundly impacted our understanding of biological processes and the underlying mechanisms that control cellular functions [4], [5], [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' However, scRNA-seq requires dissociation of the tissue [8], losing the information about the spatial context and physical relationship between cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' To overcome these limitations, there has been recent ad- vancements in spatially resolved transcriptomics methods [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Spatial transcriptomics methods measure gene expres- sion in locations, hereafter referred to as spots, coupled with their two- or three-dimensional position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' These methods vary in two axes: spatial resolution and gene throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' On one hand, technologies such as Multiplexed Error- Robust Fluorescence In-Situ Hybridization (MERFISH) and In-Situ Sequencing (ISS), achieve cellular or even subcellular resolution [10] through cell segmentation [11], [12], but are limited to measuring up to a couple of hundred pre- selected genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' On the other hand, spatially resolved RNA sequencing, such as Spatial Transcriptomics [13], commer- cially available as 10x’s Visium, and Slide-seq [14], enable high-throughput gene profiling by capturing mRNAs in-situ at the cost of spots with the size of tens of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Thus, there is a trade-off between the resolution and the richness of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' A strategy to overcome these limitations is to combine scRNA-seq data with high resolution spatial data to map dissociated cells to spatial locations or more generally to combine it with low-resolution spatial data to estimate the composition of cell types and expression in each spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We refer to this task as decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Alternatively, we can attempt to enrich high-resolution data by predicting the expression of unmeasured genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' As the latter requires extrapolation to various degrees, machine learning and opti- mization methods are generally suited to the decomposition task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We will show that our tailored Optimal Transport formulation is capable of tackling both decomposition and enrichment tasks in high- and low-resolution spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Since the initial efforts to bridge this gap [15], there has been an increased interest in improvement and new method development (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' However, so far the methods rely on the genes that are captured both by scRNA-seq and spatial data without using the remaining genes captured in each modality, do not use the spatial relationships between spots in the spatial data, and usually come with high com- putation cost for large instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Neglecting the spatial con- text is equivalent to assuming random placement of spots in the space, which is in contrast to the established structure- function relationship of tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Considering only a subset of genes limits the applicability of these methods to cases where the two data sets share several informative genes, which might not be the case when different technologies are used for profiling, or when few genes are measured in the spatial data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', in MERFISH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We address these limitations by incorporating ideas from the Optimal Transport (OT) theory and adapting a Gromov-Wasserstein (GW) distance [16], [17] between scRNA-seq and spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We present DOT (Fast Cell Type Decomposition by Optimal Transport), a fast and scalable optimization framework to integrate scRNA-seq and spatial data for cell type localization by solving a multi- criteria probabilistic matching problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We summarize the main contributions of our work as follows: (i) We propose a novel formulation for mapping cell types from scRNA-seq to spots in spatial data by casting this problem to a multi-objective probabilistic matching problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Our model is applicable to both high- and low-resolution spatial data, in the form of inferring membership probabilities for the former and relative abundance of cell types in the latter, and is capable of estimating the expression of genes that are missing in the spatial data but present in the scRNA-seq data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' (ii) We adapt a generalization of OT with a Gromov- Wasserstein objective to leverage spatial information and to go beyond the use of genes common to the two modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' (iii) We introduce a scale-invariant metric based on cosine- similarity to account for differences in the scale of gene expressions in different technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' (iv) We present a very fast implementation for our model based on the Frank-Wolfe algorithm, ensuring scalabil- ity and efficient solvability in large-scale datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 2 RELATED WORK Cell type decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Several decomposition methods (also known as deconvolution methods) have been pro- posed in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' As cell type decomposition, partic- ularly in the high-resolution spatial data, is inherently a multiclass classification task, classification methods, such as Random Forests [18], can be used for tackling this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 2 However, because of the domain-specific properties of this problem, including differences in gene coverage, resolution, measurement sensitivity, and modality-specific characteris- tics, tailored approaches are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' While most of these models are designed specifically for low-resolution spatial data, some are also applicable to high- resolution spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' [19] proposed SPOTlight, which es- timates relative abundance of cell types in spots using non- negative matrix factorization regression and non-negative least squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Robust cell type decomposition (RCTD) [20] fits a statistical model by maximum-likelihood estimation, assuming a Poisson distribution for the expression of each gene at each spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' cell2location assumes a two-step Bayesian model for inferring cell type composition of spots [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Tangram [22] proposes a deep learning model to find the best placement of single cells in spots using a designed loss function and can thus carry cell type information as a byproduct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Seurat V3 workflow [23] is a widely-used toolkit for analyzing scRNA-seq data, which offers an “anchoring” technique based on mutual nearest neighbours classifier for aligning two modalities in the PCA space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Optimal Transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Optimal Transport (OT) [24] is a way to match, with minimal cost, data points/histograms be- tween two domains embedded in possibly different spaces using different variants of the Wasserstein distance [25], [26], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Over the past years, OT has been applied to various machine learning problems in a wide variety of contexts, including but not limited to generative modeling [28], [29], Wasserstein auto-encoders [30], feature aggregation [31], generalization error prediction [32], dataset denoising [33], graph matching/classification [34], and domain adaptation [35], [36], [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Recently, OT has been employed in biology, in particular to analyse single-cell data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For example, [38] model cellu- lar dynamics as an unbalanced dynamic transport, with the goal of transporting entities from one cross sectional measurement to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' [39] use OT for studying devel- opmental time courses and understanding the molecular programs that guide differentiation during development by incorporating temporal information and modeling cell growth over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Similarly, [40] employ graphical models and OT to reconstruct developmental trajectories from time courses with snapshots of cell states and lineages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 3 MODEL 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='1 Preliminaries Given a reference scRNAseq data (R for short), which is a collection of single cells each annotated with a cell type c ∈ C, and a target spatially resolved transcriptomics data (S for short), which consists of a set I of spots without cell type annotations, the goal of decomposition is to determine the composition of cell types in spots of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Note that the term “spot” can refer to one or a group of cells in certain spatial contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We denote by ni the given size (number of cells) of spot i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' When such information is not available, or when spots are at single-cell resolution, we set ni = 1 to compute the proportion or probability of cell types in each spot rather than computing the number of cells of each type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Let XR c,g denote the mean expression of gene g ∈ GR in cell type c ∈ C, where GR is the set of genes measured in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Each spot i ∈ I of S consists of spatial coordinates xi ∈ R2 or R3 and gene expressions XS i,g for g ∈ GS, where GS is the set of genes that are measured in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Further, if prior information about the expected abundance of cell types in S is available (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', estimated from a neighboring single-cell level tissue), we denote the expected abundance of cell type c ∈ C in S by rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Note that r is scaled such that � i∈I ni = � c∈C rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For convenience, we also define G = GR ∩ GS as the set of genes that are common between R and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' In the following, unless otherwise mentioned, vectors of gene expressions are assumed to be in the space of common genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' To assess dissimilarity between expression vectors a and b, we also introduce the distance function dcos(a, b) := � 1 − cos (a, b), (1) where cos (a, b) = 1 ∥a∥∥b∥⟨a, b⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Note that dcos is convex for positive vectors a and b, and is scale-invariant, in the sense that it is indifferent to the magnitudes of the vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' This is by design, since we want to assess dissimilarity between expression vectors regardless of the measurement sensitiv- ities of different technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We also note the following important property of dcos (proofs given in Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Unlike cosine dissimilarity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', 1 − cos(·, ·)), dcos is a metric distance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='2 High-Level Model Our model relies on determining a “many-to-many” map- ping Y of cell types in R to spots in S, with Yc,i denoting the proportion (or probability when ni = 1) of spot i ∈ I that is of cell type c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' A high-quality mapping should naturally match the expression of common genes across R and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We ensure this by considering the following genomic criteria: (i) Expression of genes in each spot of S should match expression of genes mapped to that spot via Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' (ii) Centroid of each cell type in R should match the cen- troid of that cell type in S as determined via Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' (iii) Distribution of expression of each gene across spots of S should be similar to the distribution of that gene across spots as mapped from R to S via Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Additionally, we may incorporate prior knowledge in the form of spatial location of spots as well as expected abun- dance of cell types using the following auxiliary criteria: (iv) Spots that are both adjacent in space and have similar expression profiles should attain similar cell type pro- files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' (v) If prior information about abundance of cell types in S is available (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', when R and S correspond to adjacent tissues), abundance of cell types mapped to S should match with the given abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The genomic objectives naturally take precedence over the auxiliary objectives, especially when a large number of genes are common between R and S, but the auxiliary objectives are useful when the common genes are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Note that objective (v) is meant to provide additional control over the abundance of cell types in the spatial data, but can be ignored if prior information about the abundance of cell types is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We elaborate on these objectives in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='3 Formulation Objective (i) ensures that the vector of gene expressions in spot i ∈ I (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', XS i,:) is most similar to the vector of gene ex- pressions mapped to spot i through Y (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', � c∈C Yc,iXR c,:).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' To achieve this objective, we minimize dissimilarity between these vectors by using di(Y ) := dcos(XS i,:, � c∈C Yc,iXR c,:).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' (2) Objective (ii) is in nature similar to objective (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Here, we would like to minimize dissimilarity between centroid of cell type c ∈ C in R (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', XR c,:) and centroid of cell type c in S as determined via Y (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', 1 ρc � i∈I Yc,iXS i,:).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Given the scale-invariance property of dcos, we can drop 1/ρc and measure the dissimilarity between these centroids using the following distance function dc(Y ) := dcos(XR c,:, ρ−1 c � i∈I Yc,iXS i,:) = dcos(XR c,:, � i∈I Yc,iXS i,:).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' (3) Our goal in objective (iii) is to match distribution of expression of gene g ∈ G in S (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', XS :,g) with the one mapped to S through Y (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', � c∈C Yc,:XR c,g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Hence, we minimize dissimilarity between these vectors by using dg(Y ) := dcos(XS :,g, � c∈C Yc,:XR c,g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' (4) To achieve objective (iv), we borrow ideas from Optimal Transport theory and the Gromov-Wasserstein metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Let M R and M S be metrics in R and S, respectively, in that M R c,k defines distance between cell types c and k, while M S i,j defines distance between spots i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Note that these distances are defined for each dataset independently;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' hence, we can use the entire features in each set: the entire genome in R, including the genes not measured in S, and the un- common/common genes as well as the spatial coordinates in S (see Section 4 for how these matrices are computed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The 2-Gromov-Wasserstein distance [16] between R and S for given mapping Y , denoted dGW(Y ), is defined in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Minimizing dGW(Y ) ensures that similar pair of spots in S (with respect to their locations and expressions) are not assigned to dissimilar pair of cell types in R, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' dGW(Y ) := � � � � � i∈I � j∈I � c∈C � k∈C � M R c,k − M S i,j �2 Yc,iYk,j (5) Let ρc := � i∈I Yc,i denote the abundance of cell type c in S as determined by mapping Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' As noted by [41], we may simplify (5) as stated in Proposition 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Define parameter ¯mi := � j∈I(M S i,j)2nj and auxiliary variables ¯mc := � k∈C(M R c,k)2ρk and Z := M RY M S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' GW distance function in (5) is equivalent to dGW(Y ) = �� c∈C � i∈I Yc,i( ¯mc + ¯mi − 2Zc,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' (6) Objective (v) provides optional control over abundance of cell types mapped to S, when prior information about expected abundance of cell types is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We employ Jensen-Shannon divergence between ρ and r to measure their dissimilarity dA(Y ) := 1 2DKL � ρ ���� ρ + r 2 � + 1 2DKL � r ���� ρ + r 2 � , (7) where DKL (p∥q) = � j pj log(pj/qj) denotes the Kull- back–Leibler divergence [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' In addition, to avoid overfit- ting, we may require that all cell types are at least minimally represented in the mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' To achieve this goal, we define dR(Y ) := − � c∈C log(ρc) = DKL (¯r∥ρ) , (8) where ¯rc = 1 for all c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Equation (8) is in fact a Nash fairness [43] objective and its logarithmic form ensures presence of all cell types (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', ρc > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We treat these criteria as objectives in a multi-objective optimization problem and to achieve them simultaneously (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', produce a Pareto-optimal solution), we optimize Y against a linear combination of these objectives as formu- lated below, hereafter referred to DOT model: min � i∈I nidi(Y ) + λC � c∈C ρcdc(Y ) + λG � g∈G dg(Y ) + λGWdGW(Y ) + λAdA(Y ) + λRdR(Y ) (9) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Y ∈ R|C|×|I| + , ρ ∈ R|C| (10) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' � c∈C Yc,i = ni ∀i ∈ I, (11) � i∈I Yc,i = ρc ∀c ∈ C, (12) where λC, λG, λGW, λA and λR are the user-defined penalty weights, and coefficients ni and ρc in (9) balance the scales of deviations in spots and cell types, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Unlike the conventional OT formulations, DOT does not require the cell type abundances in S (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', ρ) to be strictly equal to their expected abundances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', r), and rather penalizes their deviation in the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 4 ALGORITHM We propose a solution to the DOT model based on the Frank-Wolfe (FW) algorithm [44], [45], which is a first-order method for solving non-linear optimization problems of the form minx∈X f(x), where f : Rn → R is a (potentially non-convex) continuously differentiable function over the convex and compact set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' FW operates by replacing the non-linear objective function f with its linear approximation ˜f(x) = f(x(0))+∇xf(x(0))⊤(x−x(0)) at a trial point x(0) ∈ X, and solving a simpler problem ˆx = arg minx∈X ˜f(x) to produce an “atom” solution ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The algorithm then iterates by taking a convex combination of x(0) and ˆx to produce the next trial point x(1), which remains feasible thanks to convexity of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The FW algorithm is described in Algorithm 1, in which f(Y ) is the objective function in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='1 Distance Matrices Distance matrices M R and M S incorporate the features that are not shared across R and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' To compute M R c,k, we calculate the dissimilarity between the centroids of cell types c and k considering all genes in R (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', XR c,: = (XR c,g)g∈GR for each c ∈ C) M R c,k = dcos(XR c,:, XR k,:).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 4 Algorithm 1: Frank-Wolfe algorithm for DOT 1 Initialization: Setup distance matrices M R and M S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 2 Set t = 0 and find an initial map Y (0) (see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 3 while not converged do 4 Compute gradient ∆(t) = ∇Y f(Y (t)) (see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='2) 5 for each spot i ∈ I do 6 Find current best cell type ˆc = arg minc∈C{∆(t) c,i} 7 Compute atom solution ˆY (t) ˆc,i = ni and ˆY (t) c,i = 0 for c ̸= ˆc 8 Update Y (t+1) = Y (t) + 2 2+t( ˆY (t) − Y (t)) 9 t ← t + 1 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' M R is a metric in the domain of R since dcos is a metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The matrix M S captures the dissimilarity of S spots in terms of their locations and expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Let D1 i,j and D2 i,j represent distance of spots (i, j) with respect to their locations and expressions, respectively, as defined below: D1 i,j = 1condition �∥xi − xj∥ > ¯d � D2 i,j = dcos � XS i,:, XS j,: � , where ¯d is a given distance threshold, and D2 i,j is computed with respect to all genes in S (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', GS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Finally, we take M S to be the average of D1 and D2: M S = (D1 + D2)/2 (13) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' M S is a metric in the domain of S, since both D1 and D2 are metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' To see why this definition of M S makes sense, we first note that cell types, by definition, are distinct subpopula- tions in the scRNA-seq data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Therefore, it is reasonable to assume that their centroids are dissimilar (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', Mc,k ≈ 1 for c ̸= k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' This yields the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Let α = � i∈I � j∈I(1−M S i,j)2ninj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Assuming that cell types are relatively distinct, so that M R c,k ≈ 1, for c, k ∈ C, c ̸= k, then dGW(Y ) ≈ � α + � i∈I � j∈I � 2M S i,j − 1 � ⟨Y:,i, Y:,j⟩ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Observe that ⟨Y:,i, Y:,j⟩ measures similarity between cell type profiles of spots i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Therefore, dGW (i) rewards ⟨Y:,i, Y:,j⟩ when 2M S i,j −1 ≈ +1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', encourages adjacent spots to attain similar cell types if their expressions are similar) and (ii) penalizes ⟨Y:,i, Y:,j⟩ when 2M S i,j − 1 ≈ −1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', prevents distant spots from attaining similar cell types if their expressions are different).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Moreover, (iii) dGW is indifferent to pair (i, j) when 2M S i,j − 1 ≈ 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', if i and j are distant or different in expressions, but not both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='2 Producing an Atom Solution While the DOT model is not separable, its linear approxi- mation can be decomposed to |I| independent subproblems, one for each spot i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' This is because, unlike conventional OT formulations, we do not require the distribution of cell types (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', ρ) to be equal to their expected distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', r), but have penalized their deviations in the objective function using dA (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The subproblem i then becomes min � ⟨Y:,i, ∆(t) :,i ⟩ : Y:,i ∈ R|C| + , � c∈C Yc,i = ni � which, in turn, is a simple sorting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' This property of Algorithm 1 enables it to efficiently tackle problems with large number of spots in the spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='3 Convergence Under suitable conditions, FW converges to an optimal so- lution in linear rate when optimizing a convex function over a polytope domain [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Given the non-convex objective function in (9), Algorithm 1 instead obtains a first-order sta- tionary point at a rate of O(1/ √ t) [47], [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We numerically assess the convergence of Algorithm 1 at iteration t using the so-called “FW-gap” [45] δ(t) := � i∈I � c∈C(Y (t) c,i − ˆY (t) c,i )∆(t) c,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We also implemented acceleration techniques such as av- eraging gradients [49] and away steps [46], [50], but did not observe practical gains compared to the vanilla FW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Moreover, while it is common practice to use entropic or other strongly-convex regularizations in OT to facilitate producing the atom solutions, we did not incorporate such regularizations because an atom solution can be produced easily in our formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 5 PRACTICAL ENHANCEMENTS In this section, we introduce practical enhancements to incorporate the domain-specific properties of the problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='1 Cell Heterogeneity While cell types are distinct subpopulations of cells, signif- icant variations may naturally exist within each cell type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' This means, a single vector XR c,: may not properly represent the distribution of cells within this cell type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Consequently, mapping cell types solely based on the centroids of cell types can be error-prone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' To capture the intrinsic heterogeneity of cell types, we cluster each cell type into predefined κ smaller groups using an unsupervised learning method, and produce a total of κ|C| centroids to replace the original |C| centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' With this definition of centroids, we treat all terms except dA and dR as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For dA and dR, since prior information about cell types (and not sub-clusters) are available, we keep ρ to represent the abundance of original cell types by setting ρc = � k∈Kc � i∈I Yk,i, where Kc denotes the set of sub-clusters of cell type c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Finally, once Y is obtained, � k∈Kc Yk,i determines probability that spot i is of cell type c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='2 Sparse Mapping As previously discussed, spatial data are either high- resolution (single-cell level) or low-resolution (multicell level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' In the case of high-resolution spatial data, given that each spot corresponds to an individual cell (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', ni = 1), it is desirable to produce sparse allocations, in the sense that we prefer Yc,i close to 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' In general, assuming that Yc,i ∈ {0, ni}, then (11) implies that Yc,i = ni for exactly one cell type c and is zero for all other cell types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Consequently, for binary Y we obtain dcos � XS i,:, � c∈C Yc,iXR c,: � = 1 ni � c∈C Yc,idcos � XS i,:, XR c,: � , which is linear in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' As linear objectives promote sparse (or corner point) solutions, we may control the level of sparsity of the mapping by introducing a parameter θ ∈ [0, 1] and redefining di(Y ) as di(Y ) =(1 − θ)dcos � XS i,:, � c∈C Yc,iXR c,: � + θ ni � c∈C Yc,idcos � XS i,:XR c,: � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' (14) Note that a higher value for θ yields a sparser solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Indeed, with θ = 1 and zero weights assigned to other objectives, the optimal mapping will be completely binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 6 RESULTS We compared the performance of our method, abbreviated DOT, against five state of the art models in the litera- ture: SPOTlight [19], RCTD [20], cell2location [21], Tangram [22], and Seurat [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We designed three exper- iments to evaluate the performance of DOT from different perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Briefly, in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='2, we evaluate the per- formance of models in predicting the cell type of single- cell level spots in high-resolution spatial data, followed by cell type decomposition in multicell spots in low-resolution spatial data in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Finally, in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='4, we evaluate capability of DOT in estimating the expression of genes that are missing in the spatial data but present in the reference single-cell data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We performed experiments on data coming from (i) the primary motor cortex of the mouse brain, (ii) the primary somatosensory cortex of the mouse brain, (iii) the develop- ing human heart, and (iv) the human breast cancer, specifics of which are presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='1 Experimental Setup 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='1 Parameter Setting For DOT, we set penalty weights λC = 1 and λG = |n|/|G| to balance the scales of different objectives, where |n| := � i∈I ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' This is because both � i∈I nidi(Y ) and � c∈C rcdc(Y ) are in the range of 0 and |n|, while 0 ≤ � g∈G dg(Y ) ≤ |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For the GW objective, it is not difficult to verify that 0 ≤ dGW(Y ) ≤ |n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' However, although spatial information contributes to the accuracy of cell type mapping, meaning that λGW > 0 is desirable, a large value for λGW may dominate the genomic objectives di(Y ), dc(Y ) and dg(Y ), thus reduce accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' A middle-ground is to set a small positive value for λGW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' In our computations, we found that λGW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='1 works best in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Whenever prior information about expected abundance of cell types is available, we set λA = 1 and λR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We computed ρc, the expected abundance of cell type c, based on the observed fraction of cell type c in the reference scRNA- seq data multiplied by |n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We set the sparsity parameter θ = 1 for high resolution spatial data, and set θ = 0 for low resolution spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' To capture heterogeneity of cell types, we clustered each cell type into κ = 10 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The distance threshold ¯d is computed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For each spot we computed its Euclidean distance to 8 closest spots in space1, yielding 8|I| values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We then took ¯d as the 99th percentile of these values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For RCTD, SPOTlight, Tangram, and C2L we used the default parameters suggested by the authors with the fol- lowing exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For RCTD we set the parameter UMI_min to 50 to prevent the model from removing too many cells from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Given the large number of cell types in the mouse MOp datasets, for SPOTlight we reduced the number of cells per cell type to 100 to enhance the computa- tion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Similarly, as Tangram was not able to produce results in a reasonable time for the MOp instances, we randomly selected 500 cells per cell type to reduce the com- putation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For C2L, we used 20000 epochs to balance computation performance and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For Seurat and SingleR, we followed the package documentations, with functions used with default parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For RF we used the implementation provided in the R package ranger [51] with all parameters set at their default values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='2 Performance Metrics We compared the predictive performance of DOT against the other methods using three metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Accuracy in the context of high-resolution spatial data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', when each spot corresponds to an individual cell) is the proportion of correctly classified spots (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', sum of the main diagonal in the confusion matrix) over all spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' To assess the accuracy of membership probabilities produced by each model, we compared the models using Brier Score, also known as mean squared error: Brier Score = |I|−1 � i∈I � c∈C(Yc,i − Pc,i)2, where Pc,i = 1 if spot i is of cell type c and Pc,i = 0 otherwise, and Yc,i is the predicted probability that spot i is of cell type c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' As Brier Score is a strictly proper scoring rule for measuring the accuracy of probabilistic predictions [52], a model with lower Brier Score produces better-calibrated probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Besides the cell type that each spot is annotated with, we can produce a cell type probability distribution for each spot by considering the cell type of its neighboring spots, using a Gaussian smoothing kernel of the form ˜Pc,i =( � j∈I Ki,j)−1 � j∈I Ki,jPc,j, where Ki,j = exp �−∥xi − xj∥2/2σ2� and σ is the kernel width parameter which we set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='5 ¯d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Note that as spot j becomes closer to spot i, its label contributes more to the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We used 8 closest neighbors to mimic the number of adjacent tiles in a 2D regular grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 6 probability distribution at spot i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Using these probabilities, we also introduce the Spatial Jensen-Shannon (SJS) divergence to compare the probability distributions assigned to spots (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', Y ) with the smoothed probabilities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', ˜P ) SJS = |I|−1 � i∈I JS(Y:,i, ˜P:,i), where JS(Y:,i, ˜P:,i) is the Jensen-Shannon divergence be- tween probability distributions Y:,i and ˜P:,i with base 2 logarithm [42], also defined in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='2 Experiment 1: Cell Type Prediction in High- Resolution Spatial Data Our goal with our first set of experiments is to evaluate the performance of different models in determining the probability distribution of cell types at each spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Since the identity of the cell type represented by the spot is known in our high resolution spatial data, we can use this information as ground-truth when evaluating the performance of the different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' In addition to deconvolution methods, we used SingleR [53], a method to define cell type from single-cell resolution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Given the multiclass classifica- tion nature of this task, we also used RF [18] as a multiclass classifier baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We use the high-resolution MERFISH spatial data of the primary motor cortex region (MOp) of the mouse brain [54], which contains the spatial information of 280,186 cells across 75 samples (Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' With each sample, we created a reference scRNA-seq data using all the 280,186 cells, except the cells contained in the sample, and the 254 genes to estimate the centroids of the 99 reference cell types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We further created 15 high resolution spatial datasets for each sample (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', a total of 1125 spatial datasets) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' To simulate the effect of number of shared features between the spatial and scRNA-seq data, we assumed that only a subset of the 254 genes are available in the spatial data by selecting the first |G| genes, where |G| ∈ {50, 75, 100, 125, 150} (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', 20%, 30%, 40%, 50%, 60% of genes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Moreover, to simulate the effect of differences in measurement sensitivities of different technologies, we introduced random noise in the spatial data by multiplying the expression of gene g in spot i by 1+βi,g, where βi,g ∼ U(−ϕ, ϕ) with ϕ ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We compare the predictive performance of DOT to Seurat, RCTD, Tangram, SingleR and RF in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We removed SPOTlight and C2L from these plots due to their clear under-performance in the high resolution spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We observe that not only does DOT dominate the three alter- natives in assigning correct cell types to the spots (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 1a), but also produces well-calibrated probabilities (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 1b) and better captures the relationships between cell types in space (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 1c), owing to its capacity to incorporate the spatial information in dGW through the distance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We also observe that even with very few genes in common between the spatial data and the reference scRNA-seq data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', |G| ≤ 75), DOT is able to reliably determine the cell type of spots in the space with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' In contrast, RCTD fails to produce results due to lack of shared information, and Seurat and Tangram produce results with low accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Additionally, we observe that DOT is more immune to fluctuations in expressions in the spatial data, implying the effectiveness of our dcos distance function in accounting for ϕ = 0 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='25 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='5 50 75 100 125 150 50 75 100 125 150 50 75 100 125 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='8 Accuracy (a) ϕ = 0 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='25 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='5 50 75 100 125 150 50 75 100 125 150 50 75 100 125 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='8 Brier Score (b) ϕ = 0 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='25 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='5 50 75 100 125 150 50 75 100 125 150 50 75 100 125 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='7 SJS (c) DOT Seurat RCTD Tangram RF SingleR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 1: Predictive performance of the algorithms in the high- resolution spatial data across different number of genes in the spatial data (x-axis) and different noise factors (ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Points represent the median of 75 values, and the shaded ar- eas correspond to their interquartile interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Note: SingeR does not produce probabilities and is compared based on Accuracy only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' differences in measurement scales of different technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' In terms of algorithmic performance (Table 1), DOT takes on average 441 seconds to solve each instance, which is an order of magnitude faster than RCTD, Tangram, and RF, and is comparable to Seurat and SingleR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='3 Experiment 2: Cell Type Decomposition in Low- Resolution Spatial Data We evaluated the performance of models on low-resolution spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For these experiments, since there is no ground truth for real multicell spatial data such as Visium and Slide- seq, we resort to producing ground truth low-resolution spatial data by pooling the adjacent cells in the high res- olution spatial data of primary motor cortex of the mouse brain (MOp), primary somatosensory cortex of the mouse brain (SSp), and the developing human heart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' A1 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='1 illustrates a sample low-resolution spatial data obtained from a MERFISH MOp tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Unlike the high-resolution spatial data, the ground truth Pc,i now corresponds to relative abundance of cell type c in spot i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We can therefore assess the performance of each model by comparing the probability distributions P:,i and the estimated probabilities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', Y:,i) using Brier Score or Jensen- Shannon metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='1 Experiments on the Mouse MOp To produce ground truth for MOp, using the common subclass annotations between MERFISH MOp and scRNA- seq MOp [55] (see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='1), for each of the 75 MER- FISH MOp samples, we randomly assigned each cell in the MERFISH MOp data to a cell in the scRNA-seq MOp data of the same subclass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' we lowered the resolution of spatial data by splitting each sample into regular grids of 7 Experiment Resolution Instances DOT Seurat RCTD Tangram SPOTlight C2L SingleR RF MOp High 1125 441 380 4748 10141 7884 3310 303 7427 MOp Low 75 457 1086 4705 8250 52825 6119 — — SSp Low 1 4 21 117 248 705 364 — — Heart Low 1 8 11 185 88 316 398 — — TABLE 1: Average computation times (in seconds) of different models across different experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='8 Jensen−Shannon Brier Score DOT Seurat RCTD SPOTlight C2L Tangram Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 2: Predictive performance of the algorithms in the low- resolution spatial data across 75 samples of MOp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Each point in the scatter plots denoting the average performance across all spots in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' length 100µm to mimic the size and inter-distance of spots in low-resolution spatial transcriptomics, such as Visium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Finally, we aggregated the expression profiles of cells within each tile as the expression profile of the respective spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 2 compares the performance of DOT against RCTD, SPOTlight, cell2location (C2L), Tangram and Seurat in determining the cell type composition of the multicell spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We observe that DOT outperforms other models with respect to both metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' As presented in Table 1, DOT took on average 457 seconds to solve an instance, which proved to be more than twice faster than Seurat, and orders of mag- nitude faster than RCTD, SPOTlight, C2L and Tangram, further highlighting the superiority of DOT in terms of both accuracy and computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='2 Experiments on the Mouse SSp and the Developing Human Heart We also carried out experiments on data from the SSp region of mouse brain as well as the developing human heart to evaluate the performance of models on tissues of different structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We employed single-cell level spatial data coming from osmFISH technology [56] to produce multicell data for SSp (Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For the developing human heart, we used subcellular spatial data generated by the ISS platform [57] (Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We tested the performance of the six deconvolution methods on these two samples, results of which are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Each subplot shows the distribution of prediction error based on the Jensen-Shannon divergence at each spot in the spatial data, with the average value over all spots given on top of each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' DOT outperforms other models in the human heart sample and is among the best-performing models in the mouse SSp sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Moreover, performance of DOT is not sensitive to different regions/cell type of the tissue (compare to Tangram and Seurat in SSp and RCTD in human heart).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' These results further highlight the competitive performance of DOT and its robustness in identifying the cell type com- position of spots across different tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='4 Experiment 3: Gene Expression Estimation in High- Resolution Spatial Data Recall that, in high-resolution spatial data, only a few genes are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Hence, as a result of mapping scRNA-seq to spatial data, we can estimate the expression of genes that were not measured in the spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Therefore, in our final set of experiments, we evaluate the performance of DOT in estimating the expression of missing genes in the high resolution spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For this experiment, we use the spatial data from breast cancer tumor microenvironment produced by the 10X Xenium In Situ technology [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The dataset is unique in that it contains both high-resolution (Xenium) and low-resolution (Visium) spatial data of serial sections from the same tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The high-resolution data con- tains spatial information of 313 genes across 167,782 single- cell spots, while the low-resolution data contains the spatial information of around 18,000 genes across 4,992 multicell spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The dataset also contains the dissociated scRNA-seq data coming from a tissue adjacent to the tissues used for high- and low-resolution spatial data, which contains the measurements of 18,000 genes across 30,365 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Therefore, we can use Visium as a proxy for ground truth to validate the distribution of genes that are not measured in Xenium, and are mapped by DOT from scRNA-seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For this experiment, we matched the common capture areas of high- and low-resolution spatial data using the Hematoxylin-Eosin (H&E) images accompanying these spa- tial data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' A2), which corresponded to 134,664 cells in the high-resolution and 3,928 spots in the low-resolution spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Given that the task here is to complete the expression of missing genes in the high-resolution spatial data, we first performed community detection on the graph of shared nearest neighbors of cells in scRNA-seq using the Leiden implementation in [23], which is common practice in single-cell analysis and is used as a first step towards cell type identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' This resulted in 218 clusters, and we then mapped the centroids of these clusters to the high- resolution spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' (We also tried as high as 1000 fine- grained clusters but got essentially the same results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=') We also turned off the cell-type-related objectives since we are not mapping cell types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Although the datasets are extremely 8 DOT (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='308) SPOTlight (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='456) Tangram (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='484) C2L (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='294) RCTD (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='298) Seurat (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='404) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='00 JS DOT (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='392) SPOTlight (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='548) Tangram (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='408) C2L (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='541) RCTD (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='607) Seurat (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='468) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='00 JS Mouse SSp Human Heart Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 3: Distribution of performance of models on each individual spot in the low-resolution spatial data of Mouse SSp (top) and developing human heart (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' large, DOT was able to perform the mapping in less than two hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We start by evaluating the performance of DOT on genes that are present in the high-resolution spatial data as a true ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The qualitative comparisons of gene maps of eight genes associated with breast cancer [59] produced by DOT with those of high-resolution (ground truth) and low- resolution data (approximate ground truth) can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' As can be observed, the expression maps produced by DOT match almost perfectly with the ground truth expres- sion maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Both DOT and the ground truth high-resolution spatial data also match the low resolution gene expression maps almost perfectly, which further validate the quality of mapping produced by DOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Note that due to the single- cell resolution of the high-resolution spatial data, expression levels are higher at high resolution, thus the brighter colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Nonetheless, the spatial patterns match between all three rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Next, we compare the expression map of genes that are not present in the high-resolution spatial data but are estimated by DOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 5a illustrates the expression maps of five genes associated with breast cancer that are not measured in the high-resolution spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For a quanti- tative comparison of expression maps, given that there is no one-to-one mapping between single-cell spots in the high- resolution and multicell spots in the low-resolution spatial data, we split the tissue into a 10 by 10 grid, and aggregated the expression of each gene within each tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Consequently, we obtained two 100 by 18,000 matrices, one for the low- resolution spatial data and another for DOT, with rows corresponding to tiles and columns corresponding to genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 5b compares the column-wise cosine similarities across different genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' These results further confirm the ability of DOT in reliably estimating the expression of missing genes in high-resolution spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 7 CONCLUSION Single-cell RNA-seq and spatially-resolved imag- ing/sequencing technologies, the cutting edge technologies in transcriptomic data generation, each provide a partial picture in understanding the organization of complex tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' To obtain a full picture, computational methods aim at combining data from these two modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We present DOT, a fast and scalable optimization framework based on Optimal Transport theory, for assigning cell types to tissue locations by leveraging the spatial information as well as both joint and distinct genes across scRNA-seq and spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Using experiments on data from mouse brain and human heart, we show that DOT predicts the cell type composition of spots in spatial data with high accuracy, outperforming the state of the art methods both in terms of predictive performance and computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' APPENDIX A IMPLEMENTATION DETAILS OF THE FW ALGORITHM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='1 Initial Solution A good quality initial solution plays a critical role in fast convergence of FW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Given the multi-objective nature of our model, we produce an initial solution as convex combina- tion of two solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' In the first solution, for each spot i we first find cell type ˆc = arg minc∈C{dcos � XS i,:, XR c,: � } and set Yc,i = ni if c = ˆc and Yc,i = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Note that this solution is optimal for the sparse case when di is the only objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' In the second solution, we simply set Yc,i = niρc/|I| for each i and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Note that this solution is optimal for dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We then set the initial solution as the convex combination of these two solutions, with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='9 weight assigned to the first solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='2 Derivatives To find the derivatives of di(Y ) and dc(Y ), defined in (2) and (3), we introduce auxiliary quantities ¯ XS := Y ⊤XR and ¯ XR := Y XS to denote the expressions mapped through Y to spots and cell types, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Derivatives for di(Y ) and dc(Y ) can then be calculated as: ∂di ∂Yc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='i = 1 ∥XS i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=':∥⟨XR c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=':,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' T S i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=':⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' ∂dc ∂Yc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='i = 1 ∥XRc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=':∥⟨XS i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=':,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' T R c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=':⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 9 CEACAM6 (Xenium) POSTN (Xenium) ITGAX (Xenium) VWF (Xenium) KRT15 (Xenium) FOXA1 (Xenium) GATA3 (Xenium) TACSTD2 (Xenium) CEACAM6 (DOT) POSTN (DOT) ITGAX (DOT) VWF (DOT) KRT15 (DOT) FOXA1 (DOT) GATA3 (DOT) TACSTD2 (DOT) CEACAM6 (Visium) POSTN (Visium) ITGAX (Visium) VWF (Visium) KRT15 (Visium) FOXA1 (Visium) GATA3 (Visium) TACSTD2 (Visium) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 4: Expression map of eight breast cancer markers measured in both Xenium (ground truth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' top) and Visium (low- resolution proxy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' bottom), and as mapped from scRNA-seq to Xenium using DOT (estimated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Brighter means higher expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' SCGB2A2 (DOT) KRT17 (DOT) CDH2 (DOT) SFRP2 (DOT) MT−ND1 (DOT) SCGB2A2 (Visium) KRT17 (Visium) CDH2 (Visium) SFRP2 (Visium) MT−ND1 (Visium) (a) 0% 10% 20% 30% 40% 50% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='00 Expression map similarity % Genes (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 5: (a) Expression map of five breast cancer markers that are measured in Visium (bottom) but are missing in Xenium and are mapped from scRNA-seq using DOT (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' (b) Cosine similarity between expression maps of Visium and DOT for the genes that are not measured in Xenium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' where T S i,g = −1 2di(Y ) � XS i,g ∥ ¯ XS i,:∥ − ¯XS i,g ∥ ¯ XS i,:∥3 ⟨XS i,:, ¯ XS i,:⟩ � , T R c,g = −1 2dc(Y ) � XR c,g ∥ ¯ XRc,:∥ − ¯XR c,g ∥ ¯ XRc,:∥3 ⟨XR c,:, ¯ XR c,:⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Derivative of ρcdc(Y ) then can be computed using the product rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Similarly, we may derive the derivative for dg(Y ) via ∂dg ∂Yc,i = −1 2dg(Y ) XR c,g ∥XS:,g∥ � XS i,g ∥ ¯ XS:,g∥ − Yc,i ∥ ¯ XS:,g∥3 ⟨XS :,g, ¯ XS :,g⟩ � Taking into account the simplification from Proposi- tion 2, noting that ¯mc and Zc,i are functions of Y while ¯mi is constant, we can show that ∂dGW ∂Yc,i = 1 2dGW(Y )(2 ¯mc + ¯mi − 4Zc,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Finally, the derivatives for dA and dR, defined in (7) and (8) respectively, can be calculated as: ∂dA ∂Yc,i = 1 2 log � 2ρc ρc + rc � , ∂dR ∂Yc,i = − 1 ρc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' APPENDIX B DATASETS B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='1 Mouse Primary Motor Cortex (MOp) MERFISH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For high-resolution spatial transcriptomics, we used the spatially resolved cell atlas of the MOp recently generated using multiplexed error-robust fluorescence in situ hybridization (MERFISH) technology and made pub- licly available by [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The processed dataset contains nor- malized RNA counts of 254 genes and coordinates of the boundaries of a total of 280,186 segmented cells across 75 samples in the MOp of two adult mice, with the number of cells within each sample ranging from 1000 to 7500 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We computed the (x, y) coordinates of the center of each cell by taking the average of the coordinates of its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The study also identifies 99 trasncriptionally distinct cell types 10 0 500 1000 1500 2000 2500 0 500 1000 1500 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' A1: Synthetic multicell spatial data from MERFISH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Dots show individual cells and tiles represent multicell spots (darker tile means denser spot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' by community detection applied on a cell similarity graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The clustering resulted in 39 excitatory neuronal cell types (clusters), 42 inhibitory neuronal cell types, 14 non-neuronal cell types, and four other cell types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' scRNA-seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The corresponding scRNA-seq data comes from a cell atlas of the MOp [55], which contains the mRNA expression of the full range of genes for more than 500,000 individual cells across several omics layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We used the scRNA-seq dataset scRNA_10x_v2_A generated at the Allen Institute, which contains 145,748 cells and 100 cell types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' After removing the unannotated cells and low quality cell types (as categorized in the study), we retrieved 124,330 cells and 90 distinct cell types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For computational efficiency, we also selected the top 5,000 variable genes according to their means and variances [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' A1 illustrates a sample low-resolution spatial data obtained from a MERFISH MOp tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='2 Mouse Primary Somatosensory Cortex (SSp) Similar to MOp, another well-studied tissue area is the pri- mary somatosensory cortex area (SSp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Here, we used high- resolution spatial data coming from the osmFISH platform [56], which contains measurements of 33 genes across 4,837 cells, as well as annotations based on 11 major cell types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For reference scRNA-seq data with matched cell types, we used the annotations independently generated by [60] using 5,392 single cells in the same SSp region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='3 Developing Human Heart For the developing human heart, we used subcellular spatial data generated by the ISS platform [57], which contains tissue sections from human embryonic cardiac samples collected at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We selected the PCW6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='5 slide which contains measurements of 69 genes across 17,454 cells as well as annotations of 12 major cell types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The same study also provides scRNA-seq data for similar slide, which contains matched cell types for 3,253 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' A2: Common region (cyan) in the capture areas of Visium (dashed blue lines) and Xenium (dark orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The pink region is the H&E image accompanying Visium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='4 Human Breast Cancer Breast cancer is a complex disease with significant cel- lular and molecular heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The breast cancer tu- mor microvenvironment dataset generated in [58] contains both single-cell and multicell data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The single-cell reso- lution spatial dataset is produced by the recent 10X Xe- nium In Situ technology, and contains two replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We used Xenium_FFPE_Human_Breast_Cancer_Rep1, which contains the spatial information of 313 genes for 167,782 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' The low-resolution multicell spatial dataset is produced by the 10X Visium Spatial Transcriptomics technology, which contains the spatial information of 18,000 genes for 4,992 multicell spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For the reference scRNA- seq data, we used the Single Cell Gene Expression Flex (FRP) data generated from a tissue section adjacent to the tissue sections used for Visium and Xenium work- flows, which contains expression of 18,000 genes across 30,365 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' APPENDIX C C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='1 Proof of Proposition 1 Proof: Note that ���� a ∥a∥ − b ∥b∥ ���� 2 = ∥a∥2 ∥a∥2 + ∥b∥2 ∥b∥2 − 2 ⟨a, b⟩ ∥a∥∥b∥ = 2 − 2 cos(a, b) ⇒ dcos(a, b) = � 1 − cos(a, b) = √ 2∥a/∥a∥ − b/∥b∥∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' This shows that dcos is a metric since ∥ · ∥ is a metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' We can easily show that cosine dissimilarity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=', 1 − cos(·, ·)) is not a metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' For instance, consider a = (1, 0, 0), b = (0, 1, 0) and c = (x, x, √ 1 − 2x2) for arbitrary x ∈ ( 1 2, 1 √ 2), and let f denote the cosine dissimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Then f(a, b) = 1 − cos(a, b) = 1, and f(a, c) = f(c, b) = 1 − x, which violates the triangular inequality since f(a, c) + f(c, b) = 2−2x < 1 = f(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' It is not difficult to see that dcos(a, c)+ dcos(c, b) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' 11 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='2 Proof of Proposition 2 Proof: Rewrite g(Y ) as g(Y ) = � i∈I � j∈I � c∈C � k∈C � M R c,k − M S i,j �2 Yc,iYk,j = � c∈C � i∈I Yc,i � k∈C � j∈I Yk,j � (M R c,k)2 + (M S i,j)2 − 2M R c,kM S i,j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Expanding the inner summations we obtain: � k∈C � j∈I Yk,j(M R c,k)2 = � k∈C (M R c,k)2 � j∈I Yk,j = ¯mc � k∈C � j∈I Yk,j(M S i,j)2 = � j∈I (M S i,j)2 � k∈C Yk,j = ¯mi � k∈C � j∈I Yk,jM R c,kM S i,j = � M RY M S� c,i = Zc,i, where we have used � j∈I Yk,j = ρk and � k∈C Yk,j = nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Substituting these equations into g(Y ) gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='3 Proof of Proposition 3 Proof: Provided that M R c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='k = 1 for c ̸= k and M R c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='c = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' we obtain g(Y ) = � i∈I � j∈I � c∈C � M S i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='j �2 Yc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='iYc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='j + � i∈I � j∈I � c∈C � k∈C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='k̸=c � 1 − M S i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='j �2 Yc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='iYk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='j = � i∈I � j∈I � c∈C �� M S i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='j �2 − � 1 − M S i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='j �2� Yc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='iYc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='j + � i∈I � j∈I � c∈C � k∈C � 1 − M S i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='j �2 Yc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='iYk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='j = � i∈I � j∈I � 2M S i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='j − 1 � ⟨Y:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' Y:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='j⟩ + α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' where α = � i∈I � j∈I � 1 − M S i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='j �2 � c∈C � k∈C Yc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='iYk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='j = � i∈I � j∈I � 1 − M S i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='j �2 ninj since � c∈C Yc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='i = ni and � k∈C Yk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content='j = nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' ETHIC STATEMENT The human biological samples were sourced ethically and their research use was in accord with the terms of the informed consents under an IRB/EC approved protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' All animal studies were ethically reviewed and carried out in accordance with European Directive 2010/63/EEC and the GSK Policy on the Care, Welfare and Treatment of Animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQftv3n/content/2301.01682v1.pdf'} +page_content=' REFERENCES [1] C.' metadata={'source': 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processes that are impacted not only by cellular dynamics and the ability of +pathogens to effectively reproduce and spread, but also by population-level dynamics and the effectiveness of miti- +gation measures. A timely exchange of information related to the spread of novel pathogens, stay-at-home orders, and +other containment measures can be effective at containing an infectious disease, particularly during in the early stages +when testing infrastructure, vaccines, and other medical interventions may not be available at scale. Using a multiplex +epidemic model that consists of an information layer (modeling information exchange between individuals) and a spa- +tially embedded epidemic layer (representing a human contact network), we study how random and targeted disruptions +in the information layer (e.g., errors and intentional attacks on communication infrastructure) impact outbreak dynam- +ics. We calibrate our model to the early outbreak stages of the SARS-CoV-2 pandemic in 2020. Mitigation campaign +can still be effective under random disruptions, such as failure of information channels between a few individuals. +However, targeted disruptions or sabotage of hub nodes that exchange information with a large number of individuals +can abruptly change outbreak characteristics such as the time to reach the peak infection. Our results emphasize the +importance of using a robust communication infrastructure that can withstand both random and targeted disruptions. +Online communication platforms and exposure notifica- +tion apps can help slow down and contain the spread of +an infectious disease1. Individuals who have been made +aware of an outbreak are likely to adapt their behavior to +reduce their risk of being infected. To study the interplay +between infectious disease outbreaks and corresponding +changes in individual contact behaviors, Granell et al.2 in- +troduced an epidemic model that accounts for the spread +of awareness through an information layer that is coupled +to a human contact network. Building upon their model of +awareness diffusion, our work studies the impact of ran- +dom and targeted disruptions in the information layer on +the overall outbreak dynamics. +I. +INTRODUCTION +The study of epidemic processes in networks has pro- +vided many insights into the interplay between structure and +dynamics.3,4 The aim of many works in this area has been +to analyze the impact of different structural features such as +clustering5, community structure6,7, hub nodes, and scale- +free degree distributions8 on the evolution of susceptible- +infected-susceptible (SIS) and susceptible-infected-recovered +(SIR) models and their extensions.9–11 Connections between +epidemic processes and percolation contributed to the devel- +opment of analytical methods that are useful to analyze epi- +demic transitions and determine outbreak size.12–16 Along +a)hoseingmasoomy@gmail.com +b)tomchou@ucla.edu +c)l.boettcher@fs.de +with progress in understanding epidemic processes in static +single-layer networks, developments in the study of tempo- +ral networks17, multilayer networks18,19, and other structures +describing higher-order interactions20–23 have allowed for the +integration of time-varying and non-binary interactions. +Before research turned to epidemic models in multilayer +networks, interactions between disease and behavioral dy- +namics have been studied mainly in single-layer networks24 +and well-mixed populations.25–28 In an extension of the +classical SIS model, the so-called susceptible-infected-alert- +susceptible (SIAS) model, a new compartment was used to +study the effect of “alert” individuals that are surrounded by +a certain number of infecteds on disease dynamics.29,30 The +SIAS model has been implemented using a two-layer net- +work31 with a contact layer and an information-dissemination +layer to find optimal information dissemination strategies that +help contain an outbreak. +The interplay between behavioral effects and network dy- +namics has also been analyzed in terms of a multiplex struc- +ture where information on an outbreak diffuses in an infor- +mation layer.2,32 In a multiplex network, all of the interlayer +edges are edges between nodes and their counterparts in other +layers. As in the SIAS model, individuals in the information +layer can be either aware or unaware of a disease. Aware- +ness then translates into a reduced infection rate. The origi- +nal awareness model has been modified in various ways. One +study used a threshold model in the information layer and +identified awareness cascades.33 Other research investigated +the effects of dynamically varying transmission rates34, cou- +pled SIR and unaware-aware-unaware (UAU) dynamics with +and without latency35,36, SIS and UAU dynamics that propa- +gate at different speeds37, and higher-order interactions38. For +a detailed overview of models of coevolving spreading pro- +cesses in networks, we refer the reader to Ref. 39. +arXiv:2301.00748v1 [physics.soc-ph] 2 Jan 2023 + +Impact of random and targeted disruptions on information diffusion during outbreaks +2 +In this work, we study coevolving susceptible-exposed- +infected-recovered-deceased (SEIRD) and UAU dynamics on +a multiplex network that consists of an epidemic layer and an +information layer. The exposed compartment in our model +accounts for latency (i.e., the time difference between infec- +tion and becoming infectious). Different variants of SEIRD +models have been used to mechanistically describe the spread +of an infectious disease for which the latency period between +time of infection to time of becoming infectious cannot be +neglected9,40–42. Examples of such infectious diseases include +measles, smallpox, and SARS-CoV-2. +One of the main goals of this work is to provide insight into +the impact of disruptions in the information diffusion layer +on the overall outbreak dynamics. We therefore study differ- +ent edge removal protocols that describe random and targeted +disruptions. In Sec. II, we define the disease and awareness +model, develop a heterogeneous mean-field model, define ran- +dom and targeted edge removal protocols, and briefly describe +the structure of the considered networks. In Sec. III, we first +discuss a baseline simulation that uses model parameters that +are aligned with empirical data on the outbreak of SARS- +CoV-2 in early 2020. We then use this baseline simulation +as a reference to study the impact of disruptions in the in- +formation diffusion layer on three disease severity measures: +(i) final outbreak size, (ii) maximum proportion of infectious +nodes on a given day (i.e., the height of the infection peak), +and (iii) the time until the infection peak is reached. +II. +METHODS +A. +Epidemic model with information diffusion +We study the interplay between information diffusion and +epidemic dynamics in a multiplex network with two layers +[see Fig. 1(a)]. +In the first layer, individuals exchange information (e.g., +through online social media or messaging services) on the +prevalence of a certain disease in the overall population ac- +cording to the unaware-aware-unaware (UAU) model.2 Indi- +viduals in the “information layer” (IL) can be in two states. +They are either unaware (U) or aware (A) of the disease and +do not necessarily have to be in close proximity (in terms of +connectivity) to exchange information. Unaware nodes can +become aware in two ways. First, if an unaware node is in +contact with an aware node, it becomes aware at rate λ. Sec- +ond, nodes that have been infected and experience symptoms +become aware at rate κ. Given that certain individuals forget +or do not adhere to intervention measures after a certain time, +we also account for transitions from aware to unaware at rate +δ. A schematic of UAU dynamics is shown in Fig. 1(b). +In the second layer, we model an epidemic outbreak +using the susceptible-exposed-infected-recovered-deceased +(SEIRD) model. In the “epidemic layer” (EL), nodes can be in +states S (susceptible), E (exposed), I (infected), R (recovered), +and D (deceased). We distinguish between two infection rates, +β u and β a, that describe the rates at which susceptible nodes +become infected if they are unaware and aware, respectively. +The disease transmission rate associated with aware individu- +als is assumed to be strictly lower than the disease transmis- +sion rate associated with unaware individuals (i.e., β a < β u), +accounting for the decreased likelihood of an aware individ- +ual to become infected. We assume a latent rate σ, resolu- +tion rate γ, and infection fatality ratio f that are independent +of the awareness status. This assumption is valid for infec- +tious diseases for which no medication is available that pos- +itively affects recovery, even if a person is aware of an in- +fection before developing symptoms. For example, during the +early outbreak stages of SARS-CoV-2, there was very little in- +formation available on how to medically support patients that +were aware of their infection, but did not show symptoms yet. +Non-pharmaceutical interventions such as contact restrictions, +mask mandates, and quarantine are often the only possibility +to combat novel pathogens.1 +According to the described UAU and SEIRD dynamics, +nodes can be in the following states: (U,S), (A,S), (U,E), +(A,E), (U,I), (A,I), (U,R), (A,R), and (U,D). The first en- +try in each tuple describes the awareness state (either U or A) +while the second entry describes vital and disease states (S, E, +I, R, and D). Deceased nodes are not aware. +B. +Heterogeneous mean-field theory +In accordance with Ref. 43, we formulate a heterogeneous +mean-field theory of SEIRD-UAU dynamics. We use xjyk ≡ +xjyk(t) (x ∈ {u,a},y ∈ {s,e,i,r,d}) to denote the proportion of +nodes in state XjYk (X ∈ {U,A},Y ∈ {S,E,I,R,D}) with de- +grees j and k in the IL and EL at time t, respectively. For ex- +ample, ujsk ≡ u jsk(t) denotes the proportion of unaware and +susceptible nodes with degrees j and k in the IL and EL at time +t, respectively. Henceforth, we will not explicitly include the +time dependence in the notation xjyk for the sake of notational +brevity. +The proportions of susceptible, exposed, infected, recov- +ered, and deceased nodes are +sk = +J +∑ +j=1 +(ujsk +ajsk), +(1) +ek = +J +∑ +j=1 +(ujek +ajek), +(2) +ik = +J +∑ +j=1 +(u jik +ajik), +(3) +rk = +J +∑ +j=1 +(u jrk +ajrk), +(4) +dk = +J +∑ +j=1 +ujdk , +(5) +where J is the maximum (or cut-off) degree in the IL. Simi- +larly, we find that the proportions of unaware and aware nodes + +Impact of random and targeted disruptions on information diffusion during outbreaks +3 +FIG. 1. Model schematic. (a) Information layer and epidemic layer. Nodes in the information layer are either unaware (U) or aware (A) while +nodes in the epidemic layer can be in one of five different states: susceptible (S), exposed (E), infected (I), recovered (R), and deceased (D). +Edge removal that is caused by disruptions in the information layer is indicated by the scissor symbol. (b) Unaware nodes become aware at rate +λ if they are adjacent to an aware node. If unaware nodes are infected, they can also become aware at rate κ. Aware nodes transition back to +an unaware state at rate δ. (c) Infectious nodes transmit a disease to unaware and aware susceptible nodes at rates β u and β a, respectively. To +account for a reduction in infectiousness risk of aware nodes, we assume the value of the disease transmission rate β u associated with unaware +nodes is strictly larger than the value of the disease transmission rate β a associated with aware nodes (β u > β a). Once susceptible nodes have +been infected, they enter an exposed state and become infectious at rate σ. The characteristic time scale σ−1 corresponds to the latency period +of the disease. Infected nodes either die or recover at rates fγ and (1− f)γ, respectively. +are +uj = +K +∑ +k=1 +(ujsk +ujek +u jik +ujrk +dk), +(6) +aj = +K +∑ +k=1 +(ajsk +ajek +a jik +ajrk), +(7) +where K is the maximum (or cut-off) degree in the EL. These +quantities satisfy the normalization conditions +K +∑ +k=1 +(sk +ek +ik +rk +dk) = 1, +(8) +J +∑ +j=1 +(uj +aj) = 1. +(9) +Assuming an uncorrelated network44, the rate equations of +the heterogeneous mean-field model are +dujsk +dt +=−λ ju jsk +⟨˜k⟩ ∑ +j′ +j′aj′ −β u ku jsk +⟨k⟩ ∑ +k′ +k′ik′ +δajsk , (10) +dajsk +dt +=λ ju jsk +⟨˜k⟩ ∑ +j′ +j′aj′ −β a ka jsk +⟨k⟩ ∑ +k′ +k′ik′ −δajsk , +(11) +dujek +dt +=−λ ju jek +⟨˜k⟩ ∑ +j′ +j′aj′ +β u ku jsk +⟨k⟩ ∑ +k′ +k′ik′ +(12) +−σujek +δajek +and +dajek +dt +=λ jujek +⟨˜k⟩ ∑ +j′ +j′aj′ +β a kajsk +⟨k⟩ ∑ +k′ +k′ik′ +(13) +−σajek −δa jek , +dujik +dt +=−λ ju jik +⟨˜k⟩ ∑ +j′ +j′aj′ +σu jek −γujik +(14) +−κu jik +δajik , +da jik +dt +=λ jujik +⟨˜k⟩ ∑ +j′ +j′aj′ +σajek −γajik +(15) ++κu jik −δajik , +dujrk +dt +=−λ ju jrk +⟨˜k⟩ ∑ +j′ +j′aj′ +(1− f)γujik +δa jrk , +(16) +dajrk +dt +=λ jujrk +⟨˜k⟩ ∑ +j′ +j′aj′ +(1− f)γajik −δajrk , +(17) +dujdk +dt +=fγ(uj +aj)ik , +(18) +where ⟨k⟩ and ⟨˜k⟩ denote the mean degrees of the EL and IL, +respectively. +C. +Networks +In our numerical experiments, we use a Barabási–Albert +(BA) network45 to model the information layer of the two- +layer structure underlying SEIRD-UAU dynamics. +Such +networks exhibit scale-free degree distributions p(k) ∝ k−γ + +(a) +(b) +Information layer +aware neighbor +already infected +C +E +Epidemic layerImpact of random and targeted disruptions on information diffusion during outbreaks +4 +(a) +(b) +FIG. 2. Multiplex networks. Information layer (top layer) with BA structure and and epidemic layer (bottom layer) with GIRG structure +determined by exponents α = 2, τ = 2.5 (a) and α = 2, τ = 3.5 (b). In the BA network, each new node has m = 2 edges that connect it to +existing nodes using linear preferential attachment. We use blue and orange edges in the epidemic layer to indicate short-range and long-range +connections, respectively. An edge connecting two nodes i, j is considered a short-range connection if the corresponding positions xi,x j satisfy +∥xi − xj∥ < 7. Otherwise, it is considered a long-range connection. The numbers of nodes in panels (a) and (b) are N = 921 and N = 973, +respectively. +(γ > 0), and are often found in social and technological +systems.46–50 Note that other distributions such as log-normal +distributions may also provide good descriptions of empirical +degree distributions in seemingly scale-free networks.51 In the +epidemic layer, we use a geometric inhomogeneous random +graph (GIRG)52, a spatial network that has found applications +in representing spatially embedded metapopulation structures +in COVID-19 models.53 +1. +Barabási–Albert network +Barabási–Albert networks45 are constructed using a prefer- +ential attachment procedure in which new nodes that are itera- +tively added to an existing network have a higher likelihood of +being attached to nodes that have higher numbers of connec- +tions. A mean field analysis of the BA model and correspond- +ing numerical results show that the exponent of the power-law +degree distribution is γ ≈ 3.54 +To construct the BA network that we will use in our sim- +ulations, we start with a star graph with one root node and +two leave nodes and iteratively add new nodes until we reach +N nodes. Each new node has m = 2 edges that connect it to +existing nodes using linear preferential attachment. A visual- +ization of such a BA information layer network with N ≈ 103 +is given in the top row of Fig. 2. In our simulations, we use +a BA network with a larger node number of N ≈ 104 that is +constructed in the same way as the ILs in Fig. 2. +2. +Geometric inhomogeneous random graph +The GIRG model52,55 produces a spatially embedded scale- +free random network. In this model, N points are first se- +lected uniformly at random in the n-dimensional hypercube +Kn = [0,1]n. +We denote the randomly selected point po- +sitions by xi ∈ Kn (1 ≤ i ≤ N) and assign each of them a +weight wi whose value is drawn from a power-law distribution +˜p(w) = (τ −2)w−τ (w ≥ 1,τ ≥ 2).52,55 Note that the distribu- +tion ˜p(w) is normalized such that its mean value is equal to +1. Pairs of nodes i, j with positions xi,xj are adjacent with +probability +Πij = 1−exp +� +− +� +wi w j +∥xi −xj∥n +�α� +, +(19) +where ∥xi − xj∥ denotes the Euclidean distance between +points xi and x j. The resulting degrees ki (1 ≤ i ≤ N) are +also distributed according to a power law with exponent τ. +According to Eq. 19, the exponent α tunes the distance and +weight dependence of Πij. For α = 0, the probability that +two nodes i, j are adjacent is independent of their distance +|xi − xj|. That is, Πij = 1 − e−1 for all i, j. By increasing α, +the distance-dependence of Πij strongly influences the struc- +ture of the network so that only nearby nodes are likely to be +adjacent. The bottom row of Fig. 2 shows GIRGs for various +parameters. +For small exponents τ ≥ 2, the number of nodes with large +weight values increases. According to Eq. 19, nodes with +large weights are more likely to be connected than nodes with + +Impact of random and targeted disruptions on information diffusion during outbreaks +5 +small weights. The abundance of these large-weight nodes, +which are the hubs of the underlying scale-free network, im- +pacts the global structure of GIRG. By decreasing τ, many +long-range connections are added to the GIRG. In the bottom +row of Fig. 2, we observe that smaller values of τ are associ- +ated with a larger proportion of long-range connections. +D. +Edge removal +To model disruptions in the IL, we consider two different +edge-removal protocols: (i) random edge removal and (ii) tar- +geted edge removal. In both protocols, we select ˜N ≤ N nodes +and denote the proportion of selected nodes by q = ˜N/N. For +each selected node, we remove each of its edges with proba- +bility p. Values of p,q > 0 correspond to disruptions in the +IL that slow down the information spread. For p = q = 1, +there are no awareness dynamics and the epidemic progresses +without interference from the information layer. +In random edge removal, ˜N nodes are selected uniformly +at random while we select ˜N hub nodes (i.e., nodes with the +largest degrees) in targeted edge removal. Such random and +targeted disruptions have been studied to provide insight into +the ability of different types of networks to withstand errors +and intentional attacks.65 It has been shown that structural fea- +tures of scale-free networks such as the size of the largest con- +nected component are very sensitive to intentional attacks (or +sabotage).66,67 +We next explore how variations in p,q ∈ [0,1] impact the +total proportion of infections i∗ = 1 − s∗, peak infection (i.e., +the maximum proportion of the population that was infected +on any day), and the time between the beginning of the out- +break until peak infection is reached. +III. +RESULTS +First consider a baseline case of SEIRD-UAU dynamics +without edge removal (i.e., pq = 0) in two different multi- +plex networks. Both multiplex networks are connected and +have the same BA information layer (see Sec. II C 1). In the +epidemic layer, we set τ = 3.5 and τ = 2.5 to model contact +networks with different proportions of long-range connections +(see Fig. 2). In the remainder of this work, we will refer to the +networks with τ = 2.5 and τ = 3.5 as long-range and short- +range networks, respectively. In both networks, we set α = 2 +[see Eq. (19)]. All stochastic simulations are implemented us- +ing Gillespie’s algorithm.68–70 +A. +Baseline +We have chosen the model parameters that we use in the +baseline simulation in accordance with empirical data on the +outbreak of SARS-CoV-2 in the beginning of 2020. For ex- +ample, for the two multiplex networks that we use in our sim- +ulations, we have set the infection rate of unaware nodes to +β u = 0.17,0.6 day−1 to obtain a basic reproduction number R0 +FIG. 3. +Stochastic simulation of baseline scenario without +information-layer disruption (i.e., pq = 0). (a,b) Proportions of sus- +ceptible (s(t)), exposed (e(t)), infected (i(t)), recovered (r(t)), and +deceased (d(t)) nodes at time t. The exponent τ in the epidemic +layer in panels (a,c) and (b,d) is set to 3.5 (short range) and 2.5 +(long range), respectively. The corresponding numbers of nodes are +N = 10049 and N = 10025. Solid colored lines represent mean val- +ues that are based on 10 i.i.d. realizations (thin grey lines). +FIG. 4. Heterogeneous mean-field solution of baseline scenario with- +out information-layer disruption (i.e., pq = 0). (a,b) Proportions of +susceptible (s(t)), exposed (e(t)), infected (i(t)), recovered (r(t)), +and deceased (d(t)) nodes at time t. The exponent τ in the epi- +demic layer in panels (a,c) and (b,d) is set to 3.5 (short range) and +2.5 (long range), respectively. The corresponding numbers of nodes +are N = 10049 and N = 10025. + +1.0 +a +(b) +0.8 +0.6 +0.4 +0.2 +0.0 +0 +50 +100 +150 +0 +50 +100 +150 +1.0 +(d) +(c) +0.8 +u(t) +Proportion +a(t) +0.6 +s(t) +(+)a +0.4 +i(t) +r(t) +30 × d(t) +0.2 +0.0 +0.0 +2.5 +5.0 +7.5 +10.0 +0.0 +2.5 +5.0 +7.5 +10.0 +Time [days] +Time [days]1.0 +b +0.8 +Proportion +0.6 +0.4 +0.2 +0.0 +0 +50 +100 +150 +0 +50 +100 +150 +1.0 +(d) +0.8 +u(t) +Proportion +a(t) +0.6 +s(t) +e(t) +0.4 +i(t) +r(t) +30 × d(t) +0.2 +0.0 +0.0 +2.5 +5.0 +7.5 +10.0 +0.0 +2.5 +5.0 +7.5 +10.0 +Time [days] +Time [days]Impact of random and targeted disruptions on information diffusion during outbreaks +6 +Parameter +Symbol +Value +Units +Comments/references +Infection rate (unaware) +β u +0.17,0.6 +day−1 +inferred from R0 ≈ 2−4 for a given γ56,57 +Infection rate (aware) +β a +0.2β u +day−1 +58 +Latent rate +σ +1/5 +day−1 +59 +Resolution rate +γ +1/14 +day−1 +60,61 +Infection fatality ratio +f +1% +... +62,63 +Awareness rate (infected) +κ +1 +day−1 +64 +Base awareness rate +λ +0.5κ +day−1 +64 +Unawareness rate +δ +1/30 +day−1 +64 +TABLE I. Overview of model parameters. We use infection rates β u = 0.17 day−1 and β u = 0.6 day−1 for GIRG networks with τ = 2.5 (long +range) and τ = 3.5 (short range), respectively. +of about 2−4.56,57 Given a latency period of about 5 days59, +we set the latent rate to σ = 1/5 day−1. The resolution rate is +set to γ = 1/14 day−1, and we use an infection fatality ratio f +of 1%.60–63 Other model parameters that are associated with +UAU dynamics are as in Ref. 64. We provide an overview of +all parameters and corresponding references in Tab. I. +Figure 3 shows the stochastic evolution of the proportions +of susceptible s(t), exposed e(t), infected i(t), recovered r(t), +and deceased d(t) nodes in the EL and of unaware u(t) and +aware a(t) nodes in the IL. Initially, 10 nodes are infectious +and 1 node is aware. For networks of about N = 10000 nodes +that are used in our stochastic simulations, these initial condi- +tions correspond to i(0) ≈ 10−3 and a(0) ≈ 10−4. The simu- +lation results shown in Figs. 3(a,c) and Figs. 3(b,d) are based +on short-range (τ = 3.5) and long-range (τ = 2.5) GIRGs, re- +spectively. The evolution of the UAU dynamics in the IL is +very similar for both GIRGs. However, structural differences +between the ELs directly impact the evolution of SEIRD dy- +namics. The infected fraction peaks at ∼ 0.17 after about 38 +days in the long-range EL but peaks at ∼ 0.21 at about 51 days +in the short-range EL. Figure 3 also shows that the final epi- +demic size 1−s(t → ∞) in both networks differs significantly. +To understand what causes the different outbreak character- +istics in both networks, we examined the degree distribution +of susceptible nodes at T = 150: there are substantially more +susceptible low-degree nodes in the long-range GIRG where +τ = 2.5 compared to the short-range GIRG with τ = 3.5. Al- +though, there are more hub nodes with large degree in the +long-range GIRG, the proportion of low-degree nodes is also +larger. Hence, there are more low-degree nodes in the long- +range GIRG that are less exposed to the outbreak dynamics. +To complement the stochastic simulation results, we nu- +merically solve the heterogeneous mean-field model (10)-(18) +for the same networks and model parameters (see Tab. I). +We set the degree cut-offs to J = 210, K = 400 (τ = 2.5) +and J = 210, K = 164 (τ = 3.5). In the multiplex network +with short-range IL with τ = 3.5, the degree cut-offs corre- +spond to the maximum degrees. In the long-range EL where +τ = 2.5, the maximum degree is 856, and to keep the solu- +tion of the mean-field model computationally feasible we set +the cut-off K = 400. Initially, we set ajik(0) = pj ˜pka(0)/2, +ajsk(0) = pj ˜pka(0)/2, ujsk(0) = pj ˜pk(1 − i(0) − a(0)/2), +ujik(0) = pj ˜pk(i(0) − a(0)/2), where pj and ˜pk denote the +degree distributions in the IL and EL, respectively. Both de- +gree distributions are normalized according to ∑J +j=1 pj = 1 +and ∑K +k=1 ˜pk = 1. +Note that these initial conditions satisfy +s(0) = ∑ +j,k +� +u jsk(0)+ajsk(0) +� +(20) += ∑ +j,k +pj ˜pk +� +1−i(0) +� += 1−i(0), +(21) +i(0) = ∑ +j,k +� +u jik(0)+ajik(0) +� += ∑ +j,k +pj ˜pki(0), +(22) +u(0) = ∑ +j,k +� +u jsk(0)+ujik(0) +� +(23) += ∑ +j,k +pj ˜pk +� +1−a(0) +� += 1−a(0), +(24) +a(0) = ∑ +j,k +� +a jsk(0)+ajik(0) +� += ∑ +j,k +pj ˜pka(0). +(25) +In accordance with the initial conditions that we used in the +stochastic simulations, we set i(0) = 10−3 and a(0) = 10−4. +Figure 4 shows the corresponding numerical results. Compar- +ing Figs. 3 to 4, we observe that the heterogeneous mean-field +model captures characteristic features that arise in the evolu- +tion of stochastic SEIRD-UAU dynamics. Examples of such +features include (i) the rapid spread of awareness in the IL +and (ii) differences between both ELs in the final epidemic +size 1 − s(t → ∞). In the heterogeneous mean-field model +(10)-(18), we account only for differences in node degree and +neglect other structural features of the considered multiplex +networks. Subpopulations interact in a well-mixed manner +and susceptible nodes of the same degree have the same risk +of being infected at any given time. As a consequence of these +approximations, the mean-field model overestimates both the +number of new infections and final outbreak size compared to +the stochastic simulation results in Fig. 3. + +Impact of random and targeted disruptions on information diffusion during outbreaks +7 +FIG. 5. Random edge removal. The impact of random edge removal in the IL on disease dynamics in the EL. Epidemic size 1 − s(t → ∞) +(left column), peak infection (middle column), and time to peak infection (right column) as a function of the proportion of selected nodes q +and the corresponding edge removal probability p. The exponent τ in the ELs in top row and bottom row is set to 3.5 (short range) and 2.5 +(long range), respectively. The corresponding numbers of nodes are N = 10049 and N = 10025. Simulation results are based on 230 i.i.d. +realizations. +FIG. 6. Targeted edge removal. The impact of random edge removal in the IL on disease dynamics in the EL. Epidemic size 1 − s(t → ∞) +(left panel), peak infection (middle panel), and time to peak infection (right panel) as a function of the proportion of selected nodes q and the +corresponding edge removal probability p. The exponent τ in the ELs in top row and bottom row is set to 3.5 (short range) and 2.5 (long range), +respectively. The corresponding numbers of nodes are N = 10049 and N = 10025. Simulation results are based on 230 i.i.d. realizations. + +(a) Epidemic size (T = 3.5) +(b) Peak infection (T = 3.5) +(c) Time to peak infection (↑ = 3.5) +1.00 +0.40 +50 +0.98 +0.35 +0.96 +0.30 +40 +0.94 +0.25 +0.92 +0.20 +30 +0.00 +0.00 +0.00 +.1.00 +0.25 +1.00 +0.25 +1.00 +0.25 +0.75 +0.75 +0.75 +0.50 +0.50 +0.50 +0.50 +0.50 +0.50 +0.75 +0.75 +0.75 +0.25 +0.25 +0.25 +p +p +p +1.00 +1.00 +1.00 +0.00 +0.00 +q +0.00 +q +q +(d) Epidemic size (T = 2.5) +(e) Peak infection (T = 2.5) +(f) Time to peak infection (T = 2.5) +1.0 +0.4 +40 +0.9 +35 +0.3 +0.8 +30 +0.7 +0.2 +0.6 +25 +0.00 +0.00 +0.00 +0.25 +1.00 +0.25 +1.00 +0.25 +1.00 +0.75 +0.75 +0.75 +0.50 +0.50 +0.50 +0.50 +0.50 +0.50 +0.75 +0.75 +0.75 +0.25 +0.25 +0.25 +p +p +p +1.00 +0.00 +1.00 +0.00 +1.00 +0.00 +q +q +q(a) Epidemic size (π = 3.5) +(b) Peak infection (T = 3.5) +(c) Time to peak infection (↑ = 3.5) +1.00 +0.40 +50 +0.98 +0.35 +40 +0.96 +0.30 +0.94 +0.25 +30 +0.92 +0.20 +0.00 +0.00 +0.00 +0.25 +1.00 +0.25 +1.00 +0.00 +0.25 +0.75 +0.75 +0.25 +0.50 +0.50 +0.50 +0.50 +0.50 +0.50 +0.75 +0.75 +0.75 +0.25 +0.25 +0.75 +p +p +p +1.00 +1.00 +1.00 +0.00 +0.00 +q +q +q +1.00 +(d) Epidemic size (T = 2.5) +(e) Peak infection (T = 2.5) +(f) Time to peak infection (T = 2.5) +40 +1.0 +0.4 +35 +0.9 +0.3 +0.8 +30 +0.7 +0.2 +25 +0.6 +0.00 +0.00 +0.00 +0.25 +1.00 +0.25 +1.00 +0.00 +0.25 +0.75 +0.75 +0.25 +0.50 +0.50 +0.50 +0.50 +0.50 +0.50 +0.75 +0.75 +0.75 +0.25 +0.25 +0.75 +p +p +p +1.00 +0.00 +1.00 +0.00 +1.00 +q +q +q +1.00Impact of random and targeted disruptions on information diffusion during outbreaks +8 +B. +Impact of edge removal +We now study the impact of random and targeted edge re- +moval in the IL (see Sec. II D) on SEIRD dynamics in terms +of three disease severity measures: (i) final epidemic size, (ii) +peak infection, and (iii) time to peak infection. +1. +Random edge removal +In random edge removal, we first select a proportion of q = +˜N/N nodes in the IL uniformly at random. For each of the +selected nodes, each of its edges are removed with probability +p. +Figure 5(a,d) shows the epidemic size as a function of p,q +for both short-range and long-range GIRGs. The epidemic +size increases with p and q because larger values of p,q are +associated with fewer edges in the IL, leading to a smaller pro- +portion of aware nodes. Hence, the proportion of nodes with +a reduced infection rate β u also decreases. For the long-range +GIRG (τ = 2.5), the final epidemic size undergoes a transition +from about 0.6 for p,q ≈ 0 to about 0.9 for p,q ≈ 1. Because +the final epidemic size in the short-range GIRG (τ = 3.5) is +already about 0.9, random edge removal has relatively little +impact on this quantity. +As with the impact on final epidemic size 1−s(t → ∞), ran- +dom edge removal generates a similar-looking p,q-dependent +infection peak, as shown in Fig. 5(b,e). The time to reach peak +infection decreases with p,q since higher p,q are associated +with smaller proportions of aware nodes. Thus, the proportion +of nodes with a reduced infection rate β u also decreases, and +the epidemic spreads faster through the network. +2. +Targeted edge removal +For targeted edge removal where the ˜N selected nodes cor- +respond to the hubs (i.e., largest-degree nodes) of the IL, we +find that the overall dependence of epidemic size, peak infec- +tion, and time to peak infection on p,q is qualitatively simi- +lar to random edge removal (see Fig. 6). As in random edge +removal, the impact of targeted edge removal on the final epi- +demic size is smaller for the short-range GIRG compared to +the long-range one. A key difference in targeted edge removal +is that all studied quantities are more sensitive to variations in +q, the proportion of selected hub nodes. For example, the tran- +sition of the epidemic size for p = 1 as a function of q in tar- +geted edge removal [see Fig. 6(a,d)] is steeper than the corre- +sponding transition in random edge removal [see Fig. 5(a,d)]. +Targeted edge removal selects nodes based on their degree +rather than uniformly, and leads to more significant changes +in epidemic size, peak infection, and time to peak infection +as p ≥ 0.5. These findings are in accordance with previous +work that showed that scale-free networks break down more +easily under intentional attacks than under uniform random +failure.67 Our work provide insights into how such disruptions +in information diffusion translate into differences in disease +severity measures. +IV. +DISCUSSION +In this work, we studied the impact of disruptions in com- +munication networks on information diffusion and subse- +quently disease outcome during an outbreak. To do so, we +constructed a multiplex network that consists of two lay- +ers. The first layer, called information layer (IL), is used to +model communication between individuals (e.g., online in- +formation exchange via a social media platform). The sec- +ond layer, called epidemic layer (EL), is used to represent a +spatially embedded human contact network in which infec- +tious individuals can transmit a disease to susceptible indi- +viduals. We use this multiplex network to simulate coevolv- +ing unaware-aware-unaware (UAU) and susceptible-exposed- +infected-recovered-deceased (SEIRD) dynamics. The model +parameters that we use in our simulations have been selected +in accordance with empirical data on the early outbreak stages +of SARS-CoV-2 in the beginning of 2020. +We studied two different epidemic layers with different pro- +portions of long-range connections, representing human con- +tact networks with different contact characteristics. To illus- +trate the impact of disruptions in the IL on the evolution of an +outbreak, we utilized two different edge removal protocols: (i) +random edge removal and (ii) targeted edge removal. In both +protocols, we select a proportion q of nodes and then remove +corresponding edges with probability p. In random edge re- +moval, we select nodes in the IL uniformly at random while +we select nodes with the largest degree (i.e., hub nodes) in +targeted edge removal. Although edge removal may render +the IL disconnected, the EL is always connected in our sim- +ulations such that all nodes in the EL can potentially become +infected. Previous work has shown that scale-free networks +such as the IL in our multiplex network are more robust to +random than targeted disruptions.65–67 The reason for this ef- +fect is that by removing hub nodes of a scale-free network, a +large number of all edges in the network is being removed, +strongly impacting the connectivity properties of such a net- +work. We observe that targeted edge removal can abruptly +change outbreak characteristics such as time to peak infection, +even for small proportions of selected nodes. Our results ex- +tend those presented in previous work on random and targeted +disruptions65–67 by establishing a connection to coevolving in- +formation and epidemic diffusion. +DATA AVAILABILITY +Our +source +codes +are +publicly +available +at +https://gitlab.com/ComputationalScience/ +information-epidemic. +1T. Schneider, O. R. Dunbar, J. Wu, L. Böttcher, D. Burov, A. Garbuno- +Inigo, G. L. Wagner, S. Pei, C. Daraio, R. Ferrari, et al., “Epidemic manage- +ment and control through risk-dependent individual contact interventions,” +PLOS Computational Biology 18, e1010171 (2022). +2C. Granell, S. Gómez, and A. Arenas, “Dynamical interplay between aware- +ness and epidemic spreading in multiplex networks,” Physical Review Let- +ters 111, 128701 (2013). +3J. P. Gleeson, “Binary-state dynamics on complex networks: Pair approxi- +mation and beyond,” Physical Review X 3, 021004 (2013). + +Impact of random and targeted disruptions on information diffusion during outbreaks +9 +4R. 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Herrmann, Computational Statistical Physics (Cam- +bridge University Press, Cambridge, UK, 2021). + diff --git a/49AyT4oBgHgl3EQf2PmT/content/tmp_files/load_file.txt b/49AyT4oBgHgl3EQf2PmT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d557ba4d925fbd9122ce1a3bd51a270e9ca1ae70 --- /dev/null +++ b/49AyT4oBgHgl3EQf2PmT/content/tmp_files/load_file.txt @@ -0,0 +1,995 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf,len=994 +page_content='Impact of random and targeted disruptions on information diffusion during outbreaks Impact of random and targeted disruptions on information diffusion during outbreaks Hosein Masoomy,1, a) Tom Chou,2, b) and Lucas Böttcher3, c) 1)Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' of Physics, Shahid Beheshti University, 1983969411, Tehran, Iran 2)Depts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' of Computational Medicine and Mathematics, UCLA, Los Angeles, CA 90095 3)Centre for Human and Machine Intelligence, Frankfurt School of Finance and Management, 60322 Frankfurt am Main, Germany (Dated: 3 January 2023) Outbreaks are complex multi-scale processes that are impacted not only by cellular dynamics and the ability of pathogens to effectively reproduce and spread, but also by population-level dynamics and the effectiveness of miti- gation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' A timely exchange of information related to the spread of novel pathogens, stay-at-home orders, and other containment measures can be effective at containing an infectious disease, particularly during in the early stages when testing infrastructure, vaccines, and other medical interventions may not be available at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Using a multiplex epidemic model that consists of an information layer (modeling information exchange between individuals) and a spa- tially embedded epidemic layer (representing a human contact network), we study how random and targeted disruptions in the information layer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=', errors and intentional attacks on communication infrastructure) impact outbreak dynam- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' We calibrate our model to the early outbreak stages of the SARS-CoV-2 pandemic in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Mitigation campaign can still be effective under random disruptions, such as failure of information channels between a few individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' However, targeted disruptions or sabotage of hub nodes that exchange information with a large number of individuals can abruptly change outbreak characteristics such as the time to reach the peak infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Our results emphasize the importance of using a robust communication infrastructure that can withstand both random and targeted disruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Online communication platforms and exposure notifica- tion apps can help slow down and contain the spread of an infectious disease1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Individuals who have been made aware of an outbreak are likely to adapt their behavior to reduce their risk of being infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' To study the interplay between infectious disease outbreaks and corresponding changes in individual contact behaviors, Granell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='2 in- troduced an epidemic model that accounts for the spread of awareness through an information layer that is coupled to a human contact network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Building upon their model of awareness diffusion, our work studies the impact of ran- dom and targeted disruptions in the information layer on the overall outbreak dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' INTRODUCTION The study of epidemic processes in networks has pro- vided many insights into the interplay between structure and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='3,4 The aim of many works in this area has been to analyze the impact of different structural features such as clustering5, community structure6,7, hub nodes, and scale- free degree distributions8 on the evolution of susceptible- infected-susceptible (SIS) and susceptible-infected-recovered (SIR) models and their extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='9–11 Connections between epidemic processes and percolation contributed to the devel- opment of analytical methods that are useful to analyze epi- demic transitions and determine outbreak size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='12–16 Along a)hoseingmasoomy@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='com b)tomchou@ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='edu c)l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='boettcher@fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='de with progress in understanding epidemic processes in static single-layer networks, developments in the study of tempo- ral networks17, multilayer networks18,19, and other structures describing higher-order interactions20–23 have allowed for the integration of time-varying and non-binary interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Before research turned to epidemic models in multilayer networks, interactions between disease and behavioral dy- namics have been studied mainly in single-layer networks24 and well-mixed populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='25–28 In an extension of the classical SIS model, the so-called susceptible-infected-alert- susceptible (SIAS) model, a new compartment was used to study the effect of “alert” individuals that are surrounded by a certain number of infecteds on disease dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='29,30 The SIAS model has been implemented using a two-layer net- work31 with a contact layer and an information-dissemination layer to find optimal information dissemination strategies that help contain an outbreak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The interplay between behavioral effects and network dy- namics has also been analyzed in terms of a multiplex struc- ture where information on an outbreak diffuses in an infor- mation layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='2,32 In a multiplex network, all of the interlayer edges are edges between nodes and their counterparts in other layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' As in the SIAS model, individuals in the information layer can be either aware or unaware of a disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Aware- ness then translates into a reduced infection rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The origi- nal awareness model has been modified in various ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' One study used a threshold model in the information layer and identified awareness cascades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='33 Other research investigated the effects of dynamically varying transmission rates34, cou- pled SIR and unaware-aware-unaware (UAU) dynamics with and without latency35,36, SIS and UAU dynamics that propa- gate at different speeds37, and higher-order interactions38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' For a detailed overview of models of coevolving spreading pro- cesses in networks, we refer the reader to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='00748v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='soc-ph] 2 Jan 2023 Impact of random and targeted disruptions on information diffusion during outbreaks 2 In this work, we study coevolving susceptible-exposed- infected-recovered-deceased (SEIRD) and UAU dynamics on a multiplex network that consists of an epidemic layer and an information layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The exposed compartment in our model accounts for latency (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=', the time difference between infec- tion and becoming infectious).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Different variants of SEIRD models have been used to mechanistically describe the spread of an infectious disease for which the latency period between time of infection to time of becoming infectious cannot be neglected9,40–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Examples of such infectious diseases include measles, smallpox, and SARS-CoV-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' One of the main goals of this work is to provide insight into the impact of disruptions in the information diffusion layer on the overall outbreak dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' We therefore study differ- ent edge removal protocols that describe random and targeted disruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' II, we define the disease and awareness model, develop a heterogeneous mean-field model, define ran- dom and targeted edge removal protocols, and briefly describe the structure of the considered networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' III, we first discuss a baseline simulation that uses model parameters that are aligned with empirical data on the outbreak of SARS- CoV-2 in early 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' We then use this baseline simulation as a reference to study the impact of disruptions in the in- formation diffusion layer on three disease severity measures: (i) final outbreak size, (ii) maximum proportion of infectious nodes on a given day (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=', the height of the infection peak), and (iii) the time until the infection peak is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Epidemic model with information diffusion We study the interplay between information diffusion and epidemic dynamics in a multiplex network with two layers [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' In the first layer, individuals exchange information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=', through online social media or messaging services) on the prevalence of a certain disease in the overall population ac- cording to the unaware-aware-unaware (UAU) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='2 Indi- viduals in the “information layer” (IL) can be in two states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' They are either unaware (U) or aware (A) of the disease and do not necessarily have to be in close proximity (in terms of connectivity) to exchange information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Unaware nodes can become aware in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' First, if an unaware node is in contact with an aware node, it becomes aware at rate λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Sec- ond, nodes that have been infected and experience symptoms become aware at rate κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Given that certain individuals forget or do not adhere to intervention measures after a certain time, we also account for transitions from aware to unaware at rate δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' A schematic of UAU dynamics is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' In the second layer, we model an epidemic outbreak using the susceptible-exposed-infected-recovered-deceased (SEIRD) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' In the “epidemic layer” (EL), nodes can be in states S (susceptible), E (exposed), I (infected), R (recovered), and D (deceased).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' We distinguish between two infection rates, β u and β a, that describe the rates at which susceptible nodes become infected if they are unaware and aware, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The disease transmission rate associated with aware individu- als is assumed to be strictly lower than the disease transmis- sion rate associated with unaware individuals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=', β a < β u), accounting for the decreased likelihood of an aware individ- ual to become infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' We assume a latent rate σ, resolu- tion rate γ, and infection fatality ratio f that are independent of the awareness status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' This assumption is valid for infec- tious diseases for which no medication is available that pos- itively affects recovery, even if a person is aware of an in- fection before developing symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' For example, during the early outbreak stages of SARS-CoV-2, there was very little in- formation available on how to medically support patients that were aware of their infection, but did not show symptoms yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Non-pharmaceutical interventions such as contact restrictions, mask mandates, and quarantine are often the only possibility to combat novel pathogens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='1 According to the described UAU and SEIRD dynamics, nodes can be in the following states: (U,S), (A,S), (U,E), (A,E), (U,I), (A,I), (U,R), (A,R), and (U,D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The first en- try in each tuple describes the awareness state (either U or A) while the second entry describes vital and disease states (S, E, I, R, and D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Deceased nodes are not aware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Heterogeneous mean-field theory In accordance with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 43, we formulate a heterogeneous mean-field theory of SEIRD-UAU dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' We use xjyk ≡ xjyk(t) (x ∈ {u,a},y ∈ {s,e,i,r,d}) to denote the proportion of nodes in state XjYk (X ∈ {U,A},Y ∈ {S,E,I,R,D}) with de- grees j and k in the IL and EL at time t, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' For ex- ample, ujsk ≡ u jsk(t) denotes the proportion of unaware and susceptible nodes with degrees j and k in the IL and EL at time t, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Henceforth, we will not explicitly include the time dependence in the notation xjyk for the sake of notational brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The proportions of susceptible, exposed, infected, recov- ered, and deceased nodes are sk = J ∑ j=1 (ujsk +ajsk), (1) ek = J ∑ j=1 (ujek +ajek), (2) ik = J ∑ j=1 (u jik +ajik), (3) rk = J ∑ j=1 (u jrk +ajrk), (4) dk = J ∑ j=1 ujdk , (5) where J is the maximum (or cut-off) degree in the IL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Simi- larly, we find that the proportions of unaware and aware nodes Impact of random and targeted disruptions on information diffusion during outbreaks 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Model schematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' (a) Information layer and epidemic layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Nodes in the information layer are either unaware (U) or aware (A) while nodes in the epidemic layer can be in one of five different states: susceptible (S), exposed (E), infected (I), recovered (R), and deceased (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Edge removal that is caused by disruptions in the information layer is indicated by the scissor symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' (b) Unaware nodes become aware at rate λ if they are adjacent to an aware node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' If unaware nodes are infected, they can also become aware at rate κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Aware nodes transition back to an unaware state at rate δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' (c) Infectious nodes transmit a disease to unaware and aware susceptible nodes at rates β u and β a, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' To account for a reduction in infectiousness risk of aware nodes, we assume the value of the disease transmission rate β u associated with unaware nodes is strictly larger than the value of the disease transmission rate β a associated with aware nodes (β u > β a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Once susceptible nodes have been infected, they enter an exposed state and become infectious at rate σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The characteristic time scale σ−1 corresponds to the latency period of the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Infected nodes either die or recover at rates fγ and (1− f)γ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' are uj = K ∑ k=1 (ujsk +ujek +u jik +ujrk +dk), (6) aj = K ∑ k=1 (ajsk +ajek +a jik +ajrk), (7) where K is the maximum (or cut-off) degree in the EL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' These quantities satisfy the normalization conditions K ∑ k=1 (sk +ek +ik +rk +dk) = 1, (8) J ∑ j=1 (uj +aj) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' (9) Assuming an uncorrelated network44,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' the rate equations of the heterogeneous mean-field model are dujsk dt =−λ ju jsk ⟨˜k⟩ ∑ j′ j′aj′ −β u ku jsk ⟨k⟩ ∑ k′ k′ik′ +δajsk ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' (10) dajsk dt =λ ju jsk ⟨˜k⟩ ∑ j′ j′aj′ −β a ka jsk ⟨k⟩ ∑ k′ k′ik′ −δajsk ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' (11) dujek dt =−λ ju jek ⟨˜k⟩ ∑ j′ j′aj′ +β u ku jsk ⟨k⟩ ∑ k′ k′ik′ (12) −σujek +δajek and dajek dt =λ jujek ⟨˜k⟩ ∑ j′ j′aj′ +β a kajsk ⟨k⟩ ∑ k′ k′ik′ (13) −σajek −δa jek ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' dujik dt =−λ ju jik ⟨˜k⟩ ∑ j′ j′aj′ +σu jek −γujik (14) −κu jik +δajik ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' da jik dt =λ jujik ⟨˜k⟩ ∑ j′ j′aj′ +σajek −γajik (15) +κu jik −δajik ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' dujrk dt =−λ ju jrk ⟨˜k⟩ ∑ j′ j′aj′ +(1− f)γujik +δa jrk ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' (16) dajrk dt =λ jujrk ⟨˜k⟩ ∑ j′ j′aj′ +(1− f)γajik −δajrk ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' (17) dujdk dt =fγ(uj +aj)ik ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' (18) where ⟨k⟩ and ⟨˜k⟩ denote the mean degrees of the EL and IL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Networks In our numerical experiments, we use a Barabási–Albert (BA) network45 to model the information layer of the two- layer structure underlying SEIRD-UAU dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Such networks exhibit scale-free degree distributions p(k) ∝ k−γ (a) (b) Information layer aware neighbor already infected C E Epidemic layerImpact of random and targeted disruptions on information diffusion during outbreaks 4 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Multiplex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Information layer (top layer) with BA structure and and epidemic layer (bottom layer) with GIRG structure determined by exponents α = 2, τ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 (a) and α = 2, τ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' In the BA network, each new node has m = 2 edges that connect it to existing nodes using linear preferential attachment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' We use blue and orange edges in the epidemic layer to indicate short-range and long-range connections, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' An edge connecting two nodes i, j is considered a short-range connection if the corresponding positions xi,x j satisfy ∥xi − xj∥ < 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Otherwise, it is considered a long-range connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The numbers of nodes in panels (a) and (b) are N = 921 and N = 973, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' (γ > 0), and are often found in social and technological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='46–50 Note that other distributions such as log-normal distributions may also provide good descriptions of empirical degree distributions in seemingly scale-free networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='51 In the epidemic layer, we use a geometric inhomogeneous random graph (GIRG)52, a spatial network that has found applications in representing spatially embedded metapopulation structures in COVID-19 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Barabási–Albert network Barabási–Albert networks45 are constructed using a prefer- ential attachment procedure in which new nodes that are itera- tively added to an existing network have a higher likelihood of being attached to nodes that have higher numbers of connec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' A mean field analysis of the BA model and correspond- ing numerical results show that the exponent of the power-law degree distribution is γ ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='54 To construct the BA network that we will use in our sim- ulations, we start with a star graph with one root node and two leave nodes and iteratively add new nodes until we reach N nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Each new node has m = 2 edges that connect it to existing nodes using linear preferential attachment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' A visual- ization of such a BA information layer network with N ≈ 103 is given in the top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' In our simulations, we use a BA network with a larger node number of N ≈ 104 that is constructed in the same way as the ILs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Geometric inhomogeneous random graph The GIRG model52,55 produces a spatially embedded scale- free random network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' In this model, N points are first se- lected uniformly at random in the n-dimensional hypercube Kn = [0,1]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' We denote the randomly selected point po- sitions by xi ∈ Kn (1 ≤ i ≤ N) and assign each of them a weight wi whose value is drawn from a power-law distribution ˜p(w) = (τ −2)w−τ (w ≥ 1,τ ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='52,55 Note that the distribu- tion ˜p(w) is normalized such that its mean value is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Pairs of nodes i, j with positions xi,xj are adjacent with probability Πij = 1−exp � − � wi w j ∥xi −xj∥n �α� , (19) where ∥xi − xj∥ denotes the Euclidean distance between points xi and x j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The resulting degrees ki (1 ≤ i ≤ N) are also distributed according to a power law with exponent τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 19, the exponent α tunes the distance and weight dependence of Πij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' For α = 0, the probability that two nodes i, j are adjacent is independent of their distance |xi − xj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' That is, Πij = 1 − e−1 for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' By increasing α, the distance-dependence of Πij strongly influences the struc- ture of the network so that only nearby nodes are likely to be adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The bottom row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 2 shows GIRGs for various parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' For small exponents τ ≥ 2, the number of nodes with large weight values increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 19, nodes with large weights are more likely to be connected than nodes with Impact of random and targeted disruptions on information diffusion during outbreaks 5 small weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The abundance of these large-weight nodes, which are the hubs of the underlying scale-free network, im- pacts the global structure of GIRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' By decreasing τ, many long-range connections are added to the GIRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' In the bottom row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 2, we observe that smaller values of τ are associ- ated with a larger proportion of long-range connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Edge removal To model disruptions in the IL, we consider two different edge-removal protocols: (i) random edge removal and (ii) tar- geted edge removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' In both protocols, we select ˜N ≤ N nodes and denote the proportion of selected nodes by q = ˜N/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' For each selected node, we remove each of its edges with proba- bility p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Values of p,q > 0 correspond to disruptions in the IL that slow down the information spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' For p = q = 1, there are no awareness dynamics and the epidemic progresses without interference from the information layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' In random edge removal, ˜N nodes are selected uniformly at random while we select ˜N hub nodes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=', nodes with the largest degrees) in targeted edge removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Such random and targeted disruptions have been studied to provide insight into the ability of different types of networks to withstand errors and intentional attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='65 It has been shown that structural fea- tures of scale-free networks such as the size of the largest con- nected component are very sensitive to intentional attacks (or sabotage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='66,67 We next explore how variations in p,q ∈ [0,1] impact the total proportion of infections i∗ = 1 − s∗, peak infection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=', the maximum proportion of the population that was infected on any day), and the time between the beginning of the out- break until peak infection is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' RESULTS First consider a baseline case of SEIRD-UAU dynamics without edge removal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=', pq = 0) in two different multi- plex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Both multiplex networks are connected and have the same BA information layer (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' II C 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' In the epidemic layer, we set τ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 and τ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 to model contact networks with different proportions of long-range connections (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' In the remainder of this work, we will refer to the networks with τ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 and τ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 as long-range and short- range networks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' In both networks, we set α = 2 [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' (19)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' All stochastic simulations are implemented us- ing Gillespie’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='68–70 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Baseline We have chosen the model parameters that we use in the baseline simulation in accordance with empirical data on the outbreak of SARS-CoV-2 in the beginning of 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' For ex- ample, for the two multiplex networks that we use in our sim- ulations, we have set the infection rate of unaware nodes to β u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='17,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='6 day−1 to obtain a basic reproduction number R0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Stochastic simulation of baseline scenario without information-layer disruption (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=', pq = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' (a,b) Proportions of sus- ceptible (s(t)), exposed (e(t)), infected (i(t)), recovered (r(t)), and deceased (d(t)) nodes at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The exponent τ in the epidemic layer in panels (a,c) and (b,d) is set to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 (short range) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 (long range), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The corresponding numbers of nodes are N = 10049 and N = 10025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Solid colored lines represent mean val- ues that are based on 10 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' realizations (thin grey lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Heterogeneous mean-field solution of baseline scenario with- out information-layer disruption (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=', pq = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' (a,b) Proportions of susceptible (s(t)), exposed (e(t)), infected (i(t)), recovered (r(t)), and deceased (d(t)) nodes at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The exponent τ in the epi- demic layer in panels (a,c) and (b,d) is set to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 (short range) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 (long range), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The corresponding numbers of nodes are N = 10049 and N = 10025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 a (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 0 50 100 150 0 50 100 150 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 (d) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='8 u(t) Proportion a(t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='6 s(t) (+)a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='4 i(t) r(t) 30 × d(t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 Time [days] Time [days]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='8 Proportion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 0 50 100 150 0 50 100 150 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='8 u(t) Proportion a(t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='6 s(t) e(t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='4 i(t) r(t) 30 × d(t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='0 Time [days] Time [days]Impact of random and targeted disruptions on information diffusion during outbreaks 6 Parameter Symbol Value Units Comments/references Infection rate (unaware) β u 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='17,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='6 day−1 inferred from R0 ≈ 2−4 for a given γ56,57 Infection rate (aware) β a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='2β u day−1 58 Latent rate σ 1/5 day−1 59 Resolution rate γ 1/14 day−1 60,61 Infection fatality ratio f 1% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 62,63 Awareness rate (infected) κ 1 day−1 64 Base awareness rate λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5κ day−1 64 Unawareness rate δ 1/30 day−1 64 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Overview of model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' We use infection rates β u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='17 day−1 and β u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='6 day−1 for GIRG networks with τ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 (long range) and τ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 (short range), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' of about 2−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='56,57 Given a latency period of about 5 days59, we set the latent rate to σ = 1/5 day−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The resolution rate is set to γ = 1/14 day−1, and we use an infection fatality ratio f of 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='60–63 Other model parameters that are associated with UAU dynamics are as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' We provide an overview of all parameters and corresponding references in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Figure 3 shows the stochastic evolution of the proportions of susceptible s(t), exposed e(t), infected i(t), recovered r(t), and deceased d(t) nodes in the EL and of unaware u(t) and aware a(t) nodes in the IL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Initially, 10 nodes are infectious and 1 node is aware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' For networks of about N = 10000 nodes that are used in our stochastic simulations, these initial condi- tions correspond to i(0) ≈ 10−3 and a(0) ≈ 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The simu- lation results shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 3(a,c) and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 3(b,d) are based on short-range (τ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5) and long-range (τ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5) GIRGs, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The evolution of the UAU dynamics in the IL is very similar for both GIRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' However, structural differences between the ELs directly impact the evolution of SEIRD dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The infected fraction peaks at ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='17 after about 38 days in the long-range EL but peaks at ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='21 at about 51 days in the short-range EL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Figure 3 also shows that the final epi- demic size 1−s(t → ∞) in both networks differs significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' To understand what causes the different outbreak character- istics in both networks, we examined the degree distribution of susceptible nodes at T = 150: there are substantially more susceptible low-degree nodes in the long-range GIRG where τ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 compared to the short-range GIRG with τ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Al- though, there are more hub nodes with large degree in the long-range GIRG, the proportion of low-degree nodes is also larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Hence, there are more low-degree nodes in the long- range GIRG that are less exposed to the outbreak dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' To complement the stochastic simulation results, we nu- merically solve the heterogeneous mean-field model (10)-(18) for the same networks and model parameters (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' We set the degree cut-offs to J = 210, K = 400 (τ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5) and J = 210, K = 164 (τ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' In the multiplex network with short-range IL with τ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5, the degree cut-offs corre- spond to the maximum degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' In the long-range EL where τ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5, the maximum degree is 856, and to keep the solu- tion of the mean-field model computationally feasible we set the cut-off K = 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Initially, we set ajik(0) = pj ˜pka(0)/2, ajsk(0) = pj ˜pka(0)/2, ujsk(0) = pj ˜pk(1 − i(0) − a(0)/2), ujik(0) = pj ˜pk(i(0) − a(0)/2), where pj and ˜pk denote the degree distributions in the IL and EL, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Both de- gree distributions are normalized according to ∑J j=1 pj = 1 and ∑K k=1 ˜pk = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Note that these initial conditions satisfy s(0) = ∑ j,k � u jsk(0)+ajsk(0) � (20) = ∑ j,k pj ˜pk � 1−i(0) � = 1−i(0), (21) i(0) = ∑ j,k � u jik(0)+ajik(0) � = ∑ j,k pj ˜pki(0), (22) u(0) = ∑ j,k � u jsk(0)+ujik(0) � (23) = ∑ j,k pj ˜pk � 1−a(0) � = 1−a(0), (24) a(0) = ∑ j,k � a jsk(0)+ajik(0) � = ∑ j,k pj ˜pka(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' (25) In accordance with the initial conditions that we used in the stochastic simulations, we set i(0) = 10−3 and a(0) = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Figure 4 shows the corresponding numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Compar- ing Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 3 to 4, we observe that the heterogeneous mean-field model captures characteristic features that arise in the evolu- tion of stochastic SEIRD-UAU dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Examples of such features include (i) the rapid spread of awareness in the IL and (ii) differences between both ELs in the final epidemic size 1 − s(t → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' In the heterogeneous mean-field model (10)-(18), we account only for differences in node degree and neglect other structural features of the considered multiplex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Subpopulations interact in a well-mixed manner and susceptible nodes of the same degree have the same risk of being infected at any given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' As a consequence of these approximations, the mean-field model overestimates both the number of new infections and final outbreak size compared to the stochastic simulation results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Impact of random and targeted disruptions on information diffusion during outbreaks 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Random edge removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The impact of random edge removal in the IL on disease dynamics in the EL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Epidemic size 1 − s(t → ∞) (left column), peak infection (middle column), and time to peak infection (right column) as a function of the proportion of selected nodes q and the corresponding edge removal probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The exponent τ in the ELs in top row and bottom row is set to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 (short range) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 (long range), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The corresponding numbers of nodes are N = 10049 and N = 10025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Simulation results are based on 230 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Targeted edge removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The impact of random edge removal in the IL on disease dynamics in the EL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Epidemic size 1 − s(t → ∞) (left panel), peak infection (middle panel), and time to peak infection (right panel) as a function of the proportion of selected nodes q and the corresponding edge removal probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The exponent τ in the ELs in top row and bottom row is set to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 (short range) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5 (long range), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The corresponding numbers of nodes are N = 10049 and N = 10025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Simulation results are based on 230 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' realizations.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='00 q q q(a) Epidemic size (π = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5) (b) Peak infection (T = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5) (c) Time to peak infection (↑ = 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='00 q q q 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='00Impact of random and targeted disruptions on information diffusion during outbreaks 8 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Impact of edge removal We now study the impact of random and targeted edge re- moval in the IL (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' II D) on SEIRD dynamics in terms of three disease severity measures: (i) final epidemic size, (ii) peak infection, and (iii) time to peak infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Random edge removal In random edge removal, we first select a proportion of q = ˜N/N nodes in the IL uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' For each of the selected nodes, each of its edges are removed with probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Figure 5(a,d) shows the epidemic size as a function of p,q for both short-range and long-range GIRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The epidemic size increases with p and q because larger values of p,q are associated with fewer edges in the IL, leading to a smaller pro- portion of aware nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Hence, the proportion of nodes with a reduced infection rate β u also decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' For the long-range GIRG (τ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5), the final epidemic size undergoes a transition from about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='6 for p,q ≈ 0 to about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='9 for p,q ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Because the final epidemic size in the short-range GIRG (τ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5) is already about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='9, random edge removal has relatively little impact on this quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' As with the impact on final epidemic size 1−s(t → ∞), ran- dom edge removal generates a similar-looking p,q-dependent infection peak, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 5(b,e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The time to reach peak infection decreases with p,q since higher p,q are associated with smaller proportions of aware nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Thus, the proportion of nodes with a reduced infection rate β u also decreases, and the epidemic spreads faster through the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Targeted edge removal For targeted edge removal where the ˜N selected nodes cor- respond to the hubs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=', largest-degree nodes) of the IL, we find that the overall dependence of epidemic size, peak infec- tion, and time to peak infection on p,q is qualitatively simi- lar to random edge removal (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' As in random edge removal, the impact of targeted edge removal on the final epi- demic size is smaller for the short-range GIRG compared to the long-range one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' A key difference in targeted edge removal is that all studied quantities are more sensitive to variations in q, the proportion of selected hub nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' For example, the tran- sition of the epidemic size for p = 1 as a function of q in tar- geted edge removal [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 6(a,d)] is steeper than the corre- sponding transition in random edge removal [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' 5(a,d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Targeted edge removal selects nodes based on their degree rather than uniformly, and leads to more significant changes in epidemic size, peak infection, and time to peak infection as p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' These findings are in accordance with previous work that showed that scale-free networks break down more easily under intentional attacks than under uniform random failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='67 Our work provide insights into how such disruptions in information diffusion translate into differences in disease severity measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' DISCUSSION In this work, we studied the impact of disruptions in com- munication networks on information diffusion and subse- quently disease outcome during an outbreak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' To do so, we constructed a multiplex network that consists of two lay- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The first layer, called information layer (IL), is used to model communication between individuals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=', online in- formation exchange via a social media platform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The sec- ond layer, called epidemic layer (EL), is used to represent a spatially embedded human contact network in which infec- tious individuals can transmit a disease to susceptible indi- viduals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' We use this multiplex network to simulate coevolv- ing unaware-aware-unaware (UAU) and susceptible-exposed- infected-recovered-deceased (SEIRD) dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' The model parameters that we use in our simulations have been selected in accordance with empirical data on the early outbreak stages of SARS-CoV-2 in the beginning of 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' We studied two different epidemic layers with different pro- portions of long-range connections, representing human con- tact networks with different contact characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' To illus- trate the impact of disruptions in the IL on the evolution of an outbreak, we utilized two different edge removal protocols: (i) random edge removal and (ii) targeted edge removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' In both protocols, we select a proportion q of nodes and then remove corresponding edges with probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' In random edge re- moval, we select nodes in the IL uniformly at random while we select nodes with the largest degree (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=', hub nodes) in targeted edge removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Although edge removal may render the IL disconnected, the EL is always connected in our sim- ulations such that all nodes in the EL can potentially become infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Previous work has shown that scale-free networks such as the IL in our multiplex network are more robust to random than targeted disruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='65–67 The reason for this ef- fect is that by removing hub nodes of a scale-free network, a large number of all edges in the network is being removed, strongly impacting the connectivity properties of such a net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' We observe that targeted edge removal can abruptly change outbreak characteristics such as time to peak infection, even for small proportions of selected nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' Our results ex- tend those presented in previous work on random and targeted disruptions65–67 by establishing a connection to coevolving in- formation and epidemic diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content=' DATA AVAILABILITY Our source codes are publicly available at https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQf2PmT/content/2301.00748v1.pdf'} +page_content='com/ComputationalScience/ information-epidemic.' metadata={'source': 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+2Department of Computer Science, Technion, Israel +3Department of Mathematics, Technion, Israel +4Department of Computer Science, Aarhus University +5Department of Mathematics, Princeton University +6Google Research +January 6, 2023 +Abstract +We study several variants of a combinatorial game which is based on Cantor’s diagonal +argument. The game is between two players called Kronecker and Cantor. The names of the +players are motivated by the known fact that Leopold Kronecker did not appreciate Georg +Cantor’s arguments about the infinite, and even referred to him as a “scientific charlatan”. +In the game Kronecker maintains a list of m binary vectors, each of length n, and Cantor’s +goal is to produce a new binary vector which is different from each of Kronecker’s vectors, or +prove that no such vector exists. Cantor does not see Kronecker’s vectors but he is allowed +to ask queries of the form +“What is bit number j of vector number i?” +What is the minimal number of queries with which Cantor can achieve his goal? How much +better can Cantor do if he is allowed to pick his queries adaptively, based on Kronecker’s +previous replies? +The case when m = n is solved by diagonalization using n (non-adaptive) queries. We +study this game more generally, and prove an optimal bound in the adaptive case and nearly +tight upper and lower bounds in the non-adaptive case. +1 +Introduction +The concept of infinity has been fascinating philosophers and scientists for hundreds, perhaps +thousands of years. The work of Georg Cantor (1845 – 1918) played a pivotal role in the +mathematical treatment of the infinite. Cantor’s work is based on a simple notion which asserts +that two (possibly infinite) sets have the same size whenever their elements can be paired +in one-to-one correspondence with each other [Can74]. Despite being simple, this notion has +counter-intuitive implications: for example, a set can have the same size as a proper subset of it1; +this phenomena is nicely illustrated by Hilbert’s paradox of the Grand Hotel, see e.g. [Wik22b]. +This simple notion led Cantor to develop his theory of sets, which forms the basis of modern +mathematics. Alas, Cantor’s set theory was controversial at the start, and only later became +widely accepted: +1E.g. the natural numbers and the even numbers, via the correspondence “n �→ 2n”. +1 +arXiv:2301.01924v1 [math.CO] 5 Jan 2023 + +Figure 1: Georg Cantor (1845 – 1918) +Figure 2: Leopold Kronecker (1823 – 1891) +The objections to Cantor’s work were occasionally fierce: Leopold Kronecker’s public opposition +and personal attacks included describing Cantor as a ”scientific charlatan”, a ”renegade” and a +”corrupter of youth”. Kronecker objected to Cantor’s proofs that the algebraic numbers are +countable, and that the transcendental numbers are uncountable, results now included in a +standard mathematics curriculum. [Wik22a] +1.1 +Diagonalization +One of the most basic and compelling results in set theory is that not all infinite sets have the same +size. To prove this result, Cantor came up with a beautiful argument, called diagonalization. +This argument is routinely taught in introductory classes to mathematics, and is typically +presented as follows. Let N denote the set of natural numbers and let {0, 1}N denote the set of +all infinite binary vectors. Clearly both sets are infinite, but it turns out that they do not have +the same size: assume towards contradiction that there is a one-to-one correspondence j �→ vj, +where vj = (vj(1), vj(2), . . .) is the infinite binary vector corresponding to j ∈ N. Define a vector +u = (1 − v1(1), 1 − v2(2), . . .). +That is, u is formed by letting its j’th entry be equal to the negation of the j’th entry of vj. +Notice that this way the resulting vector u disagrees with vj on the j’th entry, and hence +u ̸= vj for all j. Thus, we obtain a binary vector which does not correspond to any of the natural +numbers via the assumed correspondence – a contradiction. +Rather than reaching a contradiction, it is instructive to take a positivist perspective according +to which diagonalization can be seen as a constructive procedure that does the following: +Given binary vectors v1, v2, . . ., find a binary vector u such that u ̸= vj for all j. +Moreover, notice that Cantor’s diagonal argument involves querying only a single entry per +each of the input vectors vj (i.e. the “diagonal” entries vj(j)). Thus, it is possible to construct u +while using only a little information about the input vectors vi’s (a single bit per vector). +In this manuscript we study a finite variant of the problem in which m binary vectors +v1, . . . , vm of length n are given and the goal is to produce a vector u which is different from all +of the vi’s, or to report that no such vector exists, while querying as few as possible entries of +the vi’s. We first study the case when m < 2n whence such a u is guaranteed to exist, and the +goal boils down to finding one, and later the case when m ≥ 2n. +2 + +v1 = 0, 1, 1, 0, 1, 0 +v2 = 1, 0, 0, 1, 1, 1 +v3 = 1, 1, 1, 0, 0, 0 +v4 = 0, 1, 0, 1, 1, 0 +v5 = 1, 1, 0, 1, 0, 1 +v6 = 0, 1, 1, 1, 1, 1 +u = 1, 1, 0, 0, 1, 0 +Figure 3: An illustration of Cantor’s diagonalization: the vector u at the bottom is not equal to +any of the vi’s at the top. +2 +The Cantor-Kronecker Game +Consider a game between two players called Kronecker and Cantor. In the game there are +two parameters m and n, where m, n are positive integers. Kronecker maintains a set V = +{v1, v2, . . . , vm} of m binary vectors, each of length n. Cantor’s goal is to produce a binary +vector u, also of length n, which differs from each vi, or to report that no such vector exists. To +do so, he is allowed to ask queries, where each query is of the form +“What is bit number j of vector number i?”, +where 1 ≤ j ≤ n, 1 ≤ i ≤ m. Kronecker is answering each query being asked. The objective +of Cantor is to minimize the number of queries enabling him to produce u, whereas Kronecker +tries to maximize the number of queries. We distinguish between two versions of the game: +• In the adaptive version Cantor presents his queries to Kronecker in a sequential manner, +and may decide on the next query as a function of Kronecker’s answers to the previous +ones. +• In the oblivious version Cantor must declare all of his queries in advance, before getting +answers to any of them. +For m ≤ n the smallest number of queries, both in the adaptive and oblivious versions, is m. +Indeed, Cantor can query bit number i of vi for all 1 ≤ i ≤ m and return a vector u whose i’th +bit differs from the i’th bit of vi, for all i. The lower bound is even simpler: if Cantor asks less +than m queries then there is some vector vi about which he has no information at the end of the +game. In this case he cannot ensure that his vector u will not be equal to this vi. +Organization. +We begin with the case where m < 2n: in the next section (Section 3) we +derive nearly tight bounds both in the adaptive and oblivious cases. We do so by exhibiting and +analyzing near optimal strategies for Cantor. Then, in Section 4 we consider the case where +m ≥ 2n and derive an optimal bound of m·n in this case (for both the oblivious and the adaptive +versions). We do so by exhibiting and analyzing an optimal strategy for Kronecker. Finally, in +Section 5 we discuss some algorithmic aspects, and conclude with some suggestions to future +research. +3 + +3 +The Cantor-Kronecker Game with m < 2n +3.1 +Adaptive Version +Theorem 3.1. Let g(n, m) denote the smallest number of queries that suffices for Cantor when +he is allowed to use adaptive strategies. Then, +g(n, m) = +� +m +m ≤ n, +2m − n +n < m < 2n. +The case 1 ≤ m ≤ n is proved in the previous section so we assume n ≤ m < 2n. +Upper Bound. +We present a strategy for Cantor which combines diagonalization with another +simple idea. To illustrate this idea let us first consider the case m = n + 1. This special case +appeared as a question in the 2022 Grossman Math Olympiad for high-school students, and so +perhaps the reader might enjoy trying to solve it before continuing reading. +Let v1, . . . , vn+1 be the input vectors. Cantor begins with querying the first bit of v1, v2, and +of v3. Getting the answers, there is a bit ε so that at least two vectors among v1, v2, v3 have +their first bit equals to ε. Cantor now defines the first bit of u to be u(1) = 1 − ε and can remove +the two vectors among v1, v2, v3 whose first bit equals ε. Now Cantor is left with at most n − 1 +vectors and can therefore set the last n − 1 coordinates of u according to the diagonalization +construction. +The general case is handled similarly by induction on n: for n = 1 since n ≤ m < 2n, also m +must be 1 and the result is trivial. +Assuming the result for n − 1, let v1, . . . , vm be the m vectors of Kronecker. First, note that +there is an integer x satisfying 1 ≤ x ≤ ⌈m/2⌉ so that n − 1 ≤ m − x < 2n−1: indeed, for x = 1 +we have m − x ≥ n − 1 and for x = ⌈m/2⌉ we have m − x ≤ m/2 < 2n−1. Starting from x = 1 +keep increasing it in steps, where in each step it is increased by 1, until it reaches ⌈m/2⌉. As +m − x changes by 1 in each step we can take the smallest x ≥ 1 that satisfies m − x < 2n−1, and +it will clearly be at most ⌈m/2⌉ and satisfy m − x ≥ n − 1 as well. +Having x as above, Cantor first queries the first bit of each of the vectors v1, v2, . . . , v2x−1. +(Note that 2x − 1 ≤ m hence this is possible). Getting the answers, there is a bit ε ∈ {0, 1} so +that at least x of the vectors have their first bit equal to ε. Cantor now defines the first bit of +his vector u to be 1 − ε, removes from the set V exactly x of the vectors whose first bit is ε, +and defines as V ′ the set of all restrictions of the remaining m − x vectors to their last n − 1 +coordinates. Note that n − 1 ≤ m − x < 2n−1. +By the induction hypothesis, Cantor can now play the game for the set V ′ producing an +appropriate vector u′ by asking at most 2(m − x) − (n − 1) additional queries. The total number +of queries is thus (2x − 1) + 2(m − x) − (n − 1) = 2m − n, as needed. The vector u obtained +by concatenating the 1-bit vector 1 − ε and the vector u′ is clearly different from each member +of V . This completes the induction step argument and finishes the proof of the upper bound. +Lower Bound. +For the lower bound, we present a strategy for Kronecker which essentially +mirrors Cantor’s strategy from the upper bound. Suppose Cantor manages to produce the +required vector u after making exactly bj queries in coordinate number j of some of the vectors vi. +Kronecker chooses his answers ensuring that for each such j, the answers for bits in the j’th +location are balanced, that is, at most ⌈bj/2⌉ of the answers are 0 and at most ⌈bj/2⌉ of the +answers are 1. +Consider the vector u produced by Cantor. For every 1 ≤ j ≤ n, there are at most ⌈bj/2⌉ +vectors vi known to be different than u in coordinate number j. Thus altogether there are at +4 + +most +n +� +j=1 +�bj +2 +� +≤ +n +� +j=1 +bj + 1 +2 +. +vectors vi that are known to Cantor to be different than u. In order to ensure u is indeed +different from each vi this number has to be at least m and hence +m ≤ +n +� +j=1 +bj + 1 +2 +. +By rearranging, this implies that the total number of queries �n +j=1 bj must be at least 2m − n, +as stated. +3.2 +Oblivious Version +Theorem 3.2. Let f(n, m) denote the smallest number of queries that suffices for Cantor when +he is restricted to use oblivious strategies. Then, +f(n, m) = +� +m +m ≤ n +m +� +log +� m +n +� ++ o +� +log +� m +n +��� +n < m < 2n. +Quantitatively, for all n < m < 2n +m · +� +log +� +m +n − log m + 1 +� +− 1 +� +≤ f(n, m) ≤ m +� +log +�2m +n +� ++ 2 log +� +log +�2m +n +�� ++ 1 +� +, +The case 1 ≤ m ≤ n is proved above so we assume n < m < 2n. +Upper Bound. +Like in the adaptive case, we present a strategy for Cantor which combines +diagonalization with another simple idea. We first illustrate this idea by handling the case m = +n + 1, and again, we encourage the reader to try and handle this case before continuing reading. +Let v1, . . . , vn+1 be the input vectors. Cantor begins with querying the first two bits of each +of v1, v2, and v3 (for a total of 6 queries). Notice that there are 22 = 4 possible combinations of +0/1 patterns on the first two bits, but at most three of them are realized by v1, v2, v3. Hence, +there must be a pair of bits ε1, ε2 which is not realized by v1, v2, nor v3: +(ε1, ε2) /∈ +�� +v1(1), v1(2) +� +, +� +v2(1), v2(2) +� +, +� +v3(1), v3(2) +�� +. +Thus, by setting u(1) = ε1 and u(2) = ε2, Cantor rules out v1, v2, v3 and is left with n−2 vectors +v3, . . . , vn+1 which can be obliviously ruled out with the last n − 2 using diagonalization. +For the general case, let d be an integer (to be determined later). Pick mutually disjoint +subsets of coordinates J1, . . . , J⌊n/d⌋ ⊆ [n], each of size d, and pick a partition of the m vectors to +⌊n/d⌋ subsets V1, . . . , V⌊n/d⌋ such that the partition is as balanced as possible (i.e. the difference +between each pair of sizes is ≤ 1). Thus, each set has size +|Vi| ≤ +� +m +⌊n/d⌋ +� +≤ 2md +n . +Cantor queries (obliviously) as follows. +For each i and each vector in Vi query all the coordinates in Ji. +5 + +Thus, the total number of queries is exactly m · d. Now, notice that if d satisfies +2d > 2md +n , +(1) +then there must exist an assignment fi : Ji → {0, 1} such that fi disagrees with each of the +vectors in Vi on at least one coordinate in Ji. Hence Cantor can output the vector u, which +agrees with each of the fi on Ji. Note that Equation 1 is satisfied iff 2d +d > 2m +n ; since m > n, +it can be verified that this inequality holds when d ≥ log( 2m +n ) + 2 log(log( 2m +n )) + 1. Thus for +d = +� +log +� 2m +n +� ++ 2 log +� +log +� 2m +n +�� ++ 1 +� +, the total number of queries is at most +m · d = m +� +log +�2m +n +� ++ 2 log +� +log +�2m +n +�� ++ 1 +� +. +Lower Bound. +The lower bound proof is based on the following simple idea. Let Ji denote +the set of coordinates of vi which Cantor queries. Thus, the total number of queries Cantor uses +is |J1| + . . . + |Jm|. Now, let fi : Ji → {0, 1} denote Kronecker’s answers for the queries on vi. +The crucial observation is that the vector u that Cantor outputs must satisfy +(∀i) : u|Ji ̸= fi. +Indeed, if u|Ji = fi for some i then Kronecker can fail Cantor by picking his i’th vector vi to be +equal to Cantor’s output u (which would be consistent with Kronecker’s answers). +We summarize the above consideration with a definition that characterizes the winning (or +losing) strategies of Cantor in the oblivious case. +Definition 3.3 (Covering Assignments). We say that a sequence of sets J1, . . . , Jm ⊆ [n] +has a covering assignment if there are m functions fi : Ji → {0, 1} such that every binary +vector v ∈ {0, 1}n agrees with one of the fi on Ji (i.e. v|Ji = fi). +Thus, Kronecker has a winning strategy if and only if the sequence of sets J1, . . . , Jm that +Cantor queries has a covering assignment. The following lemma establishes the lower bound. +Lemma 3.4. Let J1, . . . , Jm ⊆ [n] such that +|J1| + . . . + |Jm| < m · +� +log +� +m +n − log m + 1 +� +− 1 +� +. +(2) +Then, J1, . . . , Jm has a covering assignment. +Equivalently, if for each vector vi Cantor queries its entries in Ji and Equation 2 holds, then +Kronecker has a winning strategy. +Proof. Let ti = |Ji| and let t = � +i ti. Assume, without loss of generality, that t1 ≤ t2 ≤ . . . ≤ tm. +To prove a lower bound of the form md for t, where d will be specified later, we show that if t is +smaller than md then there are m functions fi : Ji → {0, 1} so that for every possible vector +v ∈ {0, 1}n there is i ≤ m so that v|Ji = fi. +We do so by explicitly constructing the fi’s (which corresponds to describing a winning +strategy for Kronecker). Starting with the set V = {0, 1}n of all possible potential vectors z, +go over the vectors vi in order. In step i we choose the function fi : Ji → {0, 1} such that +|{v ∈ V : v|Ji = fi}| is maximized. Since there are 2ti possible choices for fi, the maximizing +choice satisfies +���{v ∈ V : v|Ji = fi} +��� ≥ |V | +2ti . +6 + +After picking fi, we remove all the vectors of V that agree with fi and proceed to the next step. +Therefore, after the first i steps, the size of the set V of the remaining vectors is at most +2n +i� +j=1 +(1 − 1/2tj). +We can continue with this analysis until the size of the set V becomes smaller than 1, namely +the set becomes empty. It is a bit better, however, to apply a simpler reasoning once the size +of V becomes smaller than 2d, and only argue that at least one vector from V is eliminated +in each step. (Continuing the same analysis as before would only guarantee that V shrinks by +a factor of (1 − 1/2ti) which by the choice of d would be roughly 1 − 1/2d < 1). To simplify +the computation it is not too wasteful to apply the simpler analysis already when the size of +V becomes smaller than m/2. If this happens in the first m/2 steps then by removing a single +vector in each of the remaining steps we will eliminate all of the vectors. This means that if +2n +m/2 +� +j=1 +� +1 − 1/2tj� +≤ m +2 +then the sequence J1. . . . , Jm has a covering assignment. Since d is such that the total number of +queries is m · d, the above amounts to �m/2 +j=1 tj ≤ md/2; that is, the average tj for 1 ≤ j ≤ m/2 +is at most d. This implies that +2n +m +2 +� +j=1 +� +1 − 1 +2tj +� +≤ 2n +m +2 +� +j=1 +exp +� +− 1 +2tj +� +(1 + x ≤ exp(x) for all x ∈ R) += 2n exp +� +− +m +2 +� +j=1 +1 +2tj +� +≤ 2n exp +� +− m +2d+1 +� +, +where the last inequality follows because exp(−x) is decreasing and because +m +2 +� +j=1 +1 +2tj ≥ m +2 · +1 +2 +1 +m/2 +�m/2 +j=1 tj +≥ m +2 · 1 +2d , +which follows by convexity of the function f(x) = 2x and because t1 ≤ t2 ≤ . . . ≤ tm. +We have thus shown that if |J1| + . . . + |Jm| = m · d such that +2n exp +� +− m +2d+1 +� +≤ m +2 +then the sequence J1, . . . , Jm has a covering assignment. The last inequality surely holds provided +m +2d+1 ≥ n + 1 − log m. +That is, provided +2d+1 ≤ +m +n + 1 − log m, +or +d ≤ log +� +m +n + 1 − log m +� +− 1 +completing the proof. +7 + +4 +The Cantor-Kronecker Game with m ≥ 2n +Assume now that Kronecker’s list V consists of m ≥ 2n binary vectors of length n. In this case +V may contain all the binary vectors of length n and there is no vector Cantor can output that +is different from each vector on Kronecker’s list. In this regime it is more natural to first focus +on the decision problem in which Cantor’s goal is to decide whether V contains {0, 1}n, and if +this is not the case, to provide a vector which is not in V .2 Clearly Cantor can achieve this if he +queries all mn possible queries. Can he do better? +We first observe that mn queries are in fact needed in the oblivious case: assume that Cantor +submits only mn − 1 queries, and leaves the j’th bit of vi unqueried. Then Kronecker may set vi +to be the unique occurrence of the all ones vector 1n, and set the remaining m − 1 vectors in V +to include all 2n − 1 vectors that are different from the all ones vector. Clearly, it is necessary +for Cantor to query also the last bit of vi in order to see whether vi is the all ones vector or not. +Consequently, Cantor must query all mn queries in the oblivious case. +How about the adaptive case? A similar argument shows that for m = 2n, Kronecker can +force mn = 2nn queries also in the adaptive case, by using a list which contains each binary +vector of length n exactly once: indeed, if only mn − 1 bits are queried, then the last, yet +unqueried bit, belongs to a vector which occurs only once in V . Hence it is necessary to get the +value of this bit in order to verify that V contains all 2n vectors. +The case when m > 2n turns out to be more subtle. Nevertheless, we prove that mn queries +are necessary even in this case. We start with introducing some notation. +Notation. +Each step of the game consists of a query by Cantor followed by a response by +Kronecker. The status of the game after each such step is given by an m × n matrix L, where +L(i, j) denotes the status of the j’th bit of vi, that is: L(i, j) ∈ {0, 1, ⋆}, where L(i, j) = ⋆ means +that the j’th bit of vi was not queried yet, and otherwise L(i, j) equals the value of this bit as +answered by Kronecker. +Definition 4.1. FIXED(L) = +� +v ∈ L : v ∈ {0, 1}n� +. That is, FIXED(L) is the set of all vectors +in L that were fully queried by Cantor. +Definition 4.2. L is complete if FIXED(L) = {0, 1}n. +Definition 4.3. A subset S of 2n rows of L is useful if it either contains all the 2n binary vectors +of length n, or it can be converted to this set by replacing each ⋆-entry in S by 0 or 1. +Definition 4.4. A matrix L is unblocked if it can be completed; that is, if L has a useful subset. +Otherwise L is called blocked. +Notice that for m ≥ 2n, the m by n matrix all whose entries are ⋆ is unblocked. +As a warmup, and to get used to the definitions, let us assume first that Cantor’s queries +the vectors one by one according to their order; i.e. he first queries all the bits of v1 from left +to right, then all the bits of v2 from left to right, and so on. We use the following strategy for +Kronecker: when Cantor queries the j’th bit of vi (i.e. the value of L(i, j)), Kronecker replies +according to the following “0 first” strategy: +modified value of L(i, j) = +� +1 +If setting L(i, j) to 0 blocks L +0 +otherwise +(3) +It is not hard to verify that since Cantor queries the vectors one by one, and from left (most +significant bit) to right, the following matrix is produced: each of the first m − 2n + 1 rows will +2Later we will see that the decision and search variants are in fact equivalent. +8 + +be set to the all-zeros vector, and the last 2n − 1 rows will be set to the 2n − 1 non zero vectors +in increasing lexicographical order: starting with 0n−11 and ending with 1n. Hence Cantor is +forced to query all mn entries as in the oblivious case. +Interestingly, it turns out that, for any strategy of Cantor, the above “0 first” strategy of +Kronecker forces Cantor to make mn queries. +Theorem 4.5. Let m > 2n. Then for any strategy of Cantor, the “0 first” strategy of Kronecker +forces Cantor to make mn queries in order to determine if L contains {0, 1}n. +In the following we consider an arbitrary execution of the game, where Kronecker follows the +“0 first” strategy (and Cantor’s strategy is arbitrary). We denote by Lt the m × n matrix L after +t steps of the game; thus L0 is the initial matrix which is filled only with ⋆’s. +By the fact that if L is unblocked and L(i, j) = ⋆, then it is possible to set L(i, j) to 0 or +to 1 without blocking L, we get: +Observation 4.6. If Lt is unblocked, so is Lt+1. Hence Lmn is complete; i.e. it contains {0, 1}n. +Definition 4.7. We say that a row L(i) is essential for an unblocked matrix L if every useful +subset of L’s rows contains L(i). +Note that if Lt(i) is essential for Lt, then Ls(i) is essential for Ls for all s ≥ t. Also, if Lmn(i) +is essential for Lmn, then Lmn(i) is equal to a unique vector in {0, 1}n which is different from all +other rows of Lmn. +Lemma 4.8. Assume that Lt(i) is not essential for Lt and Lt(i, j) = ⋆. If Lt(i, j) is queried at +time t + 1, then it is set to 0, i.e. Lt+1(i, j) = 0. +Proof. By the “0 first” strategy, and the fact that if L(i) is not essential for an unblocked +matrix L, then setting L(i, j) to 0 does not block L. +By a straightforwards induction Lemma 4.8 implies: +Corollary 4.9. If Lt(i) is not essential for Lt, then Lt(i) contains no 1’s (only 0’s or ⋆’s). +Specifically, if Lmn(i) is not essential for Lmn, then Lmn(i) is the zero vector 0n. Hence, every +row of Lmn which is not the zero vector is essential, and thus it is different from all other rows +of Lmn. +Lemma 4.10. Let Lmn−1(i, j) be the last bit queried in the game. Then Lmn−1(i) is an essential +row of Lmn−1. +Proof. To simplify notation, we assume without loss of generality that j = 1. Assume towards +contradiction that Lmn−1(i) is not essential for Lmn−1. By Corollary 4.9, this implies that +Lmn−1(i) = ⋆0n−1 and Lmn(i) = 0n. (i.e. Kronecker sets Lmn−1(i, 1) to 0 at Cantor’s mn’th +query). Since Lmn is complete (Observation 4.6), this implies that Lmn−1 contains a distinct +occurrence of each of the 2n − 1 nonzero vectors of {0, 1}n, and in particular for some k ̸= i, +Lmn−1(k) is the unique row of Lmn−1 which equals 10n−1. Then, any subset S of Lmn−1 which +contains +• the row Lmn−1(i), +• the 2n − 2 non zero rows of Lmn−1 excluding Lmn−1(k), and +• some zero row of Lmn−1 (by Corollary 4.9 there are m − 2n > 0 such rows in Lmn−1), +9 + +is a useful subset of Lmn−1 which does not contain Lmn−1(k). Hence Lmn−1(k) is not essential +for Lmn−1, and by Lemma 4.8 Lmn−1(1) = 0 ̸= 1, which stands in contradiction with Lmn−1(1) = +10n−1. +Proof of Theorem 4.5. Let Lmn−1(i, j) be the last query in the game. By Lemma 4.10, Lmn−1(i), +and hence also Lmn(i), is essential, meaning that Lmn(i) is different from all other rows of Lmn. +Thus Cantor must get the value of Lmn−1(i, j) in order to reach a decision. +A remark on computational complexity. +A naive implementation of the “0 first” strategy +might take exponential time: indeed, it requires checking whether setting the queried bit to 0 +blocks the current matrix, which involves checking a potentially exponential list of constraints. +Nevertheless, we next show that this strategy in fact admits a polynomial time implementation. +Firstly, notice that the first m − 2n steps are trivially efficient, because setting L(i, j) to any +value cannot block L (since at least 2n rows of L are not queried yet). +Thus it suffices to show that in each later step, deciding whether setting L(i, j) to 0 blocks +the matrix, can be performed in time which is polynomial in mn, the size of L. Let Lt be the +matrix L after t steps of the game, t > m − 2n. Consider the bipartite graph Gt = (At, B, Et), +where At = {Lt(i) : 1 ≤ i ≤ m} is the set of rows of Lt, B = {0, 1}n, and (Lt(i), u) ∈ Et if +and only if Lt(i) can be converted to the binary vector u by replacing the ⋆’s in Lt(i) (if any) +by binary digits. Then, a subset S of Lt is useful for Lt if and only if Gt contains a perfect +matching between the vertices in At which correspond to S and B. +Assume now that we are given the graph Gt, and the corresponding matching, and let Lt(i, j) +be the entry queried by Cantor at step t+1. To check if setting Lt(i, j) to 0 blocks Lt, we remove +from Gt all the edges (Lt(i), u) in which u(j) = 0, and check if the resulted graph contains a +perfect matching. Since we are given a perfect matching Mt for Gt, and removing these edges +eliminates at most one edge from Mt, this checking can be done by executing one phase in some +classical algorithm for bipartite matching, which can be done in O(|Et|) = O(m2n) = O(m2) +time (see e.g. [Eve11]). +5 +Concluding Remarks and Future Research +We studied the Cantor-Kronecker game for different values of m and n: when m ≤ n the trivial +lower bound of m is tight (a lower bound of m follows because Cantor must query at least one +bit in each vector); when m ≥ 2n, the trivial upper bound of mn is tight (an upper bound of +mn follows because querying all the bits is clearly sufficient); when n < m < 2n the landscape +is more interesting, and in particular the bounds depend on whether Cantor is adaptive or +oblivious. +Further Research. +We conclude with suggestions for possible future research: +1. Study the Cantor-Kronecker game when there are r rounds of adaptivity: i.e. there are +r rounds in which Cantor can submit queries, and in each round the submitted queries +may depend on Kronecker’s answers to queries from previous rounds. How does the query +complexity change as a function of r? Note that r = 1 is the oblivious case and r = ∞ is +the adaptive case. (In fact r = n is already equivalent to r = ∞.) +2. Consider the following generalization of the game. Let k ≤ m, ℓ ≤ n be positive integers. +Kronecker maintains an m×n binary matrix, and Cantor queries the entries of Kronecker’s +matrix. Cantor’s goal is to find a k × ℓ matrix which does not appear as a submatrix of +Kronecker’s m × n matrix, or to decide that one does not exist. So, the original game +10 + +is when k = 1, ℓ = n. What is the query complexity as a function of k, ℓ, m, n in the +adaptive/oblivious case? For which values does Cantor have a strategy that uses strictly +less than m · n queries? +3. Find tighter bounds for the oblivious case. Specifically, notice that Cantor’s original +diagonalization provides tight bound on the number of queries needed for the oblivious +case when m ≤ n. It will be interesting to derive tight bounds and optimal strategies in +the remaining cases. As we exemplify below, this question has connections with natural +combinatorial problems. +Consider the case when m is at the other end of the scale, namely 2n−1 ≤ m < 2n. Then, +Cantor can win the game by querying nm − d bits, where d = 2n − m − 1. In fact, it +suffices that Cantor chooses his queries such that each of the d unqueried entries belongs +to a different vector: in this case any assignments of values to the unqueried entries covers +(in the sense of Definition 3.3) the m − d fully queried vectors, and at most two additional +vectors per each of the remaining d vectors (each of which contains one unqueried entry): +altogether at most (m − d) + 2d = m + d vectors. Hence, Cantor is guaranteed to win the +game provided that m + d < 2n (equivalently d ≤ 2n − m − 1). +Is the above strategy optimal? i.e., can Kronecker win the game when Cantor queries only +mn − (2n − m) bits? Informally, Kronecker has a winning strategy if, for any distribution +of the 2n − m unqueried entries, there is an assignment which covers sufficiently many +vectors. This is formalized below. +Definition 5.1 (cube(v), J-cube). Let v be a vector with possibly some unqueried entries. +cube(v) is the set of binary vectors which can be obtained by replacing the unqueried +entries in v by zeros or ones. In particular, cube(v) = {v} if v is fully queried. The cube +cube(v) is called a J-cube if J = {j : the j′th bit of v is not queried}. For j ∈ [n], a +{j}-cube is denoted by j-edge. +Assume that Cantor distributes the (2n − m) unqueried entries among vectors v1, . . . , vq. +Then Kronecker answers to the queried entries define a cube C(vi) for each vector vi. +Kronecker wins if and only if those cubes cover {0, 1}n. Hence Kronecker has a winning +strategy when Cantor uses mn − (2n − m) queries (2n−1 + 1 ≤ m < 2n) if and only if the +following holds: +Conjecture 5.2. Let d = 2n − m < 2n−1. For any collection J1, J2, . . . , Jq of nonempty +subsets of [n] satisfying �q +i=1 |Ji| = d, there are cubes C1, . . . , Cq s.t. Ci is a Ji-cube, and +|�q +i=1 Ci| ≥ d + q. +The following result of [FHK93] proves Conjecture 5.2 for the case that each Ji-cube is a +ji-edge. +Theorem 5.3 ([FHK93]). Let d < 2n−1. For any multiset D = {j1, j2, . . . , jd} of elements +of [n], {0, 1}n contains a matching {e1, . . . , ed} s.t. for i = 1, . . . , d, ei is a ji-edge. +It is also shown in [FHK93] that Conjcture 5.2 does not hold when d = 2n−1: in this case +a corresponding matching exists if and only if each element in [n] occurs an even number +of times in D. This implies that when m = 2n−1 Cantor has a winning strategy with only +mn − (2n − m) = mn − 2n−1 queries: he may query n − 1 entries per each vector, so that +at least one dimension is left unqueried in an odd number of vectors. +11 + +References +[Can74] +Georg Cantor. Ueber eine Eigenschaft des inbegriffs aller reellen algebraischen Zahlen. +Journal f¨ur die reine und angewandte Mathematik (Crelles Journal), 1(77):258–262, +1874. +[Eve11] +Shimon Even. Graph Algorithms. Cambridge University Press, New York, NY, USA, +2nd edition, 2011. +[FHK93] Alexander Felzenbaum, Ron Holzman, and Daniel J. Kleitman. Packing lines in a +hypercube. Discrete Mathematics, 117(1):107–112, 1993. +[Wik22a] Wikipedia contributors. Georg Cantor — Wikipedia, the free encyclopedia. https: +//en.wikipedia.org/wiki/Georg_Cantor, 2022. [Online; accessed 20-November- +2022]. +[Wik22b] Wikipedia contributors. Hilbert’s paradox of the Grand Hotel — Wikipedia, the +free encyclopedia. https://en.wikipedia.org/wiki/Hilbert’s_paradox_of_the_ +Grand_Hotel, 2022. [Online; accessed 20-November-2022]. +12 + diff --git a/49AzT4oBgHgl3EQf9v6Q/content/tmp_files/load_file.txt b/49AzT4oBgHgl3EQf9v6Q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aaf8f27888f2677cac39665858d905cbb1ad9c3b --- /dev/null +++ b/49AzT4oBgHgl3EQf9v6Q/content/tmp_files/load_file.txt @@ -0,0 +1,475 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf,len=474 +page_content='Diagonalization Games Noga Alon1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Olivier Bousquet6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Kasper Green Larsen4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Shay Moran2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' and Shlomo Moran2 1Departments of Mathematics and Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Tel Aviv University 2Department of Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Technion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Israel 3Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Technion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Israel 4Department of Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Aarhus University 5Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Princeton University 6Google Research January 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' 2023 Abstract We study several variants of a combinatorial game which is based on Cantor’s diagonal argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' The game is between two players called Kronecker and Cantor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' The names of the players are motivated by the known fact that Leopold Kronecker did not appreciate Georg Cantor’s arguments about the infinite, and even referred to him as a “scientific charlatan”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' In the game Kronecker maintains a list of m binary vectors, each of length n, and Cantor’s goal is to produce a new binary vector which is different from each of Kronecker’s vectors, or prove that no such vector exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Cantor does not see Kronecker’s vectors but he is allowed to ask queries of the form “What is bit number j of vector number i?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' What is the minimal number of queries with which Cantor can achieve his goal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' How much better can Cantor do if he is allowed to pick his queries adaptively, based on Kronecker’s previous replies?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' The case when m = n is solved by diagonalization using n (non-adaptive) queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' We study this game more generally, and prove an optimal bound in the adaptive case and nearly tight upper and lower bounds in the non-adaptive case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' 1 Introduction The concept of infinity has been fascinating philosophers and scientists for hundreds, perhaps thousands of years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' The work of Georg Cantor (1845 – 1918) played a pivotal role in the mathematical treatment of the infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Cantor’s work is based on a simple notion which asserts that two (possibly infinite) sets have the same size whenever their elements can be paired in one-to-one correspondence with each other [Can74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Despite being simple, this notion has counter-intuitive implications: for example, a set can have the same size as a proper subset of it1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' this phenomena is nicely illustrated by Hilbert’s paradox of the Grand Hotel, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' [Wik22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' This simple notion led Cantor to develop his theory of sets, which forms the basis of modern mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Alas, Cantor’s set theory was controversial at the start, and only later became widely accepted: 1E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' the natural numbers and the even numbers, via the correspondence “n �→ 2n”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='01924v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='CO] 5 Jan 2023 Figure 1: Georg Cantor (1845 – 1918) Figure 2: Leopold Kronecker (1823 – 1891) The objections to Cantor’s work were occasionally fierce: Leopold Kronecker’s public opposition and personal attacks included describing Cantor as a ”scientific charlatan”, a ”renegade” and a ”corrupter of youth”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Kronecker objected to Cantor’s proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' [Wik22a] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='1 Diagonalization One of the most basic and compelling results in set theory is that not all infinite sets have the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' To prove this result, Cantor came up with a beautiful argument, called diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' This argument is routinely taught in introductory classes to mathematics, and is typically presented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Let N denote the set of natural numbers and let {0, 1}N denote the set of all infinite binary vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Clearly both sets are infinite, but it turns out that they do not have the same size: assume towards contradiction that there is a one-to-one correspondence j �→ vj, where vj = (vj(1), vj(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=') is the infinite binary vector corresponding to j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Define a vector u = (1 − v1(1), 1 − v2(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' That is, u is formed by letting its j’th entry be equal to the negation of the j’th entry of vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Notice that this way the resulting vector u disagrees with vj on the j’th entry, and hence u ̸= vj for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Thus, we obtain a binary vector which does not correspond to any of the natural numbers via the assumed correspondence – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Rather than reaching a contradiction, it is instructive to take a positivist perspective according to which diagonalization can be seen as a constructive procedure that does the following: Given binary vectors v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=', find a binary vector u such that u ̸= vj for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Moreover, notice that Cantor’s diagonal argument involves querying only a single entry per each of the input vectors vj (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' the “diagonal” entries vj(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Thus, it is possible to construct u while using only a little information about the input vectors vi’s (a single bit per vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' In this manuscript we study a finite variant of the problem in which m binary vectors v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , vm of length n are given and the goal is to produce a vector u which is different from all of the vi’s, or to report that no such vector exists, while querying as few as possible entries of the vi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' We first study the case when m < 2n whence such a u is guaranteed to exist, and the goal boils down to finding one, and later the case when m ≥ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' 2 v1 = 0, 1, 1, 0, 1, 0 v2 = 1, 0, 0, 1, 1, 1 v3 = 1, 1, 1, 0, 0, 0 v4 = 0, 1, 0, 1, 1, 0 v5 = 1, 1, 0, 1, 0, 1 v6 = 0, 1, 1, 1, 1, 1 u = 1, 1, 0, 0, 1, 0 Figure 3: An illustration of Cantor’s diagonalization: the vector u at the bottom is not equal to any of the vi’s at the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' 2 The Cantor-Kronecker Game Consider a game between two players called Kronecker and Cantor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' In the game there are two parameters m and n, where m, n are positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Kronecker maintains a set V = {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , vm} of m binary vectors, each of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Cantor’s goal is to produce a binary vector u, also of length n, which differs from each vi, or to report that no such vector exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' To do so, he is allowed to ask queries, where each query is of the form “What is bit number j of vector number i?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=', where 1 ≤ j ≤ n, 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Kronecker is answering each query being asked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' The objective of Cantor is to minimize the number of queries enabling him to produce u, whereas Kronecker tries to maximize the number of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' We distinguish between two versions of the game: In the adaptive version Cantor presents his queries to Kronecker in a sequential manner, and may decide on the next query as a function of Kronecker’s answers to the previous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' In the oblivious version Cantor must declare all of his queries in advance, before getting answers to any of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' For m ≤ n the smallest number of queries, both in the adaptive and oblivious versions, is m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Indeed, Cantor can query bit number i of vi for all 1 ≤ i ≤ m and return a vector u whose i’th bit differs from the i’th bit of vi, for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' The lower bound is even simpler: if Cantor asks less than m queries then there is some vector vi about which he has no information at the end of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' In this case he cannot ensure that his vector u will not be equal to this vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' We begin with the case where m < 2n: in the next section (Section 3) we derive nearly tight bounds both in the adaptive and oblivious cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' We do so by exhibiting and analyzing near optimal strategies for Cantor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Then, in Section 4 we consider the case where m ≥ 2n and derive an optimal bound of m·n in this case (for both the oblivious and the adaptive versions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' We do so by exhibiting and analyzing an optimal strategy for Kronecker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Finally, in Section 5 we discuss some algorithmic aspects, and conclude with some suggestions to future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' 3 3 The Cantor-Kronecker Game with m < 2n 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='1 Adaptive Version Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Let g(n, m) denote the smallest number of queries that suffices for Cantor when he is allowed to use adaptive strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Then, g(n, m) = � m m ≤ n, 2m − n n < m < 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' The case 1 ≤ m ≤ n is proved in the previous section so we assume n ≤ m < 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Upper Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' We present a strategy for Cantor which combines diagonalization with another simple idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' To illustrate this idea let us first consider the case m = n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' This special case appeared as a question in the 2022 Grossman Math Olympiad for high-school students, and so perhaps the reader might enjoy trying to solve it before continuing reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Let v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , vn+1 be the input vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Cantor begins with querying the first bit of v1, v2, and of v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Getting the answers, there is a bit ε so that at least two vectors among v1, v2, v3 have their first bit equals to ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Cantor now defines the first bit of u to be u(1) = 1 − ε and can remove the two vectors among v1, v2, v3 whose first bit equals ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Now Cantor is left with at most n − 1 vectors and can therefore set the last n − 1 coordinates of u according to the diagonalization construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' The general case is handled similarly by induction on n: for n = 1 since n ≤ m < 2n, also m must be 1 and the result is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Assuming the result for n − 1, let v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , vm be the m vectors of Kronecker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' First, note that there is an integer x satisfying 1 ≤ x ≤ ⌈m/2⌉ so that n − 1 ≤ m − x < 2n−1: indeed, for x = 1 we have m − x ≥ n − 1 and for x = ⌈m/2⌉ we have m − x ≤ m/2 < 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Starting from x = 1 keep increasing it in steps, where in each step it is increased by 1, until it reaches ⌈m/2⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' As m − x changes by 1 in each step we can take the smallest x ≥ 1 that satisfies m − x < 2n−1, and it will clearly be at most ⌈m/2⌉ and satisfy m − x ≥ n − 1 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Having x as above, Cantor first queries the first bit of each of the vectors v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , v2x−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' (Note that 2x − 1 ≤ m hence this is possible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Getting the answers, there is a bit ε ∈ {0, 1} so that at least x of the vectors have their first bit equal to ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Cantor now defines the first bit of his vector u to be 1 − ε, removes from the set V exactly x of the vectors whose first bit is ε, and defines as V ′ the set of all restrictions of the remaining m − x vectors to their last n − 1 coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Note that n − 1 ≤ m − x < 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' By the induction hypothesis, Cantor can now play the game for the set V ′ producing an appropriate vector u′ by asking at most 2(m − x) − (n − 1) additional queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' The total number of queries is thus (2x − 1) + 2(m − x) − (n − 1) = 2m − n, as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' The vector u obtained by concatenating the 1-bit vector 1 − ε and the vector u′ is clearly different from each member of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' This completes the induction step argument and finishes the proof of the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Lower Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' For the lower bound, we present a strategy for Kronecker which essentially mirrors Cantor’s strategy from the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Suppose Cantor manages to produce the required vector u after making exactly bj queries in coordinate number j of some of the vectors vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Kronecker chooses his answers ensuring that for each such j, the answers for bits in the j’th location are balanced, that is, at most ⌈bj/2⌉ of the answers are 0 and at most ⌈bj/2⌉ of the answers are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Consider the vector u produced by Cantor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' For every 1 ≤ j ≤ n, there are at most ⌈bj/2⌉ vectors vi known to be different than u in coordinate number j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Thus altogether there are at 4 most n � j=1 �bj 2 � ≤ n � j=1 bj + 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' vectors vi that are known to Cantor to be different than u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' In order to ensure u is indeed different from each vi this number has to be at least m and hence m ≤ n � j=1 bj + 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' By rearranging, this implies that the total number of queries �n j=1 bj must be at least 2m − n, as stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='2 Oblivious Version Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Let f(n, m) denote the smallest number of queries that suffices for Cantor when he is restricted to use oblivious strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Then, f(n, m) = � m m ≤ n m � log � m n � + o � log � m n ��� n < m < 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Quantitatively, for all n < m < 2n m · � log � m n − log m + 1 � − 1 � ≤ f(n, m) ≤ m � log �2m n � + 2 log � log �2m n �� + 1 � , The case 1 ≤ m ≤ n is proved above so we assume n < m < 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Upper Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Like in the adaptive case, we present a strategy for Cantor which combines diagonalization with another simple idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' We first illustrate this idea by handling the case m = n + 1, and again, we encourage the reader to try and handle this case before continuing reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Let v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , vn+1 be the input vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Cantor begins with querying the first two bits of each of v1, v2, and v3 (for a total of 6 queries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Notice that there are 22 = 4 possible combinations of 0/1 patterns on the first two bits, but at most three of them are realized by v1, v2, v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Hence, there must be a pair of bits ε1, ε2 which is not realized by v1, v2, nor v3: (ε1, ε2) /∈ �� v1(1), v1(2) � , � v2(1), v2(2) � , � v3(1), v3(2) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Thus, by setting u(1) = ε1 and u(2) = ε2, Cantor rules out v1, v2, v3 and is left with n−2 vectors v3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , vn+1 which can be obliviously ruled out with the last n − 2 using diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' For the general case, let d be an integer (to be determined later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Pick mutually disjoint subsets of coordinates J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , J⌊n/d⌋ ⊆ [n], each of size d, and pick a partition of the m vectors to ⌊n/d⌋ subsets V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , V⌊n/d⌋ such that the partition is as balanced as possible (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' the difference between each pair of sizes is ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Thus, each set has size |Vi| ≤ � m ⌊n/d⌋ � ≤ 2md n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Cantor queries (obliviously) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' For each i and each vector in Vi query all the coordinates in Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' 5 Thus, the total number of queries is exactly m · d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Now, notice that if d satisfies 2d > 2md n , (1) then there must exist an assignment fi : Ji → {0, 1} such that fi disagrees with each of the vectors in Vi on at least one coordinate in Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Hence Cantor can output the vector u, which agrees with each of the fi on Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Note that Equation 1 is satisfied iff 2d d > 2m n ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' since m > n, it can be verified that this inequality holds when d ≥ log( 2m n ) + 2 log(log( 2m n )) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Thus for d = � log � 2m n � + 2 log � log � 2m n �� + 1 � , the total number of queries is at most m · d = m � log �2m n � + 2 log � log �2m n �� + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Lower Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' The lower bound proof is based on the following simple idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Let Ji denote the set of coordinates of vi which Cantor queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Thus, the total number of queries Cantor uses is |J1| + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' + |Jm|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Now, let fi : Ji → {0, 1} denote Kronecker’s answers for the queries on vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' The crucial observation is that the vector u that Cantor outputs must satisfy (∀i) : u|Ji ̸= fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Indeed, if u|Ji = fi for some i then Kronecker can fail Cantor by picking his i’th vector vi to be equal to Cantor’s output u (which would be consistent with Kronecker’s answers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' We summarize the above consideration with a definition that characterizes the winning (or losing) strategies of Cantor in the oblivious case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='3 (Covering Assignments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' We say that a sequence of sets J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , Jm ⊆ [n] has a covering assignment if there are m functions fi : Ji → {0, 1} such that every binary vector v ∈ {0, 1}n agrees with one of the fi on Ji (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' v|Ji = fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Thus, Kronecker has a winning strategy if and only if the sequence of sets J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , Jm that Cantor queries has a covering assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' The following lemma establishes the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Let J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , Jm ⊆ [n] such that |J1| + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' + |Jm| < m · � log � m n − log m + 1 � − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' (2) Then, J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , Jm has a covering assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Equivalently, if for each vector vi Cantor queries its entries in Ji and Equation 2 holds, then Kronecker has a winning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Let ti = |Ji| and let t = � i ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Assume, without loss of generality, that t1 ≤ t2 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' ≤ tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' To prove a lower bound of the form md for t, where d will be specified later, we show that if t is smaller than md then there are m functions fi : Ji → {0, 1} so that for every possible vector v ∈ {0, 1}n there is i ≤ m so that v|Ji = fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' We do so by explicitly constructing the fi’s (which corresponds to describing a winning strategy for Kronecker).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Starting with the set V = {0, 1}n of all possible potential vectors z, go over the vectors vi in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' In step i we choose the function fi : Ji → {0, 1} such that |{v ∈ V : v|Ji = fi}| is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Since there are 2ti possible choices for fi, the maximizing choice satisfies ���{v ∈ V : v|Ji = fi} ��� ≥ |V | 2ti .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' 6 After picking fi, we remove all the vectors of V that agree with fi and proceed to the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Therefore, after the first i steps, the size of the set V of the remaining vectors is at most 2n i� j=1 (1 − 1/2tj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' We can continue with this analysis until the size of the set V becomes smaller than 1, namely the set becomes empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' It is a bit better, however, to apply a simpler reasoning once the size of V becomes smaller than 2d, and only argue that at least one vector from V is eliminated in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' (Continuing the same analysis as before would only guarantee that V shrinks by a factor of (1 − 1/2ti) which by the choice of d would be roughly 1 − 1/2d < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' To simplify the computation it is not too wasteful to apply the simpler analysis already when the size of V becomes smaller than m/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' If this happens in the first m/2 steps then by removing a single vector in each of the remaining steps we will eliminate all of the vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' This means that if 2n m/2 � j=1 � 1 − 1/2tj� ≤ m 2 then the sequence J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , Jm has a covering assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Since d is such that the total number of queries is m · d, the above amounts to �m/2 j=1 tj ≤ md/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' that is, the average tj for 1 ≤ j ≤ m/2 is at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' This implies that 2n m 2 � j=1 � 1 − 1 2tj � ≤ 2n m 2 � j=1 exp � − 1 2tj � (1 + x ≤ exp(x) for all x ∈ R) = 2n exp � − m 2 � j=1 1 2tj � ≤ 2n exp � − m 2d+1 � , where the last inequality follows because exp(−x) is decreasing and because m 2 � j=1 1 2tj ≥ m 2 · 1 2 1 m/2 �m/2 j=1 tj ≥ m 2 · 1 2d , which follows by convexity of the function f(x) = 2x and because t1 ≤ t2 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' ≤ tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' We have thus shown that if |J1| + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' + |Jm| = m · d such that 2n exp � − m 2d+1 � ≤ m 2 then the sequence J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , Jm has a covering assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' The last inequality surely holds provided m 2d+1 ≥ n + 1 − log m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' That is, provided 2d+1 ≤ m n + 1 − log m, or d ≤ log � m n + 1 − log m � − 1 completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' 7 4 The Cantor-Kronecker Game with m ≥ 2n Assume now that Kronecker’s list V consists of m ≥ 2n binary vectors of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' In this case V may contain all the binary vectors of length n and there is no vector Cantor can output that is different from each vector on Kronecker’s list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' In this regime it is more natural to first focus on the decision problem in which Cantor’s goal is to decide whether V contains {0, 1}n, and if this is not the case, to provide a vector which is not in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='2 Clearly Cantor can achieve this if he queries all mn possible queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Can he do better?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' We first observe that mn queries are in fact needed in the oblivious case: assume that Cantor submits only mn − 1 queries, and leaves the j’th bit of vi unqueried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Then Kronecker may set vi to be the unique occurrence of the all ones vector 1n, and set the remaining m − 1 vectors in V to include all 2n − 1 vectors that are different from the all ones vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Clearly, it is necessary for Cantor to query also the last bit of vi in order to see whether vi is the all ones vector or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Consequently, Cantor must query all mn queries in the oblivious case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' How about the adaptive case?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' A similar argument shows that for m = 2n, Kronecker can force mn = 2nn queries also in the adaptive case, by using a list which contains each binary vector of length n exactly once: indeed, if only mn − 1 bits are queried, then the last, yet unqueried bit, belongs to a vector which occurs only once in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Hence it is necessary to get the value of this bit in order to verify that V contains all 2n vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' The case when m > 2n turns out to be more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Nevertheless, we prove that mn queries are necessary even in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' We start with introducing some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Each step of the game consists of a query by Cantor followed by a response by Kronecker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' The status of the game after each such step is given by an m × n matrix L, where L(i, j) denotes the status of the j’th bit of vi, that is: L(i, j) ∈ {0, 1, ⋆}, where L(i, j) = ⋆ means that the j’th bit of vi was not queried yet, and otherwise L(i, j) equals the value of this bit as answered by Kronecker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' FIXED(L) = � v ∈ L : v ∈ {0, 1}n� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' That is, FIXED(L) is the set of all vectors in L that were fully queried by Cantor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' L is complete if FIXED(L) = {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' A subset S of 2n rows of L is useful if it either contains all the 2n binary vectors of length n, or it can be converted to this set by replacing each ⋆-entry in S by 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' A matrix L is unblocked if it can be completed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' that is, if L has a useful subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Otherwise L is called blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Notice that for m ≥ 2n, the m by n matrix all whose entries are ⋆ is unblocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' As a warmup, and to get used to the definitions, let us assume first that Cantor’s queries the vectors one by one according to their order;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' he first queries all the bits of v1 from left to right, then all the bits of v2 from left to right, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' We use the following strategy for Kronecker: when Cantor queries the j’th bit of vi (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' the value of L(i, j)), Kronecker replies according to the following “0 first” strategy: modified value of L(i, j) = � 1 If setting L(i, j) to 0 blocks L 0 otherwise (3) It is not hard to verify that since Cantor queries the vectors one by one, and from left (most significant bit) to right, the following matrix is produced: each of the first m − 2n + 1 rows will 2Later we will see that the decision and search variants are in fact equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' 8 be set to the all-zeros vector, and the last 2n − 1 rows will be set to the 2n − 1 non zero vectors in increasing lexicographical order: starting with 0n−11 and ending with 1n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Hence Cantor is forced to query all mn entries as in the oblivious case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Interestingly, it turns out that, for any strategy of Cantor, the above “0 first” strategy of Kronecker forces Cantor to make mn queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Let m > 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Then for any strategy of Cantor, the “0 first” strategy of Kronecker forces Cantor to make mn queries in order to determine if L contains {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' In the following we consider an arbitrary execution of the game, where Kronecker follows the “0 first” strategy (and Cantor’s strategy is arbitrary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' We denote by Lt the m × n matrix L after t steps of the game;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' thus L0 is the initial matrix which is filled only with ⋆’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' By the fact that if L is unblocked and L(i, j) = ⋆, then it is possible to set L(i, j) to 0 or to 1 without blocking L, we get: Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' If Lt is unblocked, so is Lt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Hence Lmn is complete;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' it contains {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' We say that a row L(i) is essential for an unblocked matrix L if every useful subset of L’s rows contains L(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Note that if Lt(i) is essential for Lt, then Ls(i) is essential for Ls for all s ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Also, if Lmn(i) is essential for Lmn, then Lmn(i) is equal to a unique vector in {0, 1}n which is different from all other rows of Lmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Assume that Lt(i) is not essential for Lt and Lt(i, j) = ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' If Lt(i, j) is queried at time t + 1, then it is set to 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Lt+1(i, j) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' By the “0 first” strategy, and the fact that if L(i) is not essential for an unblocked matrix L, then setting L(i, j) to 0 does not block L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' By a straightforwards induction Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='8 implies: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' If Lt(i) is not essential for Lt, then Lt(i) contains no 1’s (only 0’s or ⋆’s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Specifically, if Lmn(i) is not essential for Lmn, then Lmn(i) is the zero vector 0n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Hence, every row of Lmn which is not the zero vector is essential, and thus it is different from all other rows of Lmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Let Lmn−1(i, j) be the last bit queried in the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Then Lmn−1(i) is an essential row of Lmn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' To simplify notation, we assume without loss of generality that j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Assume towards contradiction that Lmn−1(i) is not essential for Lmn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='9, this implies that Lmn−1(i) = ⋆0n−1 and Lmn(i) = 0n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Kronecker sets Lmn−1(i, 1) to 0 at Cantor’s mn’th query).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Since Lmn is complete (Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='6), this implies that Lmn−1 contains a distinct occurrence of each of the 2n − 1 nonzero vectors of {0, 1}n, and in particular for some k ̸= i, Lmn−1(k) is the unique row of Lmn−1 which equals 10n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Then, any subset S of Lmn−1 which contains the row Lmn−1(i), the 2n − 2 non zero rows of Lmn−1 excluding Lmn−1(k), and some zero row of Lmn−1 (by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='9 there are m − 2n > 0 such rows in Lmn−1), 9 is a useful subset of Lmn−1 which does not contain Lmn−1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Hence Lmn−1(k) is not essential for Lmn−1, and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='8 Lmn−1(1) = 0 ̸= 1, which stands in contradiction with Lmn−1(1) = 10n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Let Lmn−1(i, j) be the last query in the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='10, Lmn−1(i), and hence also Lmn(i), is essential, meaning that Lmn(i) is different from all other rows of Lmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Thus Cantor must get the value of Lmn−1(i, j) in order to reach a decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' A remark on computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' A naive implementation of the “0 first” strategy might take exponential time: indeed, it requires checking whether setting the queried bit to 0 blocks the current matrix, which involves checking a potentially exponential list of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Nevertheless, we next show that this strategy in fact admits a polynomial time implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Firstly, notice that the first m − 2n steps are trivially efficient, because setting L(i, j) to any value cannot block L (since at least 2n rows of L are not queried yet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Thus it suffices to show that in each later step, deciding whether setting L(i, j) to 0 blocks the matrix, can be performed in time which is polynomial in mn, the size of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Let Lt be the matrix L after t steps of the game, t > m − 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Consider the bipartite graph Gt = (At, B, Et), where At = {Lt(i) : 1 ≤ i ≤ m} is the set of rows of Lt, B = {0, 1}n, and (Lt(i), u) ∈ Et if and only if Lt(i) can be converted to the binary vector u by replacing the ⋆’s in Lt(i) (if any) by binary digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Then, a subset S of Lt is useful for Lt if and only if Gt contains a perfect matching between the vertices in At which correspond to S and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Assume now that we are given the graph Gt, and the corresponding matching, and let Lt(i, j) be the entry queried by Cantor at step t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' To check if setting Lt(i, j) to 0 blocks Lt, we remove from Gt all the edges (Lt(i), u) in which u(j) = 0, and check if the resulted graph contains a perfect matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Since we are given a perfect matching Mt for Gt, and removing these edges eliminates at most one edge from Mt, this checking can be done by executing one phase in some classical algorithm for bipartite matching, which can be done in O(|Et|) = O(m2n) = O(m2) time (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' [Eve11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' 5 Concluding Remarks and Future Research We studied the Cantor-Kronecker game for different values of m and n: when m ≤ n the trivial lower bound of m is tight (a lower bound of m follows because Cantor must query at least one bit in each vector);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' when m ≥ 2n, the trivial upper bound of mn is tight (an upper bound of mn follows because querying all the bits is clearly sufficient);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' when n < m < 2n the landscape is more interesting, and in particular the bounds depend on whether Cantor is adaptive or oblivious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Further Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' We conclude with suggestions for possible future research: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Study the Cantor-Kronecker game when there are r rounds of adaptivity: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' there are r rounds in which Cantor can submit queries, and in each round the submitted queries may depend on Kronecker’s answers to queries from previous rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' How does the query complexity change as a function of r?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Note that r = 1 is the oblivious case and r = ∞ is the adaptive case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' (In fact r = n is already equivalent to r = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Consider the following generalization of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Let k ≤ m, ℓ ≤ n be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Kronecker maintains an m×n binary matrix, and Cantor queries the entries of Kronecker’s matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Cantor’s goal is to find a k × ℓ matrix which does not appear as a submatrix of Kronecker’s m × n matrix, or to decide that one does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' So, the original game 10 is when k = 1, ℓ = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' What is the query complexity as a function of k, ℓ, m, n in the adaptive/oblivious case?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' For which values does Cantor have a strategy that uses strictly less than m · n queries?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Find tighter bounds for the oblivious case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Specifically, notice that Cantor’s original diagonalization provides tight bound on the number of queries needed for the oblivious case when m ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' It will be interesting to derive tight bounds and optimal strategies in the remaining cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' As we exemplify below, this question has connections with natural combinatorial problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Consider the case when m is at the other end of the scale, namely 2n−1 ≤ m < 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Then, Cantor can win the game by querying nm − d bits, where d = 2n − m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' In fact, it suffices that Cantor chooses his queries such that each of the d unqueried entries belongs to a different vector: in this case any assignments of values to the unqueried entries covers (in the sense of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='3) the m − d fully queried vectors, and at most two additional vectors per each of the remaining d vectors (each of which contains one unqueried entry): altogether at most (m − d) + 2d = m + d vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Hence, Cantor is guaranteed to win the game provided that m + d < 2n (equivalently d ≤ 2n − m − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Is the above strategy optimal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=', can Kronecker win the game when Cantor queries only mn − (2n − m) bits?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Informally, Kronecker has a winning strategy if, for any distribution of the 2n − m unqueried entries, there is an assignment which covers sufficiently many vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' This is formalized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='1 (cube(v), J-cube).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Let v be a vector with possibly some unqueried entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' cube(v) is the set of binary vectors which can be obtained by replacing the unqueried entries in v by zeros or ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' In particular, cube(v) = {v} if v is fully queried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' The cube cube(v) is called a J-cube if J = {j : the j′th bit of v is not queried}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' For j ∈ [n], a {j}-cube is denoted by j-edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Assume that Cantor distributes the (2n − m) unqueried entries among vectors v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , vq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Then Kronecker answers to the queried entries define a cube C(vi) for each vector vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Kronecker wins if and only if those cubes cover {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Hence Kronecker has a winning strategy when Cantor uses mn − (2n − m) queries (2n−1 + 1 ≤ m < 2n) if and only if the following holds: Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Let d = 2n − m < 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' For any collection J1, J2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , Jq of nonempty subsets of [n] satisfying �q i=1 |Ji| = d, there are cubes C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , Cq s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Ci is a Ji-cube, and |�q i=1 Ci| ≥ d + q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' The following result of [FHK93] proves Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='2 for the case that each Ji-cube is a ji-edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='3 ([FHK93]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Let d < 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' For any multiset D = {j1, j2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , jd} of elements of [n], {0, 1}n contains a matching {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , ed} s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' , d, ei is a ji-edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' It is also shown in [FHK93] that Conjcture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='2 does not hold when d = 2n−1: in this case a corresponding matching exists if and only if each element in [n] occurs an even number of times in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' This implies that when m = 2n−1 Cantor has a winning strategy with only mn − (2n − m) = mn − 2n−1 queries: he may query n − 1 entries per each vector, so that at least one dimension is left unqueried in an odd number of vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' 11 References [Can74] Georg Cantor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Ueber eine Eigenschaft des inbegriffs aller reellen algebraischen Zahlen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Journal f¨ur die reine und angewandte Mathematik (Crelles Journal), 1(77):258–262, 1874.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' [Eve11] Shimon Even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Graph Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Cambridge University Press, New York, NY, USA, 2nd edition, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' [FHK93] Alexander Felzenbaum, Ron Holzman, and Daniel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Kleitman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Packing lines in a hypercube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Discrete Mathematics, 117(1):107–112, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' [Wik22a] Wikipedia contributors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Georg Cantor — Wikipedia, the free encyclopedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' https: //en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='org/wiki/Georg_Cantor, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' accessed 20-November- 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' [Wik22b] Wikipedia contributors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' Hilbert’s paradox of the Grand Hotel — Wikipedia, the free encyclopedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content='org/wiki/Hilbert’s_paradox_of_the_ Grand_Hotel, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' accessed 20-November-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} +page_content=' 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQf9v6Q/content/2301.01924v1.pdf'} diff --git a/4NE4T4oBgHgl3EQf0w0Q/content/tmp_files/2301.05284v1.pdf.txt b/4NE4T4oBgHgl3EQf0w0Q/content/tmp_files/2301.05284v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7f52ed2e6f02d89bf4fffbbf518cb80b5f168a59 --- /dev/null +++ b/4NE4T4oBgHgl3EQf0w0Q/content/tmp_files/2301.05284v1.pdf.txt @@ -0,0 +1,1123 @@ +Numerical Study of the Rate of Convergence of Chernoff +Approximations to Solutions of the Heat Equation +K.A. Dragunova, A.A. Garashenkova, I.D. Remizov +Research and educational group “Evolution semigroups and their applications” +International Laboratory of Dynamical Systems and Applications +National Research University Higher School of Economics +Email: ivremizov@yandex.ru +MSC2020: 65M12, 47D06, 35K05, 35E15, 35C99 +Keywords: heat equation, initial value problem, operator semigroups, Chernoff approximations, +rate of convergence. +Abstract. Chernoff approximations are a flexible and powerful tool of functional analysis, which +can be used, in particular, to find numerically approximate solutions of some differential equations with +variable coefficients. For many classes of equations such approximations have already been constructed, +however, the speed of their convergence to the exact solution has not been properly studied. +We +developed a program in Python 3 that allows to model a wide class of Chernoff approximations to +a wide class of evolution equations on the real line. +After that we select the heat equation (with +already known exact solutions) as a simple yet informative model example for the study of the rate +of convergence of Chernoff approximations. Examples illustrating the rate of convergence of Chernoff +approximations to the solution of the Cauchy problem for the heat conduction equation are constructed +in the paper. Numerically we show that for initial conditions that are smooth enough the order of +approximation is equal to the order of Chernoff tangency of the Chernoff function used. +We also +consider not smooth enough initial conditions and show how H¨older class of initial condition is related +to the rate of convergence. This method of study can be applied to general second order parabolic +equation with variable coefficients by a slight modification of our Python 3 code, the full text of it is +provided in the appendix to the paper. +Contents +1 +Introduction +2 +2 +Preliminaries +2 +3 +Numerical simulation results +4 +3.1 +Problem setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +3.2 +Approximations for initial condition u0(x) = sin(x) . . . . . . . . . . . . . . . . . . . . +6 +3.3 +Approximations for initial condition u0(x) = | sin(x)|3/2 +. . . . . . . . . . . . . . . . . +7 +3.4 +Approximations for initial condition u0(x) = e−|x| . . . . . . . . . . . . . . . . . . . . . +8 +3.5 +Approximations for initial condition u0(x) = | sin(x)|5/2 +. . . . . . . . . . . . . . . . . +9 +3.6 +Approximations for initial condition u0(x) = | sin(x)|7/2 +. . . . . . . . . . . . . . . . . +9 +3.7 +Approximations for initial condition u0(x) = | sin(x)|9/2 +. . . . . . . . . . . . . . . . . +10 +4 +Discussion +11 +Appendix: Python 3 code +13 +References +14 +1 +arXiv:2301.05284v1 [math.NA] 12 Jan 2023 + +1 +Introduction +Chernoff approximations are a flexible and powerful tool of functional analysis [3, 4, 5], which +can be used, in particular, to find numerically approximate solutions of some differential equa- +tions with variable coefficients, see [2, 14] for an introduction to this topic, and also Preliminaries +section of the paper. For given linear evolution equation the method of Chernoff approximation +generates a sequence of functions un(t, x) that converge to the exact solution u(t, x) of the equa- +tion studied. For arbitrary fixed moment of time t functions x �−→ u(t, x) and x �−→ un(t, x) are +elements of some Banach space, and Chernoff’s theorem guarantees that ∥u(t, ·)−un(t, ·)∥ → 0 +as n → ∞. +To our current knowledge all contributions to a very young “theory of rates of convergence +in Chernoff’s theorem” can be found in [8, 19, 20, 7, 6] and references therein. These papers +provide estimates for the rate of convergence under some conditions but if these conditions are +not satisfied then one can say nothing about the quality of Chernoff approximations. There +are also very few “practical” research papers [10, 16] that measure the speed of convergence in +particular cases obtained via numerical simulations. In our research we continue contributions +to this field of study. +We consider initial value problem for the heat equation +� u′ +t(t, x) = u′′ +xx(t, x) for t > 0, x ∈ R1 +u(0, x) = u0(x) for x ∈ R1 +(1) +which is a good model example because its bounded solution u(t, x) is already known and given +by the formula +u(t, x) = +� +R +Φ(x − y, t)u0(y)dy, where Φ(x, t) = (2 +√ +πt)−1 exp +�−x2 +4t +� +. +Then we obtain Chernoff approximations un(t, x) to the exact solution u(t, x) for n = +1, 2, . . . , 11 and fixed time t = 1/2, and via numerical simulation and linear regression (ordinary +least squares method) discover that +sup +x∈R +|u(t, x) − un(t, x)| ≈ +�1 +n +�β +with a reasonable accuracy (R2 > 0.98). Coefficient β > 0 depends on the smoothness of initial +condition u0 and of the way of constructing the Chernoff approximations. +P.S.Prudnikov in 2020 studied [16] this question in a similar setting, but his approach does +not allow a direct generalization. Meanwhile the simulation method that we use allows to study +not only heat equation, but also equations with variable coefficients. Also we consider more +initial conditions than were studied in [16]. +Now let us provide necessary background on the topic to explain the notion of Chernoff +tangency and Chernoff operator-valued function that are important to understand how we +obtain Chernoff approximations un(t, x). +2 +Preliminaries +Let F be a Banach space. Let L (F) be a set of all bounded linear operators in F. Suppose we +have a mapping V : [0, +∞) → L (F), i.e. V (t) is a bounded linear operator V (t): F → F for +each t ≥ 0. The mapping V is called [5] a C0-semigroup, or a strongly continuous one-parameter +semigroup of operators iff it satisfies the following conditions: +1) V (0) is the identity operator I, i.e. ∀ϕ ∈ F : V (0)ϕ = ϕ; +2 + +2) V maps the addition of numbers in [0, +∞) into the composition of operators in L (F), +i.e. ∀t ≥ 0, ∀s ≥ 0 : V (t + s) = V (t) ◦ V (s), where for each ϕ ∈ F the notation (A ◦ B)(ϕ) = +A(B(ϕ)) = ABϕ is used; +3) V is continuous with respect to the strong operator topology in L (F), i.e. ∀ϕ ∈ F +function t �−→ V (t)ϕ is continuous as a mapping [0, +∞) → F. +The definition of a C0-group is obtained by the substitution of [0, +∞) by R in the paragraph +above. +It is known [5] that if (V (t))t≥0 is a C0-semigroup in Banach space F, then the set +� +ϕ ∈ F : ∃ lim +t→+0 +V (t)ϕ − ϕ +t +� +denote += +Dom(L) +is a dense linear subspace in F. The operator L defined on the domain Dom(L) by the equality +Lϕ = lim +t→+0 +V (t)ϕ − ϕ +t +is called an infinitesimal generator (or just generator to make it shorter) of the C0-semigroup +(V (t))t≥0, and notation V (t) = etL is widely used. +One of the reasons for the study of C0-semigroups is their connection with differential +equations. If Q is a set, then the function u: [0, +∞) × Q → R, u: (t, x) �−→ u(t, x) of two +variables (t, x) can be considered as a function u: t �−→ [x �−→ u(t, x)] of one variable t with +values in the space of functions of the variable x. If u(t, ·) ∈ F then one can define Lu(t, x) = +(Lu(t, ·))(x). If there exists a C0-semigroup (etL)t≥0 then the Cauchy problem for a linear +evolution equation +� u′ +t(t, x) = Lu(t, x) for t > 0, x ∈ Q +u(0, x) = u0(x) for x ∈ Q +(2) +has a unique (in sense of F, where u(t, ·) ∈ F for every t ≥ 0) solution +u(t, x) = (etLu0)(x) +depending on u0 continuously. Compare also different meanings of the solution [5], including +mild solution which solves the corresponding integral equation. Note that if there exists a +strongly continuous group (etL)t∈R then in the Cauchy problem the equation u′ +t(t, x) = Lu(t, x) +can be considered not only for t > 0, but for t ∈ R, and the solution is provided by the same +formula u(t, x) = (etLu0)(x). +Definition 1 (Introduced in [13]). +Let us say that C is Chernoff-tangent to L iff the +following conditions of Chernoff tangency (CT) hold: +(CT0). Let F be a Banach space, and L (F) be a space of all linear bounded operators +in F. Suppose that we have an operator-valued function C: [0, +∞) → L (F), or, using other +words, we have a family (C(t))t≥0 of linear bounded operators in F. Closed linear operator +L: Dom(L) → F is defined on the linear subspace Dom(L) ⊂ F which is dense in F. +(CT1) Function t �−→ C(t)f ∈ F is continuous for each f ∈ F. +(CT2) C(0) = I, i.e. C(0)f = f for each f ∈ F. +(CT3) There exists such a dense subspace D ⊂ F that for each f ∈ D there exists a limit +C′(0)f = lim +t→0 +C(t)f − f +t +. +(CT4) The closure of the operator (C′(0), D) is equal to (L, Dom(L)). +Remark 1. Let us consider one-dimensional example F = L (F) = R. Then g: [0, +∞) → +R is Chernoff-tangent to l ∈ R iff g(t) = 1 + tl + o(t) as t → +0. +Theorem 1 (P. R. Chernoff (1968), see [5, 3]). Let F and L (F) be as above. Suppose +3 + +that the operator L: F ⊃ Dom(L) → F is linear and closed, and function C takes values in +L (F). Suppose that these assumptions are fulfilled: +(E) There exists a C0-semigroup (etL)t≥0 with the infenitesimal generator (L, Dom(L)). +(CT) C is Chernoff-tangent to (L, Dom(L)). +(N) There exists such a number ω ∈ R, that ∥C(t)∥ ≤ eωt for all t ≥ 0. +Then for each f ∈ F we have (C(t/n))nf → etLf as n → ∞ with respect to norm in F +uniformly with respect to t ∈ [0, T] for each T > 0, i.e. +lim +n→∞ sup +t∈[0,T] +��etLf − (C(t/n))nf +�� = 0. +Remark 2. In our one-dimensional example (F = L (F) = R) the Chernoff theorem says +that etl = limn→∞ g(t/n)n = limn→∞(1 + tl/n + o(t/n))n, which is a simple fact of calculus. +Definition 2. Let F, L (F), L be as above. If C is Chernoff-tangent to L and the equation +limn→∞ supt∈[0,T] +��etLf − (C(t/n))nf +�� = 0 holds, then C is called a Chernoff function for the +operator L, and the (C(t/n))nf is called a Chernoff approximation expression to etLf. +Remark 3. If L is a linear bounded operator in F, then etL = �+∞ +k=0(tL)k/k! where the +series converges in the usual operator norm topology in L (F). When L is not bounded (such +as Laplacian and many other differential operators), expressing (etL)t≥0 in terms of L is not +an easy problem that is equivalent to the problem of finding (for each u0 ∈ F) the F-valued +function U that solves the Cauchy problem U ′(t) = LU(t); U(0) = u0. If one finds this solution, +then etL is obtained for each u0 ∈ F and each t ≥ 0 in the form etLu0 = U(t). +Remark 4. In the definition of the Chernoff tangency the family (C(t))t≥0 usually does +not have a semigroup composition property, i.e. C(t1 + t2) ̸= C(t1)C(t2), while (etL)t≥0 has it: +et1Let2L = e(t1+t2)L. However, each C0-semigroup (etL)t≥0 is Chernoff-tangent to its generator +L and appears to be it’s Chernoff function. When coefficients of the operator L are variable, +usually there is no simple formula for etL due to the remark 3. On the other hand, even in this +case one can find rather simple formula to construct Chernoff function C for the operator L, +because there is no need to worry about the composition property, and then obtain etL in the +form etL = limn→∞ C(t/n)n via the Chernoff theorem. +3 +Numerical simulation results +3.1 +Problem setting +Definition 3. +We say that operator-valued function C is Chernoff-tangent of order k to +operator L iff C is Chernoff-tangent to L in the sense of definition 1 and the following condition +(CT3-k) holds: +There exists such a dense subspace D ⊂ F that for each f ∈ D we have +C(t)f = +� +I + tL + 1 +2t2L2 + . . . + 1 +k!tkLk +� +f + o(tk) as t → 0. +(CT3 − k) +Remark 5. It is clear that for k = 1 condition (CT3-k) becomes just (CT3). For the semigroup +C(t) = etL condition (CT3-k) holds for all k = 1, 2, 3, . . . So one can expect that the bigger k +is the better rate of convergence C(t/n)nf → etLf as n → ∞ will be, if f belongs to the space +D. This idea was proposed in [12], where two conjectures about the convergence speed were +formulated explicitly, and one of them were recently proved in [7, 6]. For initial conditions that +are good enough and t fixed, Chernoff function with Chernoff tangency of order k by conjecture +should provide ∥u(t, ·) − un(t, ·)∥ = O(1/nk) as n → ∞. However, if f ̸∈ D then nothing is +known on the rate of convergence. In the present paper we are starting to fill this gap for +4 + +operator L given by (Lf)(x) = f ′′(x) for all x ∈ R and all bounded, infinitely smooth functions +f: R → R, and k = 1, 2. +Problem setting. In the initial value problem (2) consider Q = R, and Banach space +F = UCb(R) of all bounded, uniformly continuous functions f: R → R endowed with the +uniform norm ∥f∥ = supx∈R |f(x)|. Consider operator L given by (Lf)(x) = f ′′(x) for all +x ∈ R and all f ∈ D = C∞ +b (R) of all infinitely smooth functions R → R that are bounded with +all the derivatives. Then (2) reads as (1). Cauchy problem (1) is a constant (one, zero, zero) +coefficients particular case of the Cauchy problem considered in [15], and the corresponding +Chernoff function was found in [15]. The particular case of this Chernoff function reads as +(G(t)f)(x) = 1 +2f(x) + 1 +4f(x + 2 +√ +t) + 1 +4f(x − 2 +√ +t) +where we write G(t) instead of C(t) in order to show that C(t) is a general abstract Chernoff +function for some operator L, meanwhile G(t) is this particular above-given Chernoff function +for operator d2/dx2. It was proved in [15] that G(t) is first order Chernoff-tangent to d2/dx2. +A.Vedenin (see [19]) proposed another Chernoff function for operator L considered in [15], +and the constant coefficient particular case of this operator is d2/dx2. The particular case of +the Chernoff function obtained by A.Vedenin reads as +(S(t)f)(x) = 2 +3f(x) + 1 +6f(x + +√ +6t) + 1 +6f(x − +√ +6t), +and it was proved by A.Vedenin that S(t) is second order Chernoff-tangent to d2/dx2. +In the paper we study how supx∈R |u(t, x) − un(t, x)| depends on n while t = 1/2 is fixed +and un(t, x) is given by +un(t, x) = (C(t/n)nu0)(x) +where C ∈ {G, S}, C(t/n) is obtained by substitution of t by t/n in the formula that defines +C(t), and C(t/n)n = C(t/n)C(t/n) . . . C(t/n) is a composition of n copies of linear bounded +operator C(t/n). We consider several initial conditions u0 that are all H¨older continuous (hence +all belong to the UCb(R) space) but have different H¨older exponents. Then we remark how the +rate of tending of supx∈R |u(t, x) − un(t, x)| to zero depends on these H¨older exponents and the +order of Chernoff tangency (which is 1 for G(t), and 2 for S(t)). +Comments on computational techniques. Calculations were performed in the Python +3 environment using a program we wrote and which is available in the Appendix. All measure- +ments, for the sake of reducing computational complexity, for each value of n (varying from 1 to +11) were carried out for 1000 points uniformly dividing the segment [−π, π] or [−2π, 2π]. Initial +conditions of the form u0(x) = | sin x|α for various α ∈ {9/2, 7/2, 5/2, 3/2, 1, 3/4, 1/2, 1/4}, like +any of Chernoff approximations based on them, are periodic functions. So, the standard norm +in UCb(R), namely +d = ∥un(t, ·) − u(t, ·)∥ = sup +x∈R +|un(t, x) − u(t, x)|, +where u is the exact solution of (1) and un is the Chernoff approximation, is reached at the +interval corresponding to the period. +The program code was written with the possibility to set any operator and any initial con- +dition, i.e. without simplifying Chernoff functions and using binomial coefficients, in contrast +to the work [16] published earlier. Moreover, the initial condition does not necessarily have +to be a smooth function. The number of iterations is not limited to 11, the value n can be +changed, both upward and downward. We have chosen the optimal value n since the program +is very time consuming: via Jupyter Notebook 6.1.4 Anaconda 3 Python 3.8.3 set on personal +computer with Windows 10, CPU Intel Core i5-1035G1, 1.0-3.6 GHz, 8 Gb RAM it takes about +20 minutes to complete the program for all initial conditions with construction of graphs for +5 + +them. At the research stage of the new method (Chernoff approximations) this is acceptable, +but in the future, of course, the code will be optimized for a better speed, since this is important +in practice. Our goal is to continue research and in the future write a library that allows to +solve partial derivative equations in this way. +3.2 +Approximations for initial condition u0(x) = sin(x) +Let us first analyze the approximations for the initial condition u0(x) = sin x. +fig. 1.1, n = 1, u0(x) = sin x, t = 1 +2 +Figure 1.1 shows the exact solution, which coincides with the graph of the function y = +e−1/2 sin x, and approximate solutions for the functions S(t) (left) and G(t) (right) at n = 1. +The initial condition u0 = sin x is very good, since its derivatives of any order exist, have +no discontinuities and are bounded. And already at n = 1 the function S(t) gives a good +approximation. +Figure 1.2 below shows plots of the decreasing error of Chernoff approximations as a function +of n, where 1 ≤ n ≤ 11. On the left are plots of decreasing error for Chernoff functions S(t) +(in blue) and G(t) (in green) in regular scale, and on the right – the same plots in logarithmic +scale. The graph in the logarithmic scale allows us to estimate how much the convergence rate +for the function G(t) is less than the convergence rate for the function S(t). Here and through +all the paper we use the following notation: +d = ∥un(t, ·) − u(t, ·)∥ = sup +x∈R +|un(t, x) − u(t, x)|. +fig. 1.2, 1 ≤ n ≤ 11, u0(x) = sin x, t = 1 +2 +6 + +Approximetion of the sohution via Chemoff fmctin S(t) +Approximetion of the sohution via Chemoff fmcticm G(t) +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0.0 +-2 +2 +-2 +-1 +0.0 +.3 +13 +i +2 +3 +02 +7 +0.4 +0.4 +Chernoff approximation +Chernoff approximation +0.6 +Solution +0.6 +SolutionNorm decay +Logarithmic dependence +R2=0,9949 +Discrepancy (S(t)f)(x) +-4 +Discrepancy (G(t)f)(x) +0.025 +-5 +R"=0.9949 +0.020 +-6 +0.015 +Discrepancy (S(t)f)(x) +Discrepancy (G(t)f)(x) +-7 +0.010 +-8 +0.005 +6- +0.000 +. +10 +2 +6 +8 +10 +0'0 +0.5 +10 +1'5 +2.0 +2.5 +n +In nYou can see that the points on the right graph lie on the straight lines with good accuracy. +Using the method of least squares (in Excel) we found the equations of these lines. Rounding +off the coefficients, we see that for the blue line the equation is as follows: +ln(d) = −2.092 ln(n) − 5.0671, i.e. d = n−2.092e−5.0671 = 0.0063 +n2.092 . +Similarly, for the green line, the equation ln(d) = −1.0416 ln(n) − 3.5796, i.e. +d = n−1.0416e−3.5796 = 0.0279 +n1.0416. +Using the same approach, we study the behavior of the error for other initial conditions. +3.3 +Approximations for initial condition u0(x) = | sin(x)|3/2 +fig. 5.1, n = 4, u0(x) = | sin(x)|3/2, t = 1 +2 +fig. 5.2, 1 ≤ n ≤ 11, u0(x) = | sin(x)|3/2, t = 1 +2 +The line (green) corresponding to the decreasing error of the function G(t) in the logarithmic +scale was constructed without taking into account n = 1. +For the green line (see Fig. 5.2, right) the equation ln(d) = −0.9785 ln(n) − 2.8973, i.e. +d = n−0.9785e−2.8973 = 0.0552 +n0.9785. +Similarly, for the blue line (see Figure 5.2), the equation is as follows: ln(d) = −1.5109 ln(n)− +1.8234, i.e. d = n−1.5109e−1.8234 = 0.1615 +n1.5109. +As can be seen from Figure 5.2, the difference between the error decay rates using Chernoff +functions S(t) and G(t) for u0(x) = | sin(x)|3/2 is larger than for u0(x) = | sin x|. This is due to +the greater smoothness of u0(x) = | sin(x)|3/2. +7 + +Approximetion of the sohution via Chemoff fmcticn S(t) +Approximetion of the sohution via Chemoff fmcticn G(t) +0.62 +0.62 +0.60 +0.60 +0.58 +0.58 +0.56 +0.56 +0.54 +0.54 +4152 +0.5b +0.48 +Chernoff approximation +0.50 +Chernoff approximation +Solution +Solution +-3 +-2 +-1 +0 +1 +2 +3 +-3 +-2 +-1 +0 +1 +2 +3Norm decay +Logarithmic dependence +0.16 +=0.9976 +Discrepancy (S(t)f)(x) +2.0 +Discrepancy (G(t)f)(x) +0.14 +2.5 +0.12 +3.0 +0.10 +R2=0,9903 +Discrepancy (S(t)f)(x) + 3.5 +0.08 +Discrepancy (G(t)f)(x) +4.0 +0.06 +0.04 +4.5 +0.02 +5.0 +0.00 +5.5 +2 +4 +6 +8 +10 +0'0 +0.5 +10 +15 +2.0 +2.5 +n +In n3.4 +Approximations for initial condition u0(x) = e−|x| +Let us consider a non-smooth and non-periodic function e−|x| as an initial condition. +fig. 6.1, n = 4, u0(x) = e−|x|, t = 1 +2 +fig. 6.2, 1 ≤ n ≤ 11, u0(x) = e−|x|, t = 1 +2 +Figures 6.1 and 6.2 show plots of the exact solution, approximations to the solution, and +rates of convergence of the error to zero. As can be seen, the result is similar: the conver- +gence rate of the function S(t) is higher than that of G(t), but the order of convergence is +approximately the same, as can be seen from the fact that the lines are almost parallel. +For the green line (see Fig. 6.2, right), the equation is as follows: ln(d) = −0.9294 ln(n) − +2.3832, i.e. d = n−0.9294e−2.3832 = 0.0923 +n0.9294. +Similarly, for the blue line (see Figure 6.2) the equation is as follows: ln(d) = −1.056 ln(n)− +1.5543, i.e. d = n−1.056e−1.5543 = 0.2113 +n1.5543. +8 + +Approximetion of the sohution via Chemoff fmcticn S(t) +Approximetion of the sohution via Chemoff fumcticn G(t) +- 9'0 +0.5/ +0.5 +Q4 +0/4 +0.3 +0.3 +0.2 +0.2 +0.1 +0.1 +Chernoff approximation +Chernoff approximation +0.0 +Solution +0.0 +Solution +10.0 +7.5 +5.0 +2.5 +0.0 +25 +5.D +7.5 +1±.0 +-10.0 +7.5 +0'5- +2.5 +0.D +25 +5.D +7.5 +1±.0Norm decay +Logarithmic dependence +0.200 +1.5 +Discrepancy (S(t)f)(x) +6660-_ +Discrepancy (G(t)f)(x) +0.175 +2.0 +0.150 +2.5 +0.125 +Discrepancy (S(t)f)(x) +R +=0.9979 + 0.100 +Discrepancy (G(t)f)(x) +0.075 +3.5 +0.050 +4.0 +0.025 +4.5 +2 +6 +8 +10 +0'0 +0.5 +10 +1'5 +2.0 +2.5 +n +In n3.5 +Approximations for initial condition u0(x) = | sin(x)|5/2 +fig. 7.1, n = 4, u0(x) = | sin(x)|5/2, t = 1 +2 +fig. 7.2, 1 ≤ n ≤ 11, u0(x) = | sin(x)|5/2, t = 1 +2 +The lines (green and blue) corresponding to the decreasing error of the functions G(t) and +S(t) in the logarithmic scale was constructed without taking into account n = 1 and n = 2. +3.6 +Approximations for initial condition u0(x) = | sin(x)|7/2 +fig. 8.1, n = 4, u0(x) = | sin(x)|7/2, t = 1 +2 +9 + +Approximetion of the sohution via Chemoff fmcticn S(t) +Approximetion of the soution via Chemoff fumcticn G(t) +0.52 +0.52 +0.50 +0.50 +0.48 +0.48 +0.46 +0.46 +0.44 +0.44 +1.42 +442 +00 +Chernoff approximation +0.40 +Chernoff approximation +Solution +Solution +E- +-2 +-1 +0 +1 +2 +3 +-3 +-2 +-1 +0 +1 +2 +3Norm decay +Logarithmic dependence +0.14 +2 +Discrepancy (S(t)f)(x) +Discrepancy (G(t)f)(x) +0.12 +E- +0.10 +0.08 +-4 +Discrepancy (S(t)f)(x) +2 +p ul +=0.9963 +Discrepancy (G(t)f)(x) +0.06 +-5 +0.04 +-6 +0.02 +00°0 +2 +6 +-7 +4 +8 +10 +0'0 +0.5 +10 +15 +2.0 +2.5 +n +In nApproximetion of the sohution via Chemoff fmcticn S(t) +Approximetion of the soution via Chemoff fumcticn G(t) +0.46 +0.46 +0.44 +0.44 +0.42 +0.42 +0.40 +0.40 +0.38 +0.38 +1.36 +436 +0.14 +Chernoff approximation +0.34 +Chernoff approximation +Solution +Solution +E- +-2 +-1 +0 +1 +2 +3 +-3 +-2 +-1 +0 +1 +2 +3fig. 8.2, 1 ≤ n ≤ 11, u0(x) = | sin(x)|7/2, t = 1 +2 +The line (green) corresponding to the decreasing error of the function G(t) in the logarithmic +scale was constructed without taking into account n = 1 and n = 2. +3.7 +Approximations for initial condition u0(x) = | sin(x)|9/2 +fig. 9.1, n = 4, u0(x) = | sin(x)|9/2, t = 1 +2 +fig. 9.2, 1 ≤ n ≤ 11, u0(x) = | sin(x)|9/2, t = 1 +2 +The line (green) corresponding to the decreasing error of the function G(t) in the logarithmic +scale was constructed without taking into account n = 1 and n = 2. +10 + +Norm decay +Logarithmic dependence +0.200 +R=0.9864 +: +Discrepancy (S(t)f)(x) +-2 +Discrepancy (G(t)f)(x) +0.175 +0.150 +-3 +0.125 +Discrepancy (S(t)f)(x) +-4 +p ul +0,9753 + 0.100 +Discrepancy (G(t)f)(x) +-5 +0.075 +0.050 +-6 +0.025 +-7 +0.000 +4 +6 +8 +10 +0'0 +0.5 +10 +1'5 +2.0 +2.5 +n +In nApproximetion of the sohution via Chemoff fmcticn S(t) +Approximetion of the soution via Chemoff fumcticn G(t) +0.42 +0.42 +0.40 +0.40 +0.38 +0.38 +0.36 +0.36 +0.34 +0.34 +.32 +432 +0.30 +Chernoff approximation +Chernoff approximation +Solution +Solution +0.28 +E- +-2 +-1 +0 +1 +2 +3 +-3 +-2 +-1 +0 +1 +2 +3Norm decay +Logarithmic dependence +0.25 +Discrepancy (S(t)f)(x) +-1 +R +=0.9777 +Discrepancy (G(t)f)(x) +0.20 +-2 +E- +0.15 +Discrepancy (S(t)f)(x) +-4 +R°=0,9309 +Discrepancy (G(t)f)(x) +0.10 +-5 +-6 +0.05 +-7 +0.00 +-8 +2 +6 +10 +0'0 +0.5 +10 +1'5 +2.0 +2.5 +n +In n4 +Discussion +The table below shows experimentally (using simulation in Python 3) the orders of decreas- +ing of error depending on the smoothness class of the initial condition and the Chernoff function. +The smoothness class of the initial +condition u0 +Order of decreasing er- +ror +on +the +Chernoff +function +G(t), +which +has the 1st order of +the +Chernoff +tangent +to the operator L = +d2 +dx2 +Order of decreasing er- +ror +on +the +Chernoff +function +S(t), +which +has the 2nd order of +tangency by Chernoff +to the operator L = +d2 +dx2 +C∞, i.e. +all derivatives exist and are +bounded, u0(x) = sin(x) +-1.0416 +-2.092 +C4 1 +2 , the first, second, third, and fourth +derivatives exist and are bounded, and +the fourth is H¨older with a H¨older ex- +ponent 1/2, u0(x) = | sin(x)|9/2 +-1.0212, the regression was +done without n = 1, n = 2 +-3.1219, but the points do +not fit well on a straight +line, so the number is un- +informative +C3 1 +2 , the first, second, and third deriva- +tives exist and are bounded, and the +third is H¨older with H¨older exponent +1/2, u0(x) = | sin(x)|7/2 +-1.4013,regression +was +done without considering +n = 1, n = 2, but the +points do not lie well on +the line, so the number is +uninformative +-2.5045, but the points do +not lie well on the line, so +the number is uninforma- +tive +C2 1 +2 , the first and second derivative +exist and are bounded, while the sec- +ond derivative is H¨older continuous +with H¨older exponent 1/2, u0(x) = +| sin(x)|5/2 +-1.1433, +regression +was +done without considering +n = 1, n = 2 +-1.7923, +regression +was +done without considering +n = 1, n = 2 +C1 1 +2 , +the +first +derivative: +exists, +is +bounded +and +H¨older +continuous +with H¨older exponent 1/2, u0(x) = +| sin(x)|3/2 +-0.9785, the regression was +done without considering +n = 1 +-1.5109 +H1, the H¨older with the H¨older expo- +nent 1, u0(x) = | sin(x)| +-1.0508 +-1.0948 +H1,the H¨older with the H¨older expo- +nent 1, u0(x) = e−|x| +-0.9294 +-1.056 +H3/4, the H¨older with the H¨older expo- +nent 3/4, u0(x) = | sin(x)|3/4 +-0.815 +-0.9262 +H1/2, the H¨older with the H¨older expo- +nent 1/2, u0(x) = | sin(x)|1/2 +-0.6905 +-0.7723 +H1/4, the H¨older with the H¨older expo- +nent 1/4, u0(x) = | sin(x)|1/4 +-0.6138 +-0.6653 +We see that on the initial condition with high smoothness (first line in the table), the first +order of Chernoff tangency corresponds to a decreasing error rate of about 1/n, and the second +order – a decreasing rate of about 1/n2. This is in accordance with the conjecture from [12] +and theorem from [6]. +As the smoothness is lost (second line in the table and below), theory from [6] stops working, +and the experimental evidence is the following: the convergence speed gradually decreases and +the advantages of the Chernoff function with the second order of Chernoff tangency gradually +vanish. Let us present the results from the table graphically: +11 + +fig. 10 +The regression was carried out without taking into account the point (2.5; -1.7923). The +equation of the approximating line: y = −0.684x − 0.4467.This may be interpreted as follows: +when the smoothness class α of the initial condition u0 is not greater than the order of Chernoff +tangency then +d = ∥un(t, ·) − u(t, ·)∥ = sup +x∈R +|un(t, x) − u(t, x)| ≈ const · +�1 +n +�0.68α+0.45 +. +Meanwhile when the smoothness class α of the initial condition u0 is greater than the order of +Chernoff tangency then there is no such easy-to-state dependence but still Chernoff function +S(t) with the second order Chernoff tangency provides better approximations than Chernoff +function G(t) with the first order Chernoff tangency. +Conclusion. +The results of the numerical simulation are generally in agreement with +and confirm the theory arising from the conjecture in [12]. However, some of the points that +do not lie on straight lines exactly. This deserve closer attention: n = 11 for some initial +conditions is not sufficient to derive conclusions about the asymptotic behavior of the calculation +error. For not smooth initial conditions that we studied numerically there are not known any +theoretical bounds on the rate of convergence. And, of course, the most interesting case of +variable coefficients should be considered, understanding them as parameters analogously with +u0. So the research in this direction is far from ending. +Acknowledgements. The publication was prepared within the framework of the Academic +Fund Program at HSE University in 2021 (grant №20-04-022) by the Research and educational +group ”Evolution semigroups and their applications”. Authors are thankful to Prof. O.E.Galkin +and other members of the group for discussions of the results presented in the paper. +12 + +0.6 +S(t) +0.8 +(a)9 +1.0 +1.2 +decre +1.4 + 1.6 +1.8 +R2 = 0,9854 +2.0 +2.2 +0.5 +1.0 +1.5 +2.0 +2.5 +The smoothness class of the initial conditionAppendix: Python 3 code +13 + +L.1Moduliandvariables +In [1]: +# importing moduli +importmatplotlib.pyplotasplt +from sympy import oo +from scipy importintegrate +import numpy as np +importmath +In [2]: +# variables declaration +tau = 1/2 +n = 11 +In[3]: +1=[] +fori inrange(1,n+1): +1.append(math.log(i))1.2 Functions and operators +# almost self-contained class +classclassoffunctions(object): +def thefunction(self, x): +return x +# Chernoff function (s(t)f)(x)=(2/3)f(x) + (1/6)f(x + (6t)^(1/2)) + (1/6)f(x - (6t)^(1/2)) +defoper(inputobject:classoffunctions,t): +outputobject=inputobject +f= inputobject.thefunction +def funct(x): +return (2/3)*f(×) +(1/6)*f(× + (6*t)**(1/2)) + (1/6)*f(x- (6*t)**(1/2)) +outputobject.thefunction=funct +returnoutputobject +# Chernoff function (G(t)f)(x) = (1/4)* f(x+2t^(1/2))+(1/4) f(x-2t^(1/2))+(1/2)f(x) +defoperl(inputobject:classoffunctions,t): +outputobject=inputobiect +f=inputobject.thefunction +def funct(x): +return (1/4)*f(x+2*t**(1/2))+(1/4)*f(x-2*t**(1/2))+(1/2)*f(×) +outputobject.thefunction=funct +return outputobjectReferences +[1] Bogachev V.I., Smolyanov O.G. Real and Functional Analysis. — Springer, 2020. +[2] Butko Ya.A. The method of Chernoff approximation. Springer Proceedings in Mathematics +and Statistics. Volume 325. — Springer, Cham, 2020. Pp. 19–46. +[3] Chernoff P.R. Note on product formulas for operator semigroups.// J. Functional Analysis +2:2 (1968), 238-242. +[4] Engel K.-J., Nagel R. A. Short Course on Operator Semigroups. — N.Y. Springer Science, +Business Media, 2006. +[5] Engel K.-J., Nagel R. One-Parameter Semigroups for Linear Evolution Equations. — +Springer, 2000. +[6] Galkin O.E., Remizov I.D. Rate of Convergence of Chernoff Approximations of operator +C0-semigroups.// Mathematical Notes, to appear (2022) +[7] Galkin O. E., Remizov I. D. Upper and lower estimates for rate of convergence in Chernoff’s +product formula for semigroups of operators // https://arxiv.org/abs/2104.01249, 2021 +[8] Gomilko A., Kosowicz S., Tomilov Yu. A general approach to approximation theory of +operator semigroups. // Journal de Math´ematiques Pures et Appliqu´ees. 127 (2019), 216– +267. +14 + +# composition degree of Chernoff function (s(t)f)(x) +def degr(g, tau, n): +y=[] +obj=classoffunctions() +for n_p in range(1, n + 1): +objk=obj +obj_k.thefunction = g +for k in range(1, n_p + 1): +objk=oper(objk,tau/np) +y.append(obj_k.thefunction) +return y +# composition degree of Chernoff function (G(t)f)(x) +def degrl(g, tau, n): +y=[] +obj=classoffunctions() +for n_p in range(1, n + 1): +obj_k = obj +obj_k.thefunction =g +for k in range(1, n_p + 1): +obj k = oper1(obj k,tau/n p) +y.append(obj_k.thefunction) +return y +#normcomputation +def norm(y, sol): +[] = p +for n p in range(O, n): +d.append(np.max(np.abs(sol -y[n_pj(x)))) +return d[9] Hille E., Phillips R. S. Functional Analysis and Semi-Groups. — American Mathematical +Society, 1975. +[10] Orlov Yu.N., Sakbaev V.Zh., Smolyanov O.G. Rate of convergence of Feynman approxima- +tions of semigroups generated by the oscillator Hamiltonian. // Theoret. and Math. Phys. +172:1 (2012), 987–1000. +[11] Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. +— Springer-Verlag, 1983. +[12] Remizov I.D. On estimation of error in approximations provided by Chernoff ’s product +formula.// International Conference "ShilnikovWorkshop-2018" dedicated to the memory of +outstanding Russian mathematician Leonid Pavlovich Shilnikov (1934-2011), Lobachevsky +State University of Nizhny Novgorod, December 17-18, 2018 Book of abstracts, pp.38-41 +[13] Remizov I.D. A method of obtaining the evolution operator for the Schr¨odinger equation +Quasi-Feynman formulas.// J. Funct. Anal. 270:12, (2016), 4540-4557. +[14] Remizov I.D. Feynman and Quasi-Feynman Formulas for Evolution Equations// Doklady +Mathematics, 96:2 (2017), 433-437 +[15] Remizov I. D. Approximations to the solution of Cauchy problem for a linear evolution +equation via the space shift operator (second-order equation example)// Applied Mathe- +matics and Computation, 328, 243-246, 2018 +[16] Prudnikov P.S. Speed of convergence of Chernoff approximations for two model examples: +heat equation and transport equation// arXiv:2012.09615 [math.FA] (2020). +[17] Smolyanov O.G., Tokarev A.G.,Truman A. Hamiltonian Feynman path integrals via the +Chernoff formula.// J. Math. Phys. 43. 10 (2002), 5161-5171. +[18] Vedenin A.V. Fast converging Chernoff approximations to solution of a parabolic differ- +ential equation on a real line.// International school-conference Mathematical spring 2019, +Russia, Nizhny Novgorod 2-5 May 2019, Book of abstracts, pp 22-23 +[19] Vedenin A.V., Voevodkin V.S., Galkin V.D., Karatetskaya E.Yu., Remizov I.D. . Speed +of Convergence of Chernoff Approximations to Solutions of Evolution Equations. // Math. +Notes, 108:3 (2020), 451–456. +[20] Zagrebnov V.A. Notes on the Chernoff product formula. // J. Funct. Anal. 279:7 (2020), +108696 +15 + diff --git a/4NE4T4oBgHgl3EQf0w0Q/content/tmp_files/load_file.txt b/4NE4T4oBgHgl3EQf0w0Q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..49b6b25bfb6b6cd7d88af612c0cf68f83699e5a0 --- /dev/null +++ b/4NE4T4oBgHgl3EQf0w0Q/content/tmp_files/load_file.txt @@ -0,0 +1,810 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf,len=809 +page_content='Numerical Study of the Rate of Convergence of Chernoff Approximations to Solutions of the Heat Equation K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Dragunova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Garashenkova, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Remizov Research and educational group “Evolution semigroups and their applications” International Laboratory of Dynamical Systems and Applications National Research University Higher School of Economics Email: ivremizov@yandex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='ru MSC2020: 65M12, 47D06, 35K05, 35E15, 35C99 Keywords: heat equation, initial value problem, operator semigroups, Chernoff approximations, rate of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Chernoff approximations are a flexible and powerful tool of functional analysis, which can be used, in particular, to find numerically approximate solutions of some differential equations with variable coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' For many classes of equations such approximations have already been constructed, however, the speed of their convergence to the exact solution has not been properly studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' We developed a program in Python 3 that allows to model a wide class of Chernoff approximations to a wide class of evolution equations on the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' After that we select the heat equation (with already known exact solutions) as a simple yet informative model example for the study of the rate of convergence of Chernoff approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Examples illustrating the rate of convergence of Chernoff approximations to the solution of the Cauchy problem for the heat conduction equation are constructed in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Numerically we show that for initial conditions that are smooth enough the order of approximation is equal to the order of Chernoff tangency of the Chernoff function used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' We also consider not smooth enough initial conditions and show how H¨older class of initial condition is related to the rate of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' This method of study can be applied to general second order parabolic equation with variable coefficients by a slight modification of our Python 3 code, the full text of it is provided in the appendix to the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Contents 1 Introduction 2 2 Preliminaries 2 3 Numerical simulation results 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='1 Problem setting .' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='7 Approximations for initial condition u0(x) = | sin(x)|9/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 10 4 Discussion 11 Appendix: Python 3 code 13 References 14 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='05284v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='NA] 12 Jan 2023 1 Introduction Chernoff approximations are a flexible and powerful tool of functional analysis [3, 4, 5], which can be used, in particular, to find numerically approximate solutions of some differential equa- tions with variable coefficients, see [2, 14] for an introduction to this topic, and also Preliminaries section of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' For given linear evolution equation the method of Chernoff approximation generates a sequence of functions un(t, x) that converge to the exact solution u(t, x) of the equa- tion studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' For arbitrary fixed moment of time t functions x �−→ u(t, x) and x �−→ un(t, x) are elements of some Banach space, and Chernoff’s theorem guarantees that ∥u(t, ·)−un(t, ·)∥ → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' To our current knowledge all contributions to a very young “theory of rates of convergence in Chernoff’s theorem” can be found in [8, 19, 20, 7, 6] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' These papers provide estimates for the rate of convergence under some conditions but if these conditions are not satisfied then one can say nothing about the quality of Chernoff approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' There are also very few “practical” research papers [10, 16] that measure the speed of convergence in particular cases obtained via numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' In our research we continue contributions to this field of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' We consider initial value problem for the heat equation � u′ t(t, x) = u′′ xx(t, x) for t > 0, x ∈ R1 u(0, x) = u0(x) for x ∈ R1 (1) which is a good model example because its bounded solution u(t, x) is already known and given by the formula u(t, x) = � R Φ(x − y, t)u0(y)dy, where Φ(x, t) = (2 √ πt)−1 exp �−x2 4t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Then we obtain Chernoff approximations un(t, x) to the exact solution u(t, x) for n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' , 11 and fixed time t = 1/2, and via numerical simulation and linear regression (ordinary least squares method) discover that sup x∈R |u(t, x) − un(t, x)| ≈ �1 n �β with a reasonable accuracy (R2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Coefficient β > 0 depends on the smoothness of initial condition u0 and of the way of constructing the Chernoff approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='Prudnikov in 2020 studied [16] this question in a similar setting, but his approach does not allow a direct generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Meanwhile the simulation method that we use allows to study not only heat equation, but also equations with variable coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Also we consider more initial conditions than were studied in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Now let us provide necessary background on the topic to explain the notion of Chernoff tangency and Chernoff operator-valued function that are important to understand how we obtain Chernoff approximations un(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 2 Preliminaries Let F be a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Let L (F) be a set of all bounded linear operators in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Suppose we have a mapping V : [0, +∞) → L (F), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' V (t) is a bounded linear operator V (t): F → F for each t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' The mapping V is called [5] a C0-semigroup, or a strongly continuous one-parameter semigroup of operators iff it satisfies the following conditions: 1) V (0) is the identity operator I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' ∀ϕ ∈ F : V (0)ϕ = ϕ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 2 2) V maps the addition of numbers in [0, +∞) into the composition of operators in L (F), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' ∀t ≥ 0, ∀s ≥ 0 : V (t + s) = V (t) ◦ V (s), where for each ϕ ∈ F the notation (A ◦ B)(ϕ) = A(B(ϕ)) = ABϕ is used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 3) V is continuous with respect to the strong operator topology in L (F), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' ∀ϕ ∈ F function t �−→ V (t)ϕ is continuous as a mapping [0, +∞) → F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' The definition of a C0-group is obtained by the substitution of [0, +∞) by R in the paragraph above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' It is known [5] that if (V (t))t≥0 is a C0-semigroup in Banach space F, then the set � ϕ ∈ F : ∃ lim t→+0 V (t)ϕ − ϕ t � denote = Dom(L) is a dense linear subspace in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' The operator L defined on the domain Dom(L) by the equality Lϕ = lim t→+0 V (t)ϕ − ϕ t is called an infinitesimal generator (or just generator to make it shorter) of the C0-semigroup (V (t))t≥0, and notation V (t) = etL is widely used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' One of the reasons for the study of C0-semigroups is their connection with differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' If Q is a set, then the function u: [0, +∞) × Q → R, u: (t, x) �−→ u(t, x) of two variables (t, x) can be considered as a function u: t �−→ [x �−→ u(t, x)] of one variable t with values in the space of functions of the variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' If u(t, ·) ∈ F then one can define Lu(t, x) = (Lu(t, ·))(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' If there exists a C0-semigroup (etL)t≥0 then the Cauchy problem for a linear evolution equation � u′ t(t, x) = Lu(t, x) for t > 0, x ∈ Q u(0, x) = u0(x) for x ∈ Q (2) has a unique (in sense of F, where u(t, ·) ∈ F for every t ≥ 0) solution u(t, x) = (etLu0)(x) depending on u0 continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Compare also different meanings of the solution [5], including mild solution which solves the corresponding integral equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Note that if there exists a strongly continuous group (etL)t∈R then in the Cauchy problem the equation u′ t(t, x) = Lu(t, x) can be considered not only for t > 0, but for t ∈ R, and the solution is provided by the same formula u(t, x) = (etLu0)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Definition 1 (Introduced in [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Let us say that C is Chernoff-tangent to L iff the following conditions of Chernoff tangency (CT) hold: (CT0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Let F be a Banach space, and L (F) be a space of all linear bounded operators in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Suppose that we have an operator-valued function C: [0, +∞) → L (F), or, using other words, we have a family (C(t))t≥0 of linear bounded operators in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Closed linear operator L: Dom(L) → F is defined on the linear subspace Dom(L) ⊂ F which is dense in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' (CT1) Function t �−→ C(t)f ∈ F is continuous for each f ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' (CT2) C(0) = I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' C(0)f = f for each f ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' (CT3) There exists such a dense subspace D ⊂ F that for each f ∈ D there exists a limit C′(0)f = lim t→0 C(t)f − f t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' (CT4) The closure of the operator (C′(0), D) is equal to (L, Dom(L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Let us consider one-dimensional example F = L (F) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Then g: [0, +∞) → R is Chernoff-tangent to l ∈ R iff g(t) = 1 + tl + o(t) as t → +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Theorem 1 (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Chernoff (1968), see [5, 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Let F and L (F) be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Suppose 3 that the operator L: F ⊃ Dom(L) → F is linear and closed, and function C takes values in L (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Suppose that these assumptions are fulfilled: (E) There exists a C0-semigroup (etL)t≥0 with the infenitesimal generator (L, Dom(L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' (CT) C is Chernoff-tangent to (L, Dom(L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' (N) There exists such a number ω ∈ R, that ∥C(t)∥ ≤ eωt for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Then for each f ∈ F we have (C(t/n))nf → etLf as n → ∞ with respect to norm in F uniformly with respect to t ∈ [0, T] for each T > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' lim n→∞ sup t∈[0,T] ��etLf − (C(t/n))nf �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' In our one-dimensional example (F = L (F) = R) the Chernoff theorem says that etl = limn→∞ g(t/n)n = limn→∞(1 + tl/n + o(t/n))n, which is a simple fact of calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Let F, L (F), L be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' If C is Chernoff-tangent to L and the equation limn→∞ supt∈[0,T] ��etLf − (C(t/n))nf �� = 0 holds, then C is called a Chernoff function for the operator L, and the (C(t/n))nf is called a Chernoff approximation expression to etLf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' If L is a linear bounded operator in F, then etL = �+∞ k=0(tL)k/k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' where the series converges in the usual operator norm topology in L (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' When L is not bounded (such as Laplacian and many other differential operators), expressing (etL)t≥0 in terms of L is not an easy problem that is equivalent to the problem of finding (for each u0 ∈ F) the F-valued function U that solves the Cauchy problem U ′(t) = LU(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' U(0) = u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' If one finds this solution, then etL is obtained for each u0 ∈ F and each t ≥ 0 in the form etLu0 = U(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' In the definition of the Chernoff tangency the family (C(t))t≥0 usually does not have a semigroup composition property, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' C(t1 + t2) ̸= C(t1)C(t2), while (etL)t≥0 has it: et1Let2L = e(t1+t2)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' However, each C0-semigroup (etL)t≥0 is Chernoff-tangent to its generator L and appears to be it’s Chernoff function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' When coefficients of the operator L are variable, usually there is no simple formula for etL due to the remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' On the other hand, even in this case one can find rather simple formula to construct Chernoff function C for the operator L, because there is no need to worry about the composition property, and then obtain etL in the form etL = limn→∞ C(t/n)n via the Chernoff theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 3 Numerical simulation results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='1 Problem setting Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' We say that operator-valued function C is Chernoff-tangent of order k to operator L iff C is Chernoff-tangent to L in the sense of definition 1 and the following condition (CT3-k) holds: There exists such a dense subspace D ⊂ F that for each f ∈ D we have C(t)f = � I + tL + 1 2t2L2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' + 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='tkLk � f + o(tk) as t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' (CT3 − k) Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' It is clear that for k = 1 condition (CT3-k) becomes just (CT3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' For the semigroup C(t) = etL condition (CT3-k) holds for all k = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' So one can expect that the bigger k is the better rate of convergence C(t/n)nf → etLf as n → ∞ will be, if f belongs to the space D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' This idea was proposed in [12], where two conjectures about the convergence speed were formulated explicitly, and one of them were recently proved in [7, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' For initial conditions that are good enough and t fixed, Chernoff function with Chernoff tangency of order k by conjecture should provide ∥u(t, ·) − un(t, ·)∥ = O(1/nk) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' However, if f ̸∈ D then nothing is known on the rate of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' In the present paper we are starting to fill this gap for 4 operator L given by (Lf)(x) = f ′′(x) for all x ∈ R and all bounded, infinitely smooth functions f: R → R, and k = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Problem setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' In the initial value problem (2) consider Q = R, and Banach space F = UCb(R) of all bounded, uniformly continuous functions f: R → R endowed with the uniform norm ∥f∥ = supx∈R |f(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Consider operator L given by (Lf)(x) = f ′′(x) for all x ∈ R and all f ∈ D = C∞ b (R) of all infinitely smooth functions R → R that are bounded with all the derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Then (2) reads as (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Cauchy problem (1) is a constant (one, zero, zero) coefficients particular case of the Cauchy problem considered in [15], and the corresponding Chernoff function was found in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' The particular case of this Chernoff function reads as (G(t)f)(x) = 1 2f(x) + 1 4f(x + 2 √ t) + 1 4f(x − 2 √ t) where we write G(t) instead of C(t) in order to show that C(t) is a general abstract Chernoff function for some operator L, meanwhile G(t) is this particular above-given Chernoff function for operator d2/dx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' It was proved in [15] that G(t) is first order Chernoff-tangent to d2/dx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='Vedenin (see [19]) proposed another Chernoff function for operator L considered in [15], and the constant coefficient particular case of this operator is d2/dx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' The particular case of the Chernoff function obtained by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='Vedenin reads as (S(t)f)(x) = 2 3f(x) + 1 6f(x + √ 6t) + 1 6f(x − √ 6t), and it was proved by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='Vedenin that S(t) is second order Chernoff-tangent to d2/dx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' In the paper we study how supx∈R |u(t, x) − un(t, x)| depends on n while t = 1/2 is fixed and un(t, x) is given by un(t, x) = (C(t/n)nu0)(x) where C ∈ {G, S}, C(t/n) is obtained by substitution of t by t/n in the formula that defines C(t), and C(t/n)n = C(t/n)C(t/n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' C(t/n) is a composition of n copies of linear bounded operator C(t/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' We consider several initial conditions u0 that are all H¨older continuous (hence all belong to the UCb(R) space) but have different H¨older exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Then we remark how the rate of tending of supx∈R |u(t, x) − un(t, x)| to zero depends on these H¨older exponents and the order of Chernoff tangency (which is 1 for G(t), and 2 for S(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Comments on computational techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Calculations were performed in the Python 3 environment using a program we wrote and which is available in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' All measure- ments, for the sake of reducing computational complexity, for each value of n (varying from 1 to 11) were carried out for 1000 points uniformly dividing the segment [−π, π] or [−2π, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Initial conditions of the form u0(x) = | sin x|α for various α ∈ {9/2, 7/2, 5/2, 3/2, 1, 3/4, 1/2, 1/4}, like any of Chernoff approximations based on them, are periodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' So, the standard norm in UCb(R), namely d = ∥un(t, ·) − u(t, ·)∥ = sup x∈R |un(t, x) − u(t, x)|, where u is the exact solution of (1) and un is the Chernoff approximation, is reached at the interval corresponding to the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' The program code was written with the possibility to set any operator and any initial con- dition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' without simplifying Chernoff functions and using binomial coefficients, in contrast to the work [16] published earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Moreover, the initial condition does not necessarily have to be a smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' The number of iterations is not limited to 11, the value n can be changed, both upward and downward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' We have chosen the optimal value n since the program is very time consuming: via Jupyter Notebook 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='4 Anaconda 3 Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='3 set on personal computer with Windows 10, CPU Intel Core i5-1035G1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='6 GHz, 8 Gb RAM it takes about 20 minutes to complete the program for all initial conditions with construction of graphs for 5 them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' At the research stage of the new method (Chernoff approximations) this is acceptable, but in the future, of course, the code will be optimized for a better speed, since this is important in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Our goal is to continue research and in the future write a library that allows to solve partial derivative equations in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2 Approximations for initial condition u0(x) = sin(x) Let us first analyze the approximations for the initial condition u0(x) = sin x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='1, n = 1, u0(x) = sin x, t = 1 2 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='1 shows the exact solution, which coincides with the graph of the function y = e−1/2 sin x, and approximate solutions for the functions S(t) (left) and G(t) (right) at n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' The initial condition u0 = sin x is very good, since its derivatives of any order exist, have no discontinuities and are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' And already at n = 1 the function S(t) gives a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2 below shows plots of the decreasing error of Chernoff approximations as a function of n, where 1 ≤ n ≤ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' On the left are plots of decreasing error for Chernoff functions S(t) (in blue) and G(t) (in green) in regular scale, and on the right – the same plots in logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' The graph in the logarithmic scale allows us to estimate how much the convergence rate for the function G(t) is less than the convergence rate for the function S(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Here and through all the paper we use the following notation: d = ∥un(t, ·) − u(t, ·)∥ = sup x∈R |un(t, x) − u(t, x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2, 1 ≤ n ≤ 11, u0(x) = sin x, t = 1 2 6 Approximetion of the sohution via Chemoff fmctin S(t) Approximetion of the sohution via Chemoff fmcticm G(t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 2 2 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='3 13 i 2 3 02 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='4 Chernoff approximation Chernoff approximation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='6 Solution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='6 SolutionNorm decay Logarithmic dependence R2=0,9949 Discrepancy (S(t)f)(x) 4 Discrepancy (G(t)f)(x) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='025 5 R"=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='9949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='020 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='015 Discrepancy (S(t)f)(x) Discrepancy (G(t)f)(x) 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='010 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='005 6- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=" 10 2 6 8 10 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content="5 10 1'5 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 n In nYou can see that the points on the right graph lie on the straight lines with good accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Using the method of least squares (in Excel) we found the equations of these lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Rounding off the coefficients, we see that for the blue line the equation is as follows: ln(d) = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='092 ln(n) − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0671, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' d = n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='092e−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0671 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0063 n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='092 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Similarly, for the green line, the equation ln(d) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0416 ln(n) − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5796, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' d = n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0416e−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5796 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0279 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Using the same approach, we study the behavior of the error for other initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='3 Approximations for initial condition u0(x) = | sin(x)|3/2 fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='1, n = 4, u0(x) = | sin(x)|3/2, t = 1 2 fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2, 1 ≤ n ≤ 11, u0(x) = | sin(x)|3/2, t = 1 2 The line (green) corresponding to the decreasing error of the function G(t) in the logarithmic scale was constructed without taking into account n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' For the green line (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2, right) the equation ln(d) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='9785 ln(n) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='8973, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' d = n−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='9785e−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='8973 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0552 n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='9785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Similarly, for the blue line (see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2), the equation is as follows: ln(d) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5109 ln(n)− 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='8234, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' d = n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5109e−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='8234 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='1615 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' As can be seen from Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2, the difference between the error decay rates using Chernoff functions S(t) and G(t) for u0(x) = | sin(x)|3/2 is larger than for u0(x) = | sin x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' This is due to the greater smoothness of u0(x) = | sin(x)|3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 7 Approximetion of the sohution via Chemoff fmcticn S(t) Approximetion of the sohution via Chemoff fmcticn G(t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='54 4152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='48 Chernoff approximation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='50 Chernoff approximation Solution Solution 3 2 1 0 1 2 3 3 2 1 0 1 2 3Norm decay Logarithmic dependence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='16 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='9976 Discrepancy (S(t)f)(x) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 Discrepancy (G(t)f)(x) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='10 R2=0,9903 Discrepancy (S(t)f)(x) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='08 Discrepancy (G(t)f)(x) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content="5 2 4 6 8 10 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 10 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 n In n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='4 Approximations for initial condition u0(x) = e−|x| Let us consider a non-smooth and non-periodic function e−|x| as an initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='1, n = 4, u0(x) = e−|x|, t = 1 2 fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2, 1 ≤ n ≤ 11, u0(x) = e−|x|, t = 1 2 Figures 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2 show plots of the exact solution, approximations to the solution, and rates of convergence of the error to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' As can be seen, the result is similar: the conver- gence rate of the function S(t) is higher than that of G(t), but the order of convergence is approximately the same, as can be seen from the fact that the lines are almost parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' For the green line (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2, right), the equation is as follows: ln(d) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='9294 ln(n) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='3832, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' d = n−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='9294e−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='3832 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0923 n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='9294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Similarly, for the blue line (see Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2) the equation is as follows: ln(d) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='056 ln(n)− 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5543, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' d = n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='056e−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5543 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2113 n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=" 8 Approximetion of the sohution via Chemoff fmcticn S(t) Approximetion of the sohution via Chemoff fumcticn G(t) 9'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 Q4 0/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='1 Chernoff approximation Chernoff approximation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 Solution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 Solution 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='D 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 1±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content="5 0'5- 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='D 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='D 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 1±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0Norm decay Logarithmic dependence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 Discrepancy (S(t)f)(x) 6660-_ Discrepancy (G(t)f)(x) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='175 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='150 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='125 Discrepancy (S(t)f)(x) R =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='9979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='100 Discrepancy (G(t)f)(x) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='075 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='050 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='025 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content="5 2 6 8 10 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content="5 10 1'5 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 n In n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 Approximations for initial condition u0(x) = | sin(x)|5/2 fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='1, n = 4, u0(x) = | sin(x)|5/2, t = 1 2 fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2, 1 ≤ n ≤ 11, u0(x) = | sin(x)|5/2, t = 1 2 The lines (green and blue) corresponding to the decreasing error of the functions G(t) and S(t) in the logarithmic scale was constructed without taking into account n = 1 and n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='6 Approximations for initial condition u0(x) = | sin(x)|7/2 fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='1, n = 4, u0(x) = | sin(x)|7/2, t = 1 2 9 Approximetion of the sohution via Chemoff fmcticn S(t) Approximetion of the soution via Chemoff fumcticn G(t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='42 442 00 Chernoff approximation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='40 Chernoff approximation Solution Solution E- 2 1 0 1 2 3 3 2 1 0 1 2 3Norm decay Logarithmic dependence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='14 2 Discrepancy (S(t)f)(x) Discrepancy (G(t)f)(x) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='12 E- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='08 4 Discrepancy (S(t)f)(x) 2 p ul =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='9963 Discrepancy (G(t)f)(x) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='06 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='04 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content="02 00°0 2 6 7 4 8 10 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 10 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 n In nApproximetion of the sohution via Chemoff fmcticn S(t) Approximetion of the soution via Chemoff fumcticn G(t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='36 436 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='14 Chernoff approximation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='34 Chernoff approximation Solution Solution E- 2 1 0 1 2 3 3 2 1 0 1 2 3fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2, 1 ≤ n ≤ 11, u0(x) = | sin(x)|7/2, t = 1 2 The line (green) corresponding to the decreasing error of the function G(t) in the logarithmic scale was constructed without taking into account n = 1 and n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='7 Approximations for initial condition u0(x) = | sin(x)|9/2 fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='1, n = 4, u0(x) = | sin(x)|9/2, t = 1 2 fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2, 1 ≤ n ≤ 11, u0(x) = | sin(x)|9/2, t = 1 2 The line (green) corresponding to the decreasing error of the function G(t) in the logarithmic scale was constructed without taking into account n = 1 and n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 10 Norm decay Logarithmic dependence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='200 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='9864 : Discrepancy (S(t)f)(x) 2 Discrepancy (G(t)f)(x) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='150 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='125 Discrepancy (S(t)f)(x) 4 p ul 0,9753 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='100 Discrepancy (G(t)f)(x) 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='050 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='025 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content="000 4 6 8 10 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content="5 10 1'5 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 n In nApproximetion of the sohution via Chemoff fmcticn S(t) Approximetion of the soution via Chemoff fumcticn G(t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='34 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='32 432 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='30 Chernoff approximation Chernoff approximation Solution Solution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='28 E- 2 1 0 1 2 3 3 2 1 0 1 2 3Norm decay Logarithmic dependence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='25 Discrepancy (S(t)f)(x) 1 R =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='9777 Discrepancy (G(t)f)(x) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='20 2 E- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='15 Discrepancy (S(t)f)(x) 4 R°=0,9309 Discrepancy (G(t)f)(x) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='10 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='05 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content="00 8 2 6 10 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content="5 10 1'5 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 n In n4 Discussion The table below shows experimentally (using simulation in Python 3) the orders of decreas- ing of error depending on the smoothness class of the initial condition and the Chernoff function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' The smoothness class of the initial condition u0 Order of decreasing er- ror on the Chernoff function G(t), which has the 1st order of the Chernoff tangent to the operator L = d2 dx2 Order of decreasing er- ror on the Chernoff function S(t), which has the 2nd order of tangency by Chernoff to the operator L = d2 dx2 C∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' all derivatives exist and are bounded, u0(x) = sin(x) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0416 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='092 C4 1 2 , the first, second, third, and fourth derivatives exist and are bounded, and the fourth is H¨older with a H¨older ex- ponent 1/2, u0(x) = | sin(x)|9/2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0212, the regression was done without n = 1, n = 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='1219, but the points do not fit well on a straight line, so the number is un- informative C3 1 2 , the first, second, and third deriva- tives exist and are bounded, and the third is H¨older with H¨older exponent 1/2, u0(x) = | sin(x)|7/2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='4013,regression was done without considering n = 1, n = 2, but the points do not lie well on the line, so the number is uninformative 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5045, but the points do not lie well on the line, so the number is uninforma- tive C2 1 2 , the first and second derivative exist and are bounded, while the sec- ond derivative is H¨older continuous with H¨older exponent 1/2, u0(x) = | sin(x)|5/2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='1433, regression was done without considering n = 1, n = 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='7923, regression was done without considering n = 1, n = 2 C1 1 2 , the first derivative: exists, is bounded and H¨older continuous with H¨older exponent 1/2, u0(x) = | sin(x)|3/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='9785, the regression was done without considering n = 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5109 H1, the H¨older with the H¨older expo- nent 1, u0(x) = | sin(x)| 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0508 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0948 H1,the H¨older with the H¨older expo- nent 1, u0(x) = e−|x| 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='9294 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='056 H3/4, the H¨older with the H¨older expo- nent 3/4, u0(x) = | sin(x)|3/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='815 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='9262 H1/2, the H¨older with the H¨older expo- nent 1/2, u0(x) = | sin(x)|1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='6905 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='7723 H1/4, the H¨older with the H¨older expo- nent 1/4, u0(x) = | sin(x)|1/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='6138 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='6653 We see that on the initial condition with high smoothness (first line in the table), the first order of Chernoff tangency corresponds to a decreasing error rate of about 1/n, and the second order – a decreasing rate of about 1/n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' This is in accordance with the conjecture from [12] and theorem from [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' As the smoothness is lost (second line in the table and below), theory from [6] stops working, and the experimental evidence is the following: the convergence speed gradually decreases and the advantages of the Chernoff function with the second order of Chernoff tangency gradually vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Let us present the results from the table graphically: 11 fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 10 The regression was carried out without taking into account the point (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='7923).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' The equation of the approximating line: y = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='684x − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='4467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='This may be interpreted as follows: when the smoothness class α of the initial condition u0 is not greater than the order of Chernoff tangency then d = ∥un(t, ·) − u(t, ·)∥ = sup x∈R |un(t, x) − u(t, x)| ≈ const · �1 n �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='68α+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='45 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Meanwhile when the smoothness class α of the initial condition u0 is greater than the order of Chernoff tangency then there is no such easy-to-state dependence but still Chernoff function S(t) with the second order Chernoff tangency provides better approximations than Chernoff function G(t) with the first order Chernoff tangency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' The results of the numerical simulation are generally in agreement with and confirm the theory arising from the conjecture in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' However, some of the points that do not lie on straight lines exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' This deserve closer attention: n = 11 for some initial conditions is not sufficient to derive conclusions about the asymptotic behavior of the calculation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' For not smooth initial conditions that we studied numerically there are not known any theoretical bounds on the rate of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' And, of course, the most interesting case of variable coefficients should be considered, understanding them as parameters analogously with u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' So the research in this direction is far from ending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' The publication was prepared within the framework of the Academic Fund Program at HSE University in 2021 (grant №20-04-022) by the Research and educational group ”Evolution semigroups and their applications”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Authors are thankful to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='Galkin and other members of the group for discussions of the results presented in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='6 S(t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='8 (a)9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2 decre 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='8 R2 = 0,9854 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='5 The smoothness class of the initial conditionAppendix: Python 3 code 13 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='1Moduliandvariables In [1]: # importing moduli importmatplotlib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='pyplotasplt from sympy import oo from scipy importintegrate import numpy as np importmath In [2]: # variables declaration tau = 1/2 n = 11 In[3]: 1=[] fori inrange(1,n+1): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='append(math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='log(i))1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='2 Functions and operators # almost self-contained class classclassoffunctions(object): def thefunction(self, x): return x # Chernoff function (s(t)f)(x)=(2/3)f(x) + (1/6)f(x + (6t)^(1/2)) + (1/6)f(x - (6t)^(1/2)) defoper(inputobject:classoffunctions,t): outputobject=inputobject f= inputobject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='thefunction def funct(x): return (2/3)*f(×) +(1/6)*f(× + (6*t)**(1/2)) + (1/6)*f(x- (6*t)**(1/2)) outputobject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='thefunction=funct returnoutputobject # Chernoff function (G(t)f)(x) = (1/4)* f(x+2t^(1/2))+(1/4) f(x-2t^(1/2))+(1/2)f(x) defoperl(inputobject:classoffunctions,t): outputobject=inputobiect f=inputobject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='thefunction def funct(x): return (1/4)*f(x+2*t**(1/2))+(1/4)*f(x-2*t**(1/2))+(1/2)*f(×) outputobject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='thefunction=funct return outputobjectReferences [1] Bogachev V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=', Smolyanov O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Real and Functional Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' — Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' [2] Butko Ya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' The method of Chernoff approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Springer Proceedings in Mathematics and Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Volume 325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' — Springer, Cham, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 19–46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' [3] Chernoff P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Note on product formulas for operator semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='// J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Functional Analysis 2:2 (1968), 238-242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' [4] Engel K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=', Nagel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' A.' metadata={'source': 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+page_content=', Nagel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' One-Parameter Semigroups for Linear Evolution Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' — Springer, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' [6] Galkin O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=', Remizov I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Rate of Convergence of Chernoff Approximations of operator C0-semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='// Mathematical Notes, to appear (2022) [7] Galkin O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=', Remizov I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Upper and lower estimates for rate of convergence in Chernoff’s product formula for semigroups of operators // https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='org/abs/2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='01249, 2021 [8] Gomilko A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=', Kosowicz S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=', Tomilov Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' A general approach to approximation theory of operator semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' // Journal de Math´ematiques Pures et Appliqu´ees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 127 (2019), 216– 267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 14 # composition degree of Chernoff function (s(t)f)(x) def degr(g, tau, n): y=[] obj=classoffunctions() for n_p in range(1, n + 1): objk=obj obj_k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='thefunction = g for k in range(1, n_p + 1): objk=oper(objk,tau/np) y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='append(obj_k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='thefunction) return y # composition degree of Chernoff function (G(t)f)(x) def degrl(g, tau, n): y=[] obj=classoffunctions() for n_p in range(1, n + 1): obj_k = obj obj_k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='thefunction =g for k in range(1, n_p + 1): obj k = oper1(obj k,tau/n p) y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='append(obj_k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='thefunction) return y #normcomputation def norm(y, sol): [] = p for n p in range(O, n): d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='append(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='max(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='abs(sol -y[n_pj(x)))) return d[9] Hille E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=', Phillips R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Functional Analysis and Semi-Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' — American Mathematical Society, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' [10] Orlov Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=', Sakbaev V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=', Smolyanov O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Rate of convergence of Feynman approxima- tions of semigroups generated by the oscillator Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' // Theoret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' and Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 172:1 (2012), 987–1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' [11] Pazy A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Semigroups of Linear Operators and Applications to Partial Differential Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' — Springer-Verlag, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' [12] Remizov I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' On estimation of error in approximations provided by Chernoff ’s product formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='// International Conference "ShilnikovWorkshop-2018" dedicated to the memory of outstanding Russian mathematician Leonid Pavlovich Shilnikov (1934-2011), Lobachevsky State University of Nizhny Novgorod, December 17-18, 2018 Book of abstracts, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='38-41 [13] Remizov I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' A method of obtaining the evolution operator for the Schr¨odinger equation Quasi-Feynman formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='// J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 270:12, (2016), 4540-4557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' [14] Remizov I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Feynman and Quasi-Feynman Formulas for Evolution Equations// Doklady Mathematics, 96:2 (2017), 433-437 [15] Remizov I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Approximations to the solution of Cauchy problem for a linear evolution equation via the space shift operator (second-order equation example)// Applied Mathe- matics and Computation, 328, 243-246, 2018 [16] Prudnikov P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Speed of convergence of Chernoff approximations for two model examples: heat equation and transport equation// arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='09615 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='FA] (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' [17] Smolyanov O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=', Tokarev A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=',Truman A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Hamiltonian Feynman path integrals via the Chernoff formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='// J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 10 (2002), 5161-5171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' [18] Vedenin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Fast converging Chernoff approximations to solution of a parabolic differ- ential equation on a real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='// International school-conference Mathematical spring 2019, Russia, Nizhny Novgorod 2-5 May 2019, Book of abstracts, pp 22-23 [19] Vedenin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=', Voevodkin V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=', Galkin V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=', Karatetskaya E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=', Remizov I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Speed of Convergence of Chernoff Approximations to Solutions of Evolution Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' // Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Notes, 108:3 (2020), 451–456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' [20] Zagrebnov V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Notes on the Chernoff product formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' // J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} +page_content=' 279:7 (2020), 108696 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQf0w0Q/content/2301.05284v1.pdf'} diff --git a/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf b/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..be4f211c767fb1f6019437b92f1613f69addb9c2 --- /dev/null +++ b/59E5T4oBgHgl3EQfPQ7C/content/2301.05504v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6ccf5885cf2158e625842fc32902f8303c5ed79c4f57c1b128d6f1765ce3ef5a +size 3006063 diff --git 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Providence, RI, 02912, USA. +Correspondence: jose_james@brown.edu. +Abstract: Grasping is an incredible ability of animals using their arms and limbs in their daily life. +The human hand is an especially astonishing multi-fingered tool for precise grasping, which +helped humans to develop the modern world. The implementation of the human grasp to virtual +reality and tele robotics is always interesting and challenging at the same time. In this work, au- +thors surveyed, studied, and analyzed the human hand grasping behavior for the possibilities of +haptic grasping in the virtual and remote environment. This work is focused on the motion and +force analysis of fingers in human hand grasping scenarios and the paper describes the transition +of the human hand grasping towards a tripod haptic grasp model for effective interaction in virtu- +al reality. +Keywords: hand grasp; grasp analysis; multi-finger haptics; haptic grasp interface + +1. Introduction +The human hand is a highly skilled, prehensile, multi-fingered, perception and ma- +nipulation organ at the distal end of the arm [1]. Prehensility is the quality of an ap- +pendage or organ that has adapted for grasping or holding. In the past decade’s, re- +searchers [2] explored the various aspects of the evolution, morphology, anthropology, +and social significance of the human hand as a tool [3], as a symbol [4] and as a weapon +[5]. Humans cognitively manipulating a variety of objects in daily life using various +hand configurations, resulting from changing the position, orientation and placement of +hand and fingers based on the object properties such as its weight, shape, texture, fric- +tion, hardness etc. Such a variety of grasps is possible because of the dexterity, various +degrees of freedom, and the great control strategy. +Hands are associated with the aligning capability of the body, kinesthetic percep- +tion of the limb and the richest tactile sense. Many researchers studied the anatomy, +muscles, biomechanics, kinematics, functionalities, and skills of human hand [6,7]. These +studies helped to do more focused research on human hand prehension [8] and grasp +[9,10]. Based on all these studies hand grasp types are classified and different grasp tax- +onomies were arising in the literature [11,12], covering a broad range of domains. +As technologies like virtual reality and tele-robotics being progressed, humans +started interacting with virtual and remote environments. But the integration of hand +grasping into virtual and remote environments still challenging because of the complex +architecture behind it. The extensive research in the analysis and synthesis of human +grasps [13,14] over the past years has provided a basic theoretical framework towards +better progress in human-computer interaction [15], robotic grasping [16] and dexterous +manipulation and lead to the design of artificial robotics hands and arms for the pros- +thetic application [17]. Since the last few decades, researchers put effort to mimic the +human hand to design robotic grippers [18], etc. But still, these frameworks need more +extensive studies for the practical implementation of the direct involvement of humans +Citation: James, J.; Title. Sensors +2022, +22, +x. +https://doi.org/10.3390/xxxxx +Academic Editor: Firstname Last- +name +Received: 27 April 2022 +Accepted: date +Published: date +Publisher’s Note: MDPI stays neu- +tral with regard to jurisdictional +claims in published maps and insti- +tutional affiliations. + +Copyright: © 2022 by the authors. +Submitted for possible open access +publication under the terms and +conditions of the Creative Commons +Attribution +(CC +BY) +license +(https://creativecommons.org/license +s/by/4.0/). + +sehsorsMDPIBYSensors 2022, 22, x FOR PEER REVIEW +2 of 20 + + +to grasp objects in virtual and remote environments with multi-fingers. +The grasping force depends on the orientation of fingers, palm, and wrist [19]. The +force output on the fingertip is highly joint dependent and provides stable grasp and +precise manipulation of objects. Previous works [20], more focused on muscle activation +patterns and resultant positions/forces as a function of the joints as well as subject inde- +pendent leads to the structural variability in human hands [21]. Motion and force analy- +sis of fingers in various hand manipulation actions can be observed, learned, and ana- +lyzed to come up with better framework and devices for virtual and remote manipula- +tions [22,23] Adapting the gripping functions, manipulation capabilities, kinematics, dy- +namics, and size of the human hand, will accelerate the design of the human-like artifi- +cial arms and hands for the direct grasping interaction in the virtual world. +As a previous work, authors designed haptic interfaces for tweezer pinch grasp [24] +and tripod grasp [25]. Also implemented the hand grasp through augmented haptics by +means of custom-made attachments for virtual tools in motor skill training interfaces +[26,27]. The aim of this work is to understand human grasping and manipulations, sur- +veying different types of grasp taxonomies, study the characterization of hand in grasp- +ing, model a tripod haptic grasp and design an interface for multi-finger haptic grasping +which can offer better interactions with tasks in the virtual and remote environments. +2. Human Hand +The human hand is a prehensile, multi-fingered astonishing organ/tool of complex +engineering used to carry and manipulate objects [2]. In view of human grasping, a short +description of hand anatomy, mechanisms and kinematics will help to model a multi- +finger haptic grasp and design a haptic grasping interface. +2.1. Hand Anatomy +The human hand includes mainly three areas and five digits (fingers): Thumb, In- +dex finger, Middle finger, Ring finger, and Little finger are numbered 1-5 as shown in +Figure 1. The palm with fingers holds most pressure and support for the hand to grasp. +Fingers are the densest areas of nerve endings and the richest source of tactile feedback. +So, hands are the primary tool for a sense of touch and positioning capability. The hu- +man hand consists of 27 bones and 45 muscles with at least 23 degrees of freedom at the +joints [6] including the wrist as shown in Figure 1(a). + +Figure 1. Skeleton system and arches of human hand +The human hand can grasp objects and do daily tasks by forming bony arches: +Longitudinal arches, transverse arches, and oblique arches as drawn in Figure 1(b). Lon- + +(a)HumanHandskeletonsystem +(b)HumanHandarches +DIPjoint +PIP joint. +MCPjoint +Distal +Longitudinalarch +1 +Middle +Phalanges +(Fingers) +Obliquearch +Proximal +Metacarpaltransversearch +Metacarpals +Distaltransversearch +(Palm) +Hamate +Trapezium +Pisiform +Carpals +Trapezoid- +Scaphoid +-Triquetrum +(Wrist) +Capitate +Lunate +Carpaltransversearch +Radius +UlnaSensors 2022, 22, x FOR PEER REVIEW +3 of 20 + + +gitudinal arches shaped by the finger bones and their associated metacarpal bones, +transverse arches by the carpal bones and the distal ends of the metacarpal bones, and +oblique arches by the thumb and four fingers [7]. These arches are the basic frames for +various grasp patterns. The extrinsic and intrinsic muscles in hand controlled the motion +of the fingers and making grasping possible. Thumb together with the index and middle +finger forms the dynamic tridactyl configuration responsible for most grips which not +requiring force. The ring and little fingers are more static. So, for this multi-finger haptic +grasping interface model, authors considered the motion of the first three fingers the +thumb, index finger and middle finger. +2.2 Hand Kinematics +Hand’s numerous patterns of action was resulted from the skeleton mechanical sys- +tems and the twenty-four muscle groups regulated by the diverse motor and sensory +nerve pathways [6]. +The MCP and PIP joints exhibit a common rotation pattern. The virtual center of ro- +tation of hand is the center of curvature of the distal end of the proximal member [28]. +The lateral rotation of fingers is small in the MCP joints and decreasing towards the pha- +langeal hinge joints. The thumb has the greater mobility in the CMC articulation. Other +fingers being more arched from index to little finger. The thumb, palm, and fingers to- +gether permitted to grasp a 1.75-inch cylinder at about 45 degrees to the radioulnar axis. +Bunnell [29] considers this "an ancestral position ready for grasping limbs, weapons, or +other creatures." +The major wrist motions are extension (or dorsiflexion), flexion (or volar flexion), +radial flexion and ulnar flexion, based on the angle of rotation of the wrist. The fixation +movements and ballistic movements are also major types of movements in the hand [30]. +The hand with the fully extended arm can be rotated through almost 360 degrees with +the participation of shoulder and elbow. From palm up to palm down, the hand can be +rotated through 180 degrees, with the elbow flexed. Thumb can provide a variety of flex- +ions extension patterns of the phalanges for any given metacarpal position and due to +the relative mobility of the CMC joint, which allows the thumb to act in any plane neces- +sary to oppose the digits. In the principal opposition cases and prehensions, the plane of +the thumb action is inclined 45 to 60 degrees to the palmar plane. In lateral prehension, +the plane is approximately parallel to the palmar plane. +2.3 Hand Dynamics +Fick [31] investigated the actions and contractile forces of hand muscles and esti- +mated the summed forces of the individual muscles participating in the action. But the +measured isometric forces are only 10% of the total forces because of the effective small +moment arm upon any of the wrist or hand joints. The flexor-extensor forces in the wrist +and the prehensile forces in the hand varied with wrist angle and it reaches a maximum +at a wrist angle of about 145 degrees. So, for very strong prehensions, wrist likely to at- +tain this angle [32]. +Kamper et.al [33] was analyzed the joint angles and finger trajectories in reach-and- +grasp tasks which fit the actual finger positions with a mean error of 0.23 ± 0.25 cm and +accounted for over 98% of the variance in finger position. The direction of the thumb tra- +jectories exhibited a greater dependence on object type than the finger trajectories, but +still utilized a small percentage (<5%) of the available workspace [34]. Previous studies +of musculoskeletal models [35] observed that the role of the intrinsic muscles of the hand +as the main force-producing muscles in power grip [36]. These models were used in the +commercial software ANYBODY for the thumb and the index finger [37]. However, +these models did not address the coupling between the fingers and the interaction with +the wrist which limits the investigation of human grasping. + +Sensors 2022, 22, x FOR PEER REVIEW +4 of 20 + + +In [38] authors presented an upper limb musculature model for the full arm includ- +ing the shoulder, elbow, wrist, thumb, and index finger and provides valuable data on +the wrist-finger joint coupling and extrinsic hand muscle anatomy [39]. The force output +on the fingertip is highly joint dependent and provides stable grasp and precise manipu- +lation of objects. That's why the force transfer is important for the human hand in con- +tact and manipulating purpose to avoid the slipping and deformation of the object. Pre- +vious works focused more on muscle activation patterns and resultant positions/forces +as a function of the joints [20] as well as a subject independent lead to the structural var- +iability in human hands. +2.4 Purpose of Study +This study conducted to analyze the human hand grasping to isolate the functional +properties with the final goal to optimize haptic grasping by building simpler multi- +finger haptic grasping interfaces with at least similar grasping and manipulation capa- +bilities. This work helped to learn more about the complex engineering structure of the +human hand and leads to characterizing the multi-finger haptic grasping systems. For +instance, the proposed three-finger haptic grasping system has an independent joint ar- +chitecture for three fingers in the hand, which may be advantageous in virtual grasping. +Also, this led to identify the independence of joints needs in normal grasping tasks. +Conversely, the control of such an independent architecture is very challenging. By add- +ing synergies, we can reduce the complexity of control, but we also want to keep a cer- +tain, currently unquantified, level of dexterity. +3. Human Grasp +A grasp is a system wherein the desired object is gripped by the fingers of a human +(or robot) hand. +3.1 Human Grasp Patterns +Napier [40] categorized human grasp into two basic grips: power grasps and preci- +sion grasps (pinch grasps). In power grasp, the object is in the palm of the hand and en- +closed by the fingers which lead to large area of contact between the palm, the fingers, +and the object. In precision grasp (pinch grip), the object is held between the tip of the +thumb and finger, which offer more dexterity. Precision grasps become more relevant in +robotic and virtual grasping. The power grasp is enhanced by the precision grasp be- +tween the thumb and the distal finger pads, and it is inherently stable. Pinch grip re- +quires the six joints between the index finger and the thumb to be stabilized; it requires +more activity of the intrinsic finger muscles to maintain this balance. A large variety of +prehension patterns are identified from studies of the muscle-bone-joint anatomy and +from observation of the postures and motions of the hand. The object-contact pattern +furnishes a satisfactory basis for classification of major prehension patterns [41]. +All the ages, the human hand was a part of most creative arts of every culture [42] +to speak and convey human emotions and the hands symbolize cultural behaviors, val- +ues, and beliefs. A mudra is a symbolic gesture in the spiritual practice of Indian reli- +gions and traditional art forms performed with the hands with a specific pattern of fin- +ger configurations [43]. A canonical set of predefined hand postures and modifiers can +be used in digital human modelling to develop the standard hand posture libraries and +a universal referencing scheme and continuum of hand poses from simple posture to +complex one. Researchers [9] have studied features for force-closure grasp by human +hands and characterized into four mutually independent properties for robotic arm +grasping listed as dexterity, equilibrium, stability, and dynamic behavior. The principal +component analysis of static hand posture of several subjects provides information +about the finger joint variance and shape of the grasped object and did not consider the +hand position/orientation relative to the object placement [44]. + +Sensors 2022, 22, x FOR PEER REVIEW +5 of 20 + + +3.2 Grasp Taxonomy +Based on the various studies about human hand in the literature, hand grasp types +are classified, and different grasp taxonomies were raised in the literature [11,12] as +shown in Table 1. If the grasp object size/shape is not considered, this taxonomy might +be lowered to broad range. +Grasps are classified based on precision [45], grasped object’s size [9], shape [46], +weight, rigidity, force requirement, the position of thumb (adducted or abducted posi- +tion) and the situation. Based on the level of precision, grasps are classified as precision +grasp [10], intermediate grasp [47], and power grasp [48]. The movements of the hand in +the power grip are evoked by the arm but in the precision handling, the intrinsic move- +ments on the hand not evoked by the arm. In intermediate grasp, elements of power and +precision grasps are present in roughly the same proportion. Later studies included oth- +er grasps like hook grasp, flat hand grasp, platform grasp, push grasp [49]. In static and +stable grasps [12], the object is in a constant relation to the hand. + Based on the direction of force relative to the hand coordinate frame, applied by +the hand on the object to hold it securely [8] opposition type grasps are classified as pad +opposition, palm opposition and side opposition grasp [50]. Pad Opposition occurs be- +tween hand surfaces along a direction generally parallel to the palm, usually occurs be- +tween volar (palmar) surfaces and the fingers and thumb, near or on the pads. Palm Op- +position occurs between hand surfaces along a direction generally perpendicular to the +palm. Side Opposition occurs between hand surfaces along a direction generally trans- +verse to the palm. +The taxonomy of Cutkosky [9], which is widely used in the field of robotics, lists 15 +different grasps. Other taxonomies mentioned in works of Kamakura et al. [47], Ed- +wards et al. [10], Kapandji [45] are listed with 14, 20 and 21 grasps respectively. A simi- +lar study [51], that used a different categorization which incorporated non-prehensile +grasps. Even though there has been a considerable effort in creating statistics of human +hand use and grouping of hand grasps [11,12]. The extensive research in human grasp +analysis and taxonomies over the past years helped towards better progress in human- +computer interaction, robotic grasping, and dexterous manipulation and lead to the de- +sign of artificial robotics hands and arms for the prosthetic application. +This helped to classify all hand usage in everyday life situations [52]. Furthermore, +the taxonomy could be extended to include non-prehensile “grasps”, or for dynamic +within - hand manipulation movements. Kamakura et al. [47] classified the tripod grasps +as intermediate grasps, apart from that it was classified as a precision grasp. Several +studies have investigated classifying grasps into a discrete set of types [8,9,47], and oth- +ers have been aimed at understanding certain aspects of human hand usage [53]. The +number of fingers used for grasping increases with the size and mass of the object [54] +until a two-handed grasp is required, indicating that object size and mass are strong fac- +tors in determining the grasp type. The rigidity of the objects also influencing the grasp +[55]. + + +Sensors 2022, 22, x FOR PEER REVIEW +6 of 20 + + + +Table 1. Taxonomy of grasps and allocation of virtual fingers. +The studies in [56,13,14] analyzed the human grasping scenarios and behavior of +unstructured tasks and investigated the relationship between grasp types and object +properties and the results indicated that three-fingertip precision grasps such as thumb- +2 finger, tripod, or lateral tripod can be used to handle dexterous manipulation of a wide +range of objects. In [12] authors analyzed and compared 33 existing human grasp taxon- +omies of static and stable grasps performed by one hand and synthesized them into a +single new taxonomy called ‘The GRASP Taxonomy’. The grasps are arranged according +to opposition type, the virtual finger assignments, type in terms of power, precision or +intermediate grasp, and the position of the thumb. The classifications of micro- +interaction grasp instances [57] helped researchers in defining human hand capabilities +[58] and affordances in robotic hand design [59]. +The concept of the virtual finger [15] has also incorporated in this taxonomy as an +abstract representation through which the human brain plans grasping tasks [60]. The +virtual finger is a functional unit of several fingers work together comprised of at least +one real physical finger to reduce the degrees of the human hand to perform the grasp- +ing task. This concept replaces the analysis of the mechanical degrees of freedom of in- + +Grasp +Thumb +Virtual +Min. No. +Grasp +Thumb +Virtual +Min. No. +No. +Name +Picture +Type +Opp.Type Position +Fingers +Mod. VF +of Fingers +No. +Name +Picture +Type +Opp.Type Position +Fingers +Mod. VF +of Fingers +VF1: +VF1: 1 +VF1: +VF1: 1 +large +VF2: 2-5 +VF2: 2-5 +Extension +VF2: 2-5 +VF2: 2-5 +Diameter +Power +Palm +Abducted +VF3: +VF3: Palm +3 +18 +Type +Power +Pad +Abducted VF3: +VF3: +3 +VF1: +VF1: Palm +VF1: +VF1: 1 +Small +心 +VF2: 2-5 +VF2: 2-5 +Distal +VF2: 2-5 +VF2: 2-5 +2 +Daimeter +Power +Palm +Abducted +VF3: +VF3: +2 +19 +Type +Power +Pad +Abducted VF3: +VF3: +2 +VF1: +VF1: Palm +VF1: +VF1: 1 +Medium +VF2: 2-5 +VF2: 2-5 +Writing +VF2: 2 +VF2: 2 +m +Wrap. +Power +Palm +Abducted +VF3: +VF3: +m +20 +Tripod +Precision +Side +Abducted VF3: +VF3: 3 +m +VF1: +VF1: Palm +VF1: +VF1: 1 +Adducted +VF2: 2-5 +VF2: 2-5 +Tripod +Intermed +VF2: 3-4 +VF2: 3-4 +Thumb +Power +Palm +Adducted +VF3: 1 +VF3: 1 +3 +21 +Variation +iate +Side +AbductedVF3: +VF3: 2 +VF1: +VF1: Palm +VF1: +VF1: 1 +VF2: 2-5 +VF2: 2-5 +Parallel +VF2: 2-5 +VF2: 2-5 +5 +Light Tool +Power +Palm +Adducted +VF3: (1) +VF3: (1) +3 +22 +Extension +Precision +Pad +Adducted VF3: +VF3: +m +VF1: +VF1: 1 +VF1: +VF1: 2 +Prismatic 4 +VF2: 2-5 +VF2: 2 +Adductio +Intermed +VF2: 2 +VF2: 3 +6 +Finger +Precision +Pad +Abducted +VF3: +VF3: 3-5 +m +23 +n Grip +iate +Side +Abducted VF3: +VF3: +2 +VF1: +VF1: 1 +VF1: +VF1: 1 +Prismatic 3 +VF2: 2-4 +VF2: 2 +VF2: 2 +VF2: 2 +Finger +Precision +Pad +Abducted +VF3: +VF3: 3-4 +3 +24 +Tip Pinch +Precision +Pad +Abducted/VF3: +VF3: +2 +VF1: +VF1: 1 +VF1: +VF1: 1-2 +Prismatic 2 +VF2: 2-3 +VF2: 2 +Lateral +Intermed +VF2: 3 +VF2: 3 +Finger +Precision +Pad +Abducted +VF3: +VF3: 3 +3 +25 +podul +iate +Side +Adducted VF3: +VF3: +3 +VF1: +VF1: 1 +VF1: +VF1: 1 +Palmar +VF2: 2 +VF2: 2 +Sphere4 +VF2: 2-4 +VF2: 2-4 +6 +Pinch +Precision +Pad +Abducted +VF3: +VF3: +2 +26 +Finger +Power +Pad +Abducted/VF3: +VF3: +m +VF1: +VF1: 1 +VF1: +VF1: 1 +VF2: 2-5 +VF2: 2-5 +VF2: 2-4 +VF2: 2-4 +10 +Power Disk +Power +Palm +Abducted +VF3: +VF3: Palm +3 +27 +QuadPod +Precision +Pad +Abducted VF3: +VF3: +3 +VF1: +VF1: 1 +VF1: +VF1: 1 +Power +VF2: 2-5 +VF2: 2-5 +Sphere 3 +VF2: 2-3 +VF2: 2-3 +11 +Sphere +Power +Palm +Abducted +VF3: +VF3: Palm +3 +28 +Finger +Power +Pad +Abducted VF3: +VF3: +3 +VF1: +VF1: 1 +VF1: +VF1: 1 +Precision +VF2: 2-5 +VF2: 2-5 +Intermed +VF2: 2 +VF2: 2 +12 +Disk +Precision +Pad +Abducted +VF3: +VF3: +29 +Stick +iate +Side +Adducted VF3: +VF3:3-53 +VF1: +VF1: 1 +VF1: +1,Palm +Precision +VF2: 2-5 +VF2: 2-5 +VF2: 2-5 +VF2: 2-5 +13 +Sphere +Precision +Pad +Abducted +VF3: +VF3: +3 +30 +Palmar +Power +Palm +Adducted VF3: +VF3: +3 +VF1: +VF1: 1 +VF1: +VF1: 1 +VF2: 2-3 +VF2: 2-3 +VF2: 2 +VF2: 2 +14 +podul +Precision +Pad +Abducted +VF3: +VF3: +31 +Ring +Power +Pad +Abducted VF3: +VF3: +2 +VF1: +VF1: Palm +VF1: +VF1: 1 +VF2: 2-5 +VF2: 2-5 +Intermed +VF2: 2 +VF2: 2 +15 +Fixed Hook +Power +Palm +Adducted +VF3: +VF3: +3 +32 +Ventral +iate +Side +Adducted VF3: +VF3: 3-5 +3 +VF1: +VF1: 1 +VF1: +VF1: 1 +Intermedia +VF2: 2 +VF2: 2 +Inferier +VF2: 2 +VF2: 2 +16 +Lateral +te +Side +Adducted +VF3: +VF3: 3-5 +2 +33 +Pincer +Precision Pad +Abducted VF3: +VF3: +2 +Index +心 +VF1: +VF1: 1 +Finger +VF2: 3-5 +VF2: 3-5 +17 +Extension +Power +Palm +Adducted +VF3: 2 +VF3: 2 +3Sensors 2022, 22, x FOR PEER REVIEW +7 of 20 + + +dividual fingers by the analysis of the functional roles of forces being applied in a grasp. +The virtual fingers oppose each other in the grasp. Virtual fingers are assigned for each +grasp in the grasp taxonomy as mentioned in Table 1. Our characterization study re- +vised the existing virtual fingers allocation and replaced with new as mentioned in Table +1. In this work, the authors aim for designing a three-finger haptic grasping interface for +virtual grasping the common objects in everyday life and tools in motor skill profes- +sions. For the proposed three-finger haptic grasping interface, the thumb assigned as +𝑉𝐹1, Index finger as 𝑉𝐹2 and other three fingers as a single virtual finger 𝑉𝐹3 as ex- +plained in section 5. +4. Characterization Experiments +4.1 Overview +For designing a multi-finger haptic grasping interface, it is quintessential to study +the characteristics of the motion and force distribution of fingers in grasping activities. +This characterization study helped to propose models for multi-finger haptic rendering +and grasping haptic devices. Here the authors conducted experiments for calculating the +finger movements, trajectories, positions, orientations on different grasping activities. +Also tracked the positions and orientations of the skeleton of each finger through motion +tracking techniques. A characterization study was carried out to compute the models +and the design of the multi-finger haptic device. +4.2 Subjects and Methods +Six subjects, 3 males and 3 females between the ages of 23 and 35 years were taking +part in this grasping experiment study with 10 common objects as listed in Table 2 with +possible grasp patterns and minimum number of virtual fingers to execute the grasp. +Each subject grasps each object for 5 seconds and repeats for 5 trials results total 300 in- +stances for the data sets. This study aimed on the force distribution and orientation of +wrist, palm, and finger in all grasping scenarios. The experiment set up consists of a leap +motion sensor [61] to track the movement of fingers and a wearable glove with Force +Sensitive Resistor (FSR) [62] to measure the force on fingers while grasping different ob- +jects and a computer display with the virtual grasping interface as shown in Figure 2. + +Table 2. Selected objects with possible grasp patterns and minimum number of virtual fingers to +execute the grasp. + +Index Finger Extension +Prismatic 4 Finger +Prismatic 3 Finger +Prismatic 2 Finger +n Sphere +Tripod Variation +[Parallel Extension +Small Daimeter +Medium Wrap. +[Extension Type +Adduction Grip +sphere 4 Finger +Sphere 3 Finger +Grasp Pattern +[large Diameter +[Power Sphere +Disk +Lateral Tripod +Inferier Pincer +[Palmar Pinch +Power Disk +Fixed Hook +Distal Type +Adducted +Light Tool +Precision I +Precision +Tip Pinch +padpeno +[Lateral +Palmar +stick +Ring +min.num. of VF|3 +3 +2 +3 +3 +3 +3 +3 +3 +2 +3 +3 +3 +3 +3 +3 +2 +3 +3 +2 +3 +3 +3 +2 +2 +3 +3 +3 +3 +3 +3 +2 +3 +2 +Objects +Ball Pen +Marker Pen +Cube box +Toy Wheel +1 +Cup +Plastic bottle +Tennis ball +Credit card +Scissors +ScrewdriverSensors 2022, 22, x FOR PEER REVIEW +8 of 20 + + + +Figure 2. Experimental setup for characterization study: (a) Motion tracking setup, (b) Force track- +ing setup for hand fingers during grasping and (c) 3d printed mounting module for FSR. +The Leap Motion sensor is a small USB peripheral device which is placed on the ta- +ble surface and connected to computer interface. Subjects grasped the 10 objects in the +hemispherical workspace area of Leap motion sensor and traced the position and orien- +tation parameters of user’s hand with an average accuracy of 0.7 mm [63]. The experi- +ment set up for tracking grasping forces in fingers as shown in Figure 2(b), users wore a +glove with FSR to measure force and pressure. But the sensing ability of FSR is depend- +ent on its contact area. To overcome this, a 3d printed mounting module placed on the +sensing area of the FSR in the fingertips of gloves as shown in Figure 2(c). The values +from FSR processed into multiple linear areas using an Arduino and measured the ex- +erted forces in Newton(N). Experiment procedure in one trial consist of pick the object, +grasp for 5 seconds, and place back the object and repeat for five trials for each object. In +each trial subject’s grasp parameters such as the position, speed, orientation, and force +are tracked. Further analysis of these primary data leads to the modelling of multi-finger +grasping haptics interface. +4.3 Results and Discussions +The 30 primary parameters were tracked and used for calculating the user's hand +grasp movements and forces. Here the x-axis is defined as forward-backward, the y-axis +as right-left, and the z-axis as up-down. Roll (γ) is taken to be about the x-axis, pitch (β) +about the y-axis and yaw (α) about the z-axis. +4.3.1. Direction, Trajectory, and Rotation of hand in grasping + + +(a)Motiontrackingsetup +(b)Forcetrackingsetup +(c)3dprintedmountingmoduleforFSR(a)Directionofhand +(b)Trajectoryofhand +(c)Rotationofhand +100 +Direction.x +-0.4 +Trajectory +Yaw +Direction.y +O +Position +Pitch +Direction.z +50 +Position (cm) +0.5 +-0.6 +Angle (degree) +Roll +axis +0 +N-0.8 +& +-1 +-50 +-0.5 +1 +-100 +0 +0 +0.2 +-1 +-150 +0 +500 +1000 +1500 +Yaxis +-1 +-0.4 +-0.2 +0 +500 +1000 +1500 +Time (cs) +Xaxis +Time (cs) +(d)Boxplotofyawrotationofhand +(e)Boxplotofpitchrotationofhand +(f)Boxplotofroll rotationofhand +十 +-20 +5 +Angle (degree) +0 +Angle (degree) +-40 +-5 +-60 +-10 +Angle +20 +-80 +-15 +-100 +0 +-20 +-120 +-140 +1 +1 +1Sensors 2022, 22, x FOR PEER REVIEW +9 of 20 + + +Figure 3. Plots for the movement of hand: (a) Direction of hand, (b) Trajectory of hand, (c) Rotation +of hand, (d) Box plot of yaw rotation of the hand, (e) Box plot of pitch rotation, (f) Box plot of roll +rotation of the hand. +The direction, trajectory, and rotation of hand in grasping (based on the experiment +data set) is plotted as shown in Figure 3. The direction of the Hand in the 3D co-ordinate +axes is shown in Figure 3(a). The x, y and z components of hand direction are spanning +from -3 mm to 1 mm, -2 mm to 8 mm, and -10 mm to -6 mm respectively with standard +deviation (SD) of 0.11, 0.20 and 0.08. The trajectory of hand movement direction is plot- +ted in Figure 3(b). The movement in the y-direction is more than x and z-direction. The +ellipse region in the plot represents the trajectory of hand exactly in the grasping time. +Figure 3(c) plots the rotation of hand in the coordinate frame during grasping scenarios. +The span of the roll is more than yaw and pitch. The boxplot of hand rotation clearly ex- +plains the distribution of the yaw, pitch, and roll of hand in grasping exercises shown in +Figure 3(d)-(f). The minimum angle of hand yaw is -22.61° and the maximum is -0.86° +with an average of -11.57° and SD of 6.7°. Inter Quartile Range (IQR) of hand yaw is +9.18°. The minimum angle of hand pitch is -11.06° and the maximum is 56.04° with an +average of 20.04° and SD of 13.09°. Inter Quartile Range (IQR) of hand pitch is 19.21°. +The minimum angle of hand roll is -136.69° and the maximum is -12.90° with an average +of -53.68° and SD of 32.29°. Inter Quartile Range (IQR) of hand roll is 50.14°. +4.3.2. Position and trajectory of wrist and palm in grasping +The wrist is acted as the basement for most of the grasping scenarios. The infor- +mation about the movement of the wrist is helpful for characterizing the basement of +multi-finger gripper modules. The Position and trajectory of the wrist in grasping dur- +ing the experiments were tracked and plotted as shown in Figure 4. Here the position of +wrist n the 3D coordinate frame shows the span of position in x, y, and z-axes. It is clear +from the plot that the span of the position of the wrist in grasping is average of 50 mm. +Also, the plot of the trajectory of wrist shows a minimum workspace of movement for +the wrist in grasping is in shape of a square with each side 50 mm. The thick ellipse re- +gion represents the data exactly during the grasping scenarios after noise filtering. Scat- +ter plots in Figure 4 gives a clearer picture of the position of the wrist in the 3d space and +2D planes (x-y, y-z, and x-z). + +Figure 4. Plots for the movement of the wrist: (a) Position of the wrist, (b) Trajectory and scatter +plot of wrist, (c) Position of the wrist in the x-y plane, (d) Position of the wrist in y-z plane, and (e) +Position of the wrist in x-z plane. + +(a)Positionofwrist +(b)Trajectoryandscatterplotofwrist +200 +Position.x +Trajectory +Position.y +200 +Position +150 +Position.z +(mm) +sixe +150 +Position +100 +N100 +50 +50 +120 +100 +80 +0 +60 +200 +600 +800 +1000 +1200 +80 +0 +400 +40 +60 +Time (cs) +Yaxis +20 +Xaxis +(c)Position ofwrist inX-Yplane +(d)PositionofwristinY-Zplane +(e)PositionofwristinX-Zplane +120 +160 +160 +O +110 +00 +O +140 +140 +100 +120 +O +120 +axis +90 +8 +N +N +80 +O +80 +80 +O +70 +60 +60 +O +60 +40 +40 +30 +40 +50 +60 +70 +80 +60 +70 +80 +90 +100 +110 +120 +30 +40 +50 +60 +70 +80 +Xaxis +Yaxis +XaxisSensors 2022, 22, x FOR PEER REVIEW +10 of 20 + + +Next, to the wrist, the palm is also an important part of better grasping. In some of +the grasping types, the palm acts as an additional supportive area for the successful +grasp. So, it is important to study the movement factors of palm while grasping. This +will help to model the minimal number of virtual fingers (VF) for a general haptic grasp- +ing interface. The position and trajectory of the palm are plotted in Figure 5. It shows +that there is not much deflection in the position of palm while grasping time in three ax- +es. The trajectory of the palm shows the workspace of palm during the grasping scenari- +os. After filtering out the noise in the dataset, the maximum span in the 3D space for a +grasping activity is 30 mm. + +Figure 5. Position and trajectory of palm: (a) Position of palm, (b) Trajectory and scatter plot of +palm, (c) Position of palm in the x-y plane, (d) Position of palm in the y-z plane, and (e) Position of +palm in the x-z plane. +4.3.3. Angle of grasp +The angle of grasp is one of the common measurements used to describe grasping. +Grasp angle describes the angle of the hand in grasping in relation to the Wrist position. +The grasp angle influences comfort and easiness in grasping. A little bend in the wrist +helps to maintain the grasp angle suitable for comfortable grasping of objects to get a +proper grip. This will aid controlling the gripped objects easier. Figure 6 shows the vari- +ous analysis plots of grasp angle tracked during the experimental scenarios. + +(a)Positionofpalm +200 +(b)Trajectoryandscatterplotofpalm +Position.x +Position.y +Trajectory +150 +Position.z +200 +Position +(ww) +100 +100 +axis +Position +50 +N +0 +-100 +0 +200 +150 +50 +60 +-50 +100 +30 +40 +0 +200 +400 +600 +800 +1000 +1200 +50 +10 +20 +Time (cs) +Yaxis +Xaxis +(c)Positionofpalm inX-Yplane +(d)PositionofpalminY-Zplane +(e)Positionofpalm inX-Zplane +180 +150 +150 +160 +100 +100 +axis +80 +8 +50 +50 +120 +N +N +O +05 +O +O +0 +0 +100 +00 +O +80 +-50 +-50 +10 +20 +30 +40 +50 +60 +80 +100 +120 +140 +160 +180 +10 +20 +30 +40 +50 +60 +Xaxis +Yaxis +XaxisSensors 2022, 22, x FOR PEER REVIEW +11 of 20 + + + +Figure 6. Plots for the angle of grasp: (a) Angle of grasp, (b) Box plot of grasp angle, (c) Grasp an- +gle with a normal distribution, (d) Probability plot for normal distribution, and (e) Quantile- +Quantile plot of grasp angle. +The angle of grasp is spanning from minimum 0°to maximum of 3.12° with an av- +erage value of 1.33° and SD of 1.10 as shown in Figure 6(a). The grasp angle data are +charted as a box and whisker plot as shown in Figure 6(b). This will help to show the +shape of the distribution, its central value, and its variability. The first quartile of the +grasp angle values lies between 0° to 0.41°, second lies between 0.41° to 1.03°, third lies +between 1.03° to 2.44° and the final lies between 2.44° to 3.14°. 75% of the grasp angle is +below 2.44°. So, most of the grasp types can perform comfortably with a maximum angle +of grasp 2.44°. The ideal maximum angle of grasp for the proposed gripper module +should be between 2.5° to 4°. +Figure 6(c) plots a histogram of grasp angle in data using the number of bins equal +to the square root of the number of elements in data and fits a normal density function. +The bell curve fits the normal distribution with an SD of 1.10. 68% of the data falls with- +in one SD of the mean 1.33°. The standard deviation controls the spread of the distribu- +tion. Here the larger standard deviation indicates that the data is spread out around the +mean and the normal distribution is flat and wide. Figure 6(d) draws a normal probabil- +ity plot, comparing the distribution of the grasp angle data to the normal distribution. +The plot includes a reference line helped to judge whether the data follow a normal dis- +tribution. The plot shows that the normal line fit the data except the tails because of the +outliers. Figure 6(e) displays a quantile-quantile plot of the sample quantiles of grasp +angles versus theoretical quantiles from a normal distribution. The plot is close to linear +in the IQR, so the distribution of grasp angle is normal during the grasping. In the dura- +tion of not grasping the plot shows the distribution is not normal. +4.3.4.Sphere of grasp +The spherical grip is the most used grasp in everyday life [64]. It is important to an- +alyze the sphere of grasp in common grasping scenarios. Here the authors tracked the +center and radius of the sphere of grasp in all the grasping scenarios in the experimental +procedure. Figure 7 shows the various plots related to center and radius of the sphere of +grasp. Through the experiment, the position of the center of the sphere is spanning from +minimum (-147 mm, 48 mm, -48 mm) to maximum (181 mm, 259 mm, 85 mm) in x, y, +and z-axes. After filtering out the non-grasping samples, the span of sphere grasp is re- +duced from minimum (-19 mm, 42 mm, -3 mm) to maximum (68 mm, 150 mm, 78 mm). +The scattering and trajectory of the sphere of grasp shown in Figure 7(b). + +(a) Angle of grasp +(b)Boxplotofgraspangle +(c)Graspanglewithnormaldistribution +4 +200 +Grasp angle +3 +Normal distribution +150 +degr +100 +Angle +50 +0 +0 +0 +0 +200 +400 +600 +800 +1000 +1200 +-2 +0 +2 +4 +6 +Time (cs) +Angle (degree) +(d)Probabilityplotfornormaldistribution +(e)Quantile-Quantileplotofgraspangle +0.9999 +8 +6 +Data +Quantiles of Input Sample +Duration of not grasping +0.995 +Normal +6 +Inter Quartile Range +%% +十 +Data +Probability +4 +0.75 ++ +0.5 +2 +0.25 +0.05 +0 +0.005 +-2 +.0005 +0.0001 +0 +1 +2 +3 +4 +-2 +0 +2 +4 +Angle (degree) +StandardNormalQuantilesSensors 2022, 22, x FOR PEER REVIEW +12 of 20 + + + +Figure 7. Center and radius plots for sphere of grasp: (a) Position of center of sphere of grasp, (b) +Trajectory and scatter plot of center of sphere, (c) Radius of sphere of grasp, (d) Histogram of +sphere radius with normal distribution, and (e) Box plot of sphere radius. +The volume of grasp can be represented by analyzing the radius of the sphere of +grasp. Figure 7(c)-(e) shows the plots for the radius of the sphere of grasp. In the whole +data tracked during the experiment, the radius of the sphere of grasp spanning from +minimum 30 mm to 188 mm. In the box plot showing in Figure 7(e) 75% of the data is +less than 10 mm and 50% of data is 64 mm. Again, after filtering out the noises the radi- +us of the sphere of grasp is spanning from 31 mm to 50 mm. So, the authors targeting to +model the grasping module with a radius of the sphere of grasp is 50 mm. +4.3.5.Distance of Pinch + +Figure 8. Plots for the distance of pinch: (a) Distance of pinch, (b) Box plot of pinch distance, (c) +Pinch distance with a normal distribution, (d) Probability plot for normal distribution, and (e) +Quantile-Quantile plot of pinch distance. +The pinch distance is the distance between two fingers in grasping. Pinch gestures +are common for touch screens. The experiment interface tracked the pinch distance + +(a)Positionofcenterofsphereofgrasp +(b)Trajectoryandscatterplotofcenterofsphere +300 +Center.x +100 +Trajectory +Center.y +Position +200 +Center.z +50 +(mm) +axis +100 +Position +N +0 +0 +-50 +400 +-100 +200 +-200 +50 +100 +150 +200 +0 +500 +1000 +1500 +Yaxis +0 +-150 +-100 +-50 +0 +Time(cs) +Xaxis +(c)Radiusofsphereofgrasp +(d)Histogramofsphereradiuswithnormaldistribution +(e)Boxplotofsphereradius +250 +150 +data +bellcurve +200 +200 +Radius (mm) +100 +150 +Radius +100 +100 +50 +50 +50 +0 +0 +0 +500 +1000 +1500 +-100 +0 +100 +200 +300 +Time (cs) +Radius (mm)(a) Distance of pinch +(b)Boxplotofpinchdistance +(c)Pinchdistancewithnormaldistribution +120 +80 +Pinch distance +Frequency of occurrence +100 +100 +Normal distribution +(ww) +60 +80 +Distance (mm) +80 +Distance +60 +60 +40 +40 +40 +20 +20 +0 +0 +200 +400 +600 +800 +1000 +1200 +-100 +-50 +0 +50 +100 +150 +200 +time (cs) +Distance (mm) +(d)Probability plotfornormal distribution +(e)Quantile-Quantileplotofpinchdistance +0.9999 +300 +Data +Duration of not grasping +Quantiles of Input Sample +X +Normal +Inter Quartile Range +0.995 +200 +Data +%% +Probability +0.75 +100 +0.5 +0.25 +0 +% +9.005 +-100 +0.0005 +0.0001 +-200 +0 +50 +100 +150 +200 +-4 +-2 +0 +2 +4 +Pinch distance (mm) +StandardNormalQuantilesSensors 2022, 22, x FOR PEER REVIEW +13 of 20 + + +throughout the experiment and various plots are shown in Figure 8. The maximum +pinch distance traced in the experiment setup is 110 mm. 75% of data is less than 90 mm +and 50% is less than 70 mm as per the boxplot showed in Figure 8(b). The normal proba- +bility plot is shown in Figure 8(d) compared the distribution of the pinch distance to the +normal distribution. The reference normal line fits the data in a range of 20 mm – 100 +mm. Figure 8(e) displays a quantile-quantile plot of the sample quantiles of grasp angles +versus theoretical quantiles from a normal distribution. This quantile-quantile plot is +close to linear in the IQR, so the distribution of pinch distance is normal during the +grasping. In the duration of not grasping the plot shows the distribution is not normal. +So, the ideal values for the radius of the sphere of grasp for our proposed model are +from 20 mm to 100 mm. +4.3.6.Finger motion parameters +When a person lifts any object, his fingers align in a particular way. This must be re- +flected by the device so that the user is at ease when he uses the device. The experiment +setup traced the length and width of fingers and angular values between fingers in +grasping objects differ in size and dimensions. These angular values measured between +fingers must be replicated by the device. Unless the user feels comfortable when using +the device, its intended purpose cannot be met. A base point was marked at the center of +the outside of the palm. A line was drawn from the base point to the base of the middle +finger. This was the baseline at 0 degrees as seen in Figure 9(a). Lines were also drawn to +the base of the index and the thumb. The angles between the thumb and the index finger +and the angles between the middle and the index fingers were measured during the +grasping scenarios. + +Figure 9. Length, width, and angle of fingers. +The average length and width of fingers of the subjects took part in the experiment +shown in Figure 9(b). The average length and width of Thumb are 50 mm and 18 mm, +the index finger is 57 mm and 17 mm, the middle finger is 67 mm and 17 mm, ring finger +is 61 mm and 16 mm, and little finger is 48 mm and 14 mm. This will help the authors to +model the wearable multi-finger grasping interface to fit the user’s fingers. The meas- +ured angles were tabulated and plotted as shown in Figure 9(c), proves that angular +measurements between fingers in grasping and lifting of objects does not depend on +gender. The angle between thumb and index finger is in range of 50-60degree, index and +middle finger is in range of 20-30 degree and thumb and middle finger is in range of 70- + +(a)Length,widthandangleoffingers +(b)AverageLengthandwidthoffingersofsubjects +ODegrees +Length +Width +FingerLength +70 +Length & Width (mm) +60 +30Degrees +FingerWidth +30 +20 +10 +Thumb +Thumb +Index +Middle +Ring +Little +Range +(c)Anglesbetweenfingers +100 +90 +80 +70 +60 +90Degrees +50 +40 +30 +20 +10 +0 +Average +Maximun +Minimum +Average +Maximum +Minimum +Angle +Angle +Men +Women +middle andindex +IndexandThumb +mMiddleandThumbSensors 2022, 22, x FOR PEER REVIEW +14 of 20 + + +90 degree. This provided the data on how far apart the finger holders must be placed +when designing the prototype for comfortable grasping and lifting of objects. + +Figure 10. Position of movement of fingers +The motion parameters of fingers in grasping are so important to characterize and +model the multi-finger grasping module. Here The authors traced all the five-finger’s +movement and force distributions. The movement of the tip of five fingers of the hand in +all the grasping scenarios during the experiment was plotted in Figure 10. The move- +ment of the fingertip is minimal in both x and z-axes is around -50 mm to +50 mm. The +movement in x and z-axes are varied for fingers, especially it is very less in case of +thumb, ring finger and little finger in the range of -20 mm to +20 mm. For index and +middle finger, it is an almost same range of -50 mm to +50 mm. The movement in the y- +axis is more compared to the other axes is around 50 mm to 200 mm for all fingers. + +Figure 11. (a). Span of the trajectory of fingers and (b) Workspace triangle of three Virtual Fingers. +The span of the trajectory of all five-finger movement in space during the grasping +experiment is plotted together in Figure 11(a). This plot gave a clear idea about the ac- +tive participation of all the fingers during grasping. Thumb is more active in grasping +and gradually decreasing towards little finger. The thump, index and middle fingers +more actively participate in the grasping scenarios than the ring and little fingers. Also, + +(a)Position of thumb +(b)Position of indexfinger +(c) Position of middle finger +300 +200 +200 +Position (mm) +200 +Positions (mm) +Position (mm) +100 +100 +100 +-100 +-100 +-100 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +Samples +Samples +Samples +(d) Position of ring finger +(e) Position of little finger +150 +150 +Tip position.x +Tip position.y +Position (mm) +100 +Position (mm) +100 +Tip position.z +50 +50 +0 +-50 +-50 +0 +20 +40 +60 +80 +0 +20 +40 +60 +80 +Samples +Samples(a)Spanoftrajectoryoffingermovement +(b)WorkspacetriangleofthreeVirtualFingers +Thumb +Thumb +100 +Index finger +100 +Index finger +Middle finger +Middle finger +Ring finger +80 +Littlefinger +axis +0 +60 +N +-100 +40 +220 +axis +20 +200 +N +0. +180 +-20 +160 +-40 +axis +140 +120 +-60 +250 +100 +200 +150 +80 +100 +100 +60~ +60 +80 +50 +50 +20 +40 +Yaxis +0 +40 +-20 +0 +0 +-50 +Xaxis +-60 +-40 +XaxisSensors 2022, 22, x FOR PEER REVIEW +15 of 20 + + +the workspace of the ring and little finger aligns with the middle finger as shown in Fig- +ure 11(a). So, authors proposed to group middle, ring, and little fingers to one Virtual +Finger (VF) for multi-finger grasping interfaces. Thumb and index fingers can act as oth- +er two Virtual Fingers. The workspace triangle for three VF is shown in Figure 11(b). +4.3.7 Grasping forces on Fingers +The force values obtained through FSR sensors during the grasping exercises were +tabulated and the average force was calculated for each person. As shown in Figure 12, +forces are not more than 10N was experienced by the user. It can also be seen that the +thumb experienced the maximum force, whereas the middle and the index finger expe- +rienced somewhat similar forces. + +Figure 12. Average grasping force values on each finger. +5. Computational Model for Tripod Haptic Grasp +The multi-finger perception studies confirmed the hypothesis that the minimal con- +figuration that allows most of the grasps is the three-finger tripod grasp. Based on the +human grasp analysis, a conceptual, computational model is presented here for the +three-finger tripod haptic grasping interface. Previous researchers worked on force +models [65] and virtual linkages [66] for multi-grasp manipulations. +The concept of the virtual finger [15] has been postulated as an abstract representa- +tion through which the human brain plans are grasping tasks [60]. The virtual finger is a +functional unit of several fingers work together comprised of at least one real physical +finger (which may include the palm). This effectively reduces the many degrees of the +human hand to those that are deemed necessary to perform the grasping task. This con- +cept replaces the analysis of the mechanical degrees of freedom of individual fingers by +the analysis of the functional roles of forces being applied in a grasp. Here the concept of +the virtual finger was implemented to reduce the realistic five-finger grasping to virtual +three-finger grasping. The characterization study revised the existing virtual fingers al- +location and replaced with new tripod virtual fingers allocation. + +Average Force (N) on Each Finger +9 +8 +6 +5 +4 +3 +2 +1 +0 +7 +2 +3 +4 +5 +6 +7 +8 +10 +Thumb +indlex +MiddleSensors 2022, 22, x FOR PEER REVIEW +16 of 20 + + + +Figure 13. Haptic computation model for three-finger tripod haptic grasping interface. +The haptic computation model is shown in Figure 13 was implemented to create the +suitable force feedback for the tripod haptic grasping interface. Each finger holder is +connected to each slider and these sliders are responsible for creating forces to each fin- +ger attachment point on the grasping interface through the closed-loop belt system. +Three proxies and Haptic Interactive Points (HIP) are assigned for three fingers. Based +on the position information received from each slider, the position of proxy and HIP are +updated. Collision detection algorithms detected the collision of each fingertip with the +virtual object, these collision points act as simple virtual walls. Then applying the God +object algorithms to these three proxies and calculated the resultant forces as shown in +Figure 13. Each actuator in the slider generates forces and these forces are transferred to +the fingertip through the attached finger holder. In this proposed model for multi-finger +tripod haptic grasping, the thumb is assigned as 𝑉𝐹1, Index finger as 𝑉𝐹2 and other three +fingers as a single virtual finger 𝑉𝐹3. The proposed model including object and virtual +fingers in the virtual interface measures collision and progress of interactions in terms of +applying forces and movements of the fingers and computes force feedback to be pro- +vided by the haptic interfaces. It covers the dependence of the force feedback and the ef- +fect of finger motions on the tripod grasp. A basic grasping touch in virtual reality is de- +fined as a touch that provides vertical force feedbacks on the contact surface coincident +with the reverse direction of the fingers in the space. +Forces and moments from the individual virtual fingers are considered for the ren- +dering of resultant forces for the model. Three proxies and Haptic Interactive Points +(HIP) are assigned for three virtual fingers. Based on the position information received +from each 𝑉𝐹𝑖, the position of proxy and HIP are updated. When collision detection al- +gorithms detected the collision of each 𝑉𝐹𝑖 with the virtual object, these collision points +act as simple virtual walls. The authors assume that only forces act at the grasp points. +Let 𝑢𝑖𝑗 is the unit vector along the virtual finger i to j, 𝑣𝑖 be the vector from the object ref- +erence point o to the virtual finger grasp point. Let 𝐹𝑖 be the force exerted on objects +through each virtual finger 𝑉𝐹𝑖 and F be the vector. Based on the object shape, size, and +stiffness the applying forces on objects by the virtual fingers are different. +𝐹𝑖 = 𝑘 𝑋𝑖 +(1) +where i = 1,2,3. + + 𝐹 = [𝐹1 +𝐹2 +𝐹3]𝑇 + +(2) + +Slider3 +Slider2 +R2=-k.X2 +R3=-k.X3 +X3 +X2 +VirtualFinger3 +F3 +VirtualFinger2 +F2 +HIP +Proxy +F1 +X1 +VirtualFinger1 +R1=-k.X1 +Slider1Sensors 2022, 22, x FOR PEER REVIEW +17 of 20 + + +Grasp perception rate depends on the force applied by the fingers 𝐹𝑖, the area of +contact between the fingers and the object A, the distance between the proxy and HIP X +and the change in position of the HIP ΔX. Mathematically, the perception rate is + 𝑃 = +𝑘 𝛥𝑋 𝐹𝑖 +𝐴 + + + + (3) +where k is a constant that depends on the material of the object. Let 𝑅𝑖 be the result- +ant force exerted on each virtual finger 𝑉𝐹𝑖 and R be the vector. Then applying the God +object algorithms [67] to three proxies and calculated the resultant forces 𝑅𝑖. + 𝑅 = [𝑅1 +𝑅2 +𝑅3]𝑇 + + (4) +The relationship between the applied forces F and resultant forces R would be giv- +en by + 𝑅 = 𝑢𝐹 + + + + + + +(5) +were + + 𝑢 = [ +𝑢11 +𝑢12 +𝑢13 +𝑢21 +𝑢22 +𝑢23 +𝑢31 +𝑢32 +𝑢33 +] + (6) +The relationships between the resultant force R, the resultant moment m, and the +applied forces F are given by + 𝐹 = 𝑣 [𝑅 +𝑚]𝑇 + + (7) + +were + 𝑣 = [𝑣1 +𝑣2 +𝑣3]𝑇 + + (8) +and + 𝑚 = [𝑚1 +𝑚2 +𝑚3]𝑇 + (9) +Force feedback on the grasping scenarios along virtual fingers has two components; +the force feedback resisting the grasping motion and a frictional force (𝐹𝑟), represented +by + 𝑅 = 𝑢𝐹 + 𝐹𝑟 + + + (10) + +An experiment was done to evaluate the rendered force to the fingers through the +virtual fingers using the proposed tripod haptic grasp model. The results plotted as +shown in Figure 14. Initially, the grasping forces in the real grasping cases were tracked +using the FSR and plotted as shown in Figure 14(a). In case of the real grasping scenari- +os, the forces exerted by the thumb is in the range of 6-8N which is greater than the force +exerted by the index and middle finger. The force exerted by the index and middle fin- +gers in in the range of 2-5N and 3-5N respectively. The same experiment was carried out +to measure the rendered forces in haptic grasping interfaces during virtual grasping. +Force is measured using the FSR sensor. The FSR was placed inside the finger holder of +the gripper at the location of the fingertip contact. The sensing part of FSR facing the flat +area of the holder and the user keeps a finger on top of the FSR. The motors were actuat- +ed by the haptic rendering model. The tracked rendered forces were plotted in Figure +14(b). This graph shows that the range of rendered forces in virtual grasping is almost in +the same range of forces in real grasping scenarios. The range of force rendered at the +Thumb, index and middle finger holders is 5.7-7.5N, 2.6-4.4N and 3-4.5N respectively. +This demonstrates that the three-finger haptic grasping interfaces able to provide realis- +tic grasping haptic feedbacks to the users. Also, the experiment setup was traced the +peak force that can provide the gripper device. As plotted in Figure 14(c), the device at- +tained a peak force of 10.4N, 10.1N, and 10.2N at Thumb, index, and middle finger hold- +ers respectively. It was found that the interface can give approximately 10N force which +is greater than the force when lifting objects as found out in Section 3.1. Finally, the Fig- + +Sensors 2022, 22, x FOR PEER REVIEW +18 of 20 + + +ure 14(d) shows the performance index of three-finger haptic grasping interface in haptic +feedback and finger manipulability. + +Figure 14. Evaluation results of the haptic grasping interface: (a) Exerted force tracked in the real +grasping scenarios, (b) Force rendered in haptic grasping interface during virtual grasping, (c) +Peak force generated in the haptic grasping interface, and (d) Performance index of the haptic +grasping interface. +6. Conclusions +The aim of this research was to come up with an analysis, model, and design of a +three-finger haptic interface. A detailed literature review was carried out regards the +anatomy, kinematics, and dynamics of hand. Also, a focused survey on the different +human grasps, prehension patterns and grasp taxonomy. As part of this work, authors +carried out characterization studies in most of the major aspects of human grasping. The +position, orientation, and forces of hand, wrist, palm, and fingers were analyzed, and the +results were discussed. The characterization studies confirmed the hypothesis that the +minimal configuration that allows most of the grasps in grasp taxonomy is the three fin- +gers grasping. This detailed characterization study leads to the design of a three-finger +haptic grasping interface as an extension. +As a future work, I am planning to work on the multi-Finger grasping interface for +bimanual scenarios which provides the users a complete immersed grasping manipula- +tion in the virtual and remote environment. + +Funding: This research received no external funding. +Institutional Review Board Statement: Not applicable. +Informed Consent Statement: Not applicable. +Data Availability Statement: Not applicable. +Acknowledgments: I wholeheartedly thank the wonderful team at AMMACHI Labs for their +support, encouragement and for providing constructive criticism and valuable inputs during the +various stages of this work. +Conflicts of Interest: The author declares no conflicts of interest. The funders had no role in the +design of the study; in the collection, analyses, or interpretation of data; in the writing of the man- +uscript; or in the decision to publish the results. +References +1. +Wolff, C. The human hand (Vol. 1). Routledge, 2015 +2. +Napier, J. R., & Tuttle, R. H. Hands. Princeton University Press,1993. + +(a).Exertedforcetrackedintherealgraspingscenario. +(b).Forcerenderedinhapticgraspinginterfaceduringvirtualgrasping +10 +Thumb +10 +Index +Middle +Force +Force +2 +2 +0 +0 +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +100 +Time (s) +Time (s) +(c).Peakforceinhapticgraspinginterface. +(d).Performanceindexofhapticgraspinginterface +12 +4 +Thumb +Hapticfeedback +Index +Fingermanipulability +11 +Middle +3 +PI(x10%) +Force +10 +2 +9 +8 +0 +0 +20 +40 +60 +80 +100 +5 +10 +15 +20 +25 +30 +Time (ms) +Time (s)Sensors 2022, 22, x FOR PEER REVIEW +19 of 20 + + +3. +Susman, R. L. Hand function and tool behavior in early hominids. 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In Proceedings +of Eurohaptics conference (pp. 82-85), July 2002. + diff --git a/7NAyT4oBgHgl3EQfQfbq/content/tmp_files/load_file.txt b/7NAyT4oBgHgl3EQfQfbq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3f99dc627bde803d5b2636a5983f291e6df82a27 --- /dev/null +++ b/7NAyT4oBgHgl3EQfQfbq/content/tmp_files/load_file.txt @@ -0,0 +1,1520 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf,len=1519 +page_content='Sensors 2022, 22, x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='3390/xxxxx www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='mdpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='com/journal/sensors Article Multi-Finger Haptics: Analysis of Human Hand Grasp towards a Tripod Three-Finger Haptic Grasp model Jose James Brown University, Providence, RI, 02912, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Correspondence: jose_james@brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Abstract: Grasping is an incredible ability of animals using their arms and limbs in their daily life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The human hand is an especially astonishing multi-fingered tool for precise grasping, which helped humans to develop the modern world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The implementation of the human grasp to virtual reality and tele robotics is always interesting and challenging at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' In this work, au- thors surveyed, studied, and analyzed the human hand grasping behavior for the possibilities of haptic grasping in the virtual and remote environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This work is focused on the motion and force analysis of fingers in human hand grasping scenarios and the paper describes the transition of the human hand grasping towards a tripod haptic grasp model for effective interaction in virtu- al reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Keywords: hand grasp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' grasp analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' multi-finger haptics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' haptic grasp interface 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Introduction The human hand is a highly skilled, prehensile, multi-fingered, perception and ma- nipulation organ at the distal end of the arm [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Prehensility is the quality of an ap- pendage or organ that has adapted for grasping or holding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' In the past decade’s, re- searchers [2] explored the various aspects of the evolution, morphology, anthropology, and social significance of the human hand as a tool [3], as a symbol [4] and as a weapon [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Humans cognitively manipulating a variety of objects in daily life using various hand configurations, resulting from changing the position, orientation and placement of hand and fingers based on the object properties such as its weight, shape, texture, fric- tion, hardness etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Such a variety of grasps is possible because of the dexterity, various degrees of freedom, and the great control strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Hands are associated with the aligning capability of the body, kinesthetic percep- tion of the limb and the richest tactile sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Many researchers studied the anatomy, muscles, biomechanics, kinematics, functionalities, and skills of human hand [6,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' These studies helped to do more focused research on human hand prehension [8] and grasp [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Based on all these studies hand grasp types are classified and different grasp tax- onomies were arising in the literature [11,12], covering a broad range of domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' As technologies like virtual reality and tele-robotics being progressed, humans started interacting with virtual and remote environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' But the integration of hand grasping into virtual and remote environments still challenging because of the complex architecture behind it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The extensive research in the analysis and synthesis of human grasps [13,14] over the past years has provided a basic theoretical framework towards better progress in human-computer interaction [15], robotic grasping [16] and dexterous manipulation and lead to the design of artificial robotics hands and arms for the pros- thetic application [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Since the last few decades, researchers put effort to mimic the human hand to design robotic grippers [18], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' But still, these frameworks need more extensive studies for the practical implementation of the direct involvement of humans Citation: James, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Title.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Sensors 2022, 22, x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='3390/xxxxx Academic Editor: Firstname Last- name Received: 27 April 2022 Accepted: date Published: date Publisher’s Note: MDPI stays neu- tral with regard to jurisdictional claims in published maps and insti- tutional affiliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Copyright: © 2022 by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Submitted for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='org/license s/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='0/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' sehsorsMDPIBYSensors 2022, 22, x FOR PEER REVIEW 2 of 20 to grasp objects in virtual and remote environments with multi-fingers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The grasping force depends on the orientation of fingers, palm, and wrist [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The force output on the fingertip is highly joint dependent and provides stable grasp and precise manipulation of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Previous works [20], more focused on muscle activation patterns and resultant positions/forces as a function of the joints as well as subject inde- pendent leads to the structural variability in human hands [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Motion and force analy- sis of fingers in various hand manipulation actions can be observed, learned, and ana- lyzed to come up with better framework and devices for virtual and remote manipula- tions [22,23] Adapting the gripping functions, manipulation capabilities, kinematics, dy- namics, and size of the human hand, will accelerate the design of the human-like artifi- cial arms and hands for the direct grasping interaction in the virtual world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' As a previous work, authors designed haptic interfaces for tweezer pinch grasp [24] and tripod grasp [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Also implemented the hand grasp through augmented haptics by means of custom-made attachments for virtual tools in motor skill training interfaces [26,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The aim of this work is to understand human grasping and manipulations, sur- veying different types of grasp taxonomies, study the characterization of hand in grasp- ing, model a tripod haptic grasp and design an interface for multi-finger haptic grasping which can offer better interactions with tasks in the virtual and remote environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Human Hand The human hand is a prehensile, multi-fingered astonishing organ/tool of complex engineering used to carry and manipulate objects [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' In view of human grasping, a short description of hand anatomy, mechanisms and kinematics will help to model a multi- finger haptic grasp and design a haptic grasping interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Hand Anatomy The human hand includes mainly three areas and five digits (fingers): Thumb, In- dex finger, Middle finger, Ring finger, and Little finger are numbered 1-5 as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The palm with fingers holds most pressure and support for the hand to grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Fingers are the densest areas of nerve endings and the richest source of tactile feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' So, hands are the primary tool for a sense of touch and positioning capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The hu- man hand consists of 27 bones and 45 muscles with at least 23 degrees of freedom at the joints [6] including the wrist as shown in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Skeleton system and arches of human hand The human hand can grasp objects and do daily tasks by forming bony arches: Longitudinal arches, transverse arches, and oblique arches as drawn in Figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Lon- (a)HumanHandskeletonsystem (b)HumanHandarches DIPjoint PIP joint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' MCPjoint Distal Longitudinalarch 1 Middle Phalanges (Fingers) Obliquearch Proximal Metacarpaltransversearch Metacarpals Distaltransversearch (Palm) Hamate Trapezium Pisiform Carpals Trapezoid- Scaphoid Triquetrum (Wrist) Capitate Lunate Carpaltransversearch Radius UlnaSensors 2022,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' x FOR PEER REVIEW 3 of 20 gitudinal arches shaped by the finger bones and their associated metacarpal bones,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' transverse arches by the carpal bones and the distal ends of the metacarpal bones,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' and oblique arches by the thumb and four fingers [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' These arches are the basic frames for various grasp patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The extrinsic and intrinsic muscles in hand controlled the motion of the fingers and making grasping possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Thumb together with the index and middle finger forms the dynamic tridactyl configuration responsible for most grips which not requiring force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The ring and little fingers are more static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' So, for this multi-finger haptic grasping interface model, authors considered the motion of the first three fingers the thumb, index finger and middle finger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='2 Hand Kinematics Hand’s numerous patterns of action was resulted from the skeleton mechanical sys- tems and the twenty-four muscle groups regulated by the diverse motor and sensory nerve pathways [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The MCP and PIP joints exhibit a common rotation pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The virtual center of ro- tation of hand is the center of curvature of the distal end of the proximal member [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The lateral rotation of fingers is small in the MCP joints and decreasing towards the pha- langeal hinge joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The thumb has the greater mobility in the CMC articulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Other fingers being more arched from index to little finger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The thumb, palm, and fingers to- gether permitted to grasp a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='75-inch cylinder at about 45 degrees to the radioulnar axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Bunnell [29] considers this "an ancestral position ready for grasping limbs, weapons, or other creatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='" The major wrist motions are extension (or dorsiflexion), flexion (or volar flexion), radial flexion and ulnar flexion, based on the angle of rotation of the wrist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The fixation movements and ballistic movements are also major types of movements in the hand [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The hand with the fully extended arm can be rotated through almost 360 degrees with the participation of shoulder and elbow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' From palm up to palm down, the hand can be rotated through 180 degrees, with the elbow flexed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Thumb can provide a variety of flex- ions extension patterns of the phalanges for any given metacarpal position and due to the relative mobility of the CMC joint, which allows the thumb to act in any plane neces- sary to oppose the digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' In the principal opposition cases and prehensions, the plane of the thumb action is inclined 45 to 60 degrees to the palmar plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' In lateral prehension, the plane is approximately parallel to the palmar plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='3 Hand Dynamics Fick [31] investigated the actions and contractile forces of hand muscles and esti- mated the summed forces of the individual muscles participating in the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' But the measured isometric forces are only 10% of the total forces because of the effective small moment arm upon any of the wrist or hand joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The flexor-extensor forces in the wrist and the prehensile forces in the hand varied with wrist angle and it reaches a maximum at a wrist angle of about 145 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' So, for very strong prehensions, wrist likely to at- tain this angle [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Kamper et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='al [33] was analyzed the joint angles and finger trajectories in reach-and- grasp tasks which fit the actual finger positions with a mean error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='25 cm and accounted for over 98% of the variance in finger position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The direction of the thumb tra- jectories exhibited a greater dependence on object type than the finger trajectories, but still utilized a small percentage (<5%) of the available workspace [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Previous studies of musculoskeletal models [35] observed that the role of the intrinsic muscles of the hand as the main force-producing muscles in power grip [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' These models were used in the commercial software ANYBODY for the thumb and the index finger [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' However, these models did not address the coupling between the fingers and the interaction with the wrist which limits the investigation of human grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Sensors 2022, 22, x FOR PEER REVIEW 4 of 20 In [38] authors presented an upper limb musculature model for the full arm includ- ing the shoulder, elbow, wrist, thumb, and index finger and provides valuable data on the wrist-finger joint coupling and extrinsic hand muscle anatomy [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The force output on the fingertip is highly joint dependent and provides stable grasp and precise manipu- lation of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=" That's why the force transfer is important for the human hand in con- tact and manipulating purpose to avoid the slipping and deformation of the object." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Pre- vious works focused more on muscle activation patterns and resultant positions/forces as a function of the joints [20] as well as a subject independent lead to the structural var- iability in human hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='4 Purpose of Study This study conducted to analyze the human hand grasping to isolate the functional properties with the final goal to optimize haptic grasping by building simpler multi- finger haptic grasping interfaces with at least similar grasping and manipulation capa- bilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This work helped to learn more about the complex engineering structure of the human hand and leads to characterizing the multi-finger haptic grasping systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' For instance, the proposed three-finger haptic grasping system has an independent joint ar- chitecture for three fingers in the hand, which may be advantageous in virtual grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Also, this led to identify the independence of joints needs in normal grasping tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Conversely, the control of such an independent architecture is very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' By add- ing synergies, we can reduce the complexity of control, but we also want to keep a cer- tain, currently unquantified, level of dexterity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Human Grasp A grasp is a system wherein the desired object is gripped by the fingers of a human (or robot) hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='1 Human Grasp Patterns Napier [40] categorized human grasp into two basic grips: power grasps and preci- sion grasps (pinch grasps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' In power grasp, the object is in the palm of the hand and en- closed by the fingers which lead to large area of contact between the palm, the fingers, and the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' In precision grasp (pinch grip), the object is held between the tip of the thumb and finger, which offer more dexterity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Precision grasps become more relevant in robotic and virtual grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The power grasp is enhanced by the precision grasp be- tween the thumb and the distal finger pads, and it is inherently stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Pinch grip re- quires the six joints between the index finger and the thumb to be stabilized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' it requires more activity of the intrinsic finger muscles to maintain this balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' A large variety of prehension patterns are identified from studies of the muscle-bone-joint anatomy and from observation of the postures and motions of the hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The object-contact pattern furnishes a satisfactory basis for classification of major prehension patterns [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' All the ages, the human hand was a part of most creative arts of every culture [42] to speak and convey human emotions and the hands symbolize cultural behaviors, val- ues, and beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' A mudra is a symbolic gesture in the spiritual practice of Indian reli- gions and traditional art forms performed with the hands with a specific pattern of fin- ger configurations [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' A canonical set of predefined hand postures and modifiers can be used in digital human modelling to develop the standard hand posture libraries and a universal referencing scheme and continuum of hand poses from simple posture to complex one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Researchers [9] have studied features for force-closure grasp by human hands and characterized into four mutually independent properties for robotic arm grasping listed as dexterity, equilibrium, stability, and dynamic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The principal component analysis of static hand posture of several subjects provides information about the finger joint variance and shape of the grasped object and did not consider the hand position/orientation relative to the object placement [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Sensors 2022, 22, x FOR PEER REVIEW 5 of 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='2 Grasp Taxonomy Based on the various studies about human hand in the literature, hand grasp types are classified, and different grasp taxonomies were raised in the literature [11,12] as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' If the grasp object size/shape is not considered, this taxonomy might be lowered to broad range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Grasps are classified based on precision [45], grasped object’s size [9], shape [46], weight, rigidity, force requirement, the position of thumb (adducted or abducted posi- tion) and the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Based on the level of precision, grasps are classified as precision grasp [10], intermediate grasp [47], and power grasp [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The movements of the hand in the power grip are evoked by the arm but in the precision handling, the intrinsic move- ments on the hand not evoked by the arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' In intermediate grasp, elements of power and precision grasps are present in roughly the same proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Later studies included oth- er grasps like hook grasp, flat hand grasp, platform grasp, push grasp [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' In static and stable grasps [12], the object is in a constant relation to the hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Based on the direction of force relative to the hand coordinate frame, applied by the hand on the object to hold it securely [8] opposition type grasps are classified as pad opposition, palm opposition and side opposition grasp [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Pad Opposition occurs be- tween hand surfaces along a direction generally parallel to the palm, usually occurs be- tween volar (palmar) surfaces and the fingers and thumb, near or on the pads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Palm Op- position occurs between hand surfaces along a direction generally perpendicular to the palm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Side Opposition occurs between hand surfaces along a direction generally trans- verse to the palm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The taxonomy of Cutkosky [9], which is widely used in the field of robotics, lists 15 different grasps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Other taxonomies mentioned in works of Kamakura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' [47], Ed- wards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' [10], Kapandji [45] are listed with 14, 20 and 21 grasps respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' A simi- lar study [51], that used a different categorization which incorporated non-prehensile grasps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Even though there has been a considerable effort in creating statistics of human hand use and grouping of hand grasps [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The extensive research in human grasp analysis and taxonomies over the past years helped towards better progress in human- computer interaction, robotic grasping, and dexterous manipulation and lead to the de- sign of artificial robotics hands and arms for the prosthetic application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This helped to classify all hand usage in everyday life situations [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Furthermore, the taxonomy could be extended to include non-prehensile “grasps”, or for dynamic within - hand manipulation movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Kamakura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' [47] classified the tripod grasps as intermediate grasps, apart from that it was classified as a precision grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Several studies have investigated classifying grasps into a discrete set of types [8,9,47], and oth- ers have been aimed at understanding certain aspects of human hand usage [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The number of fingers used for grasping increases with the size and mass of the object [54] until a two-handed grasp is required, indicating that object size and mass are strong fac- tors in determining the grasp type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The rigidity of the objects also influencing the grasp [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Sensors 2022, 22, x FOR PEER REVIEW 6 of 20 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Taxonomy of grasps and allocation of virtual fingers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The studies in [56,13,14] analyzed the human grasping scenarios and behavior of unstructured tasks and investigated the relationship between grasp types and object properties and the results indicated that three-fingertip precision grasps such as thumb- 2 finger, tripod, or lateral tripod can be used to handle dexterous manipulation of a wide range of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' In [12] authors analyzed and compared 33 existing human grasp taxon- omies of static and stable grasps performed by one hand and synthesized them into a single new taxonomy called ‘The GRASP Taxonomy’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The grasps are arranged according to opposition type, the virtual finger assignments, type in terms of power, precision or intermediate grasp, and the position of the thumb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The classifications of micro- interaction grasp instances [57] helped researchers in defining human hand capabilities [58] and affordances in robotic hand design [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The concept of the virtual finger [15] has also incorporated in this taxonomy as an abstract representation through which the human brain plans grasping tasks [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The virtual finger is a functional unit of several fingers work together comprised of at least one real physical finger to reduce the degrees of the human hand to perform the grasp- ing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This concept replaces the analysis of the mechanical degrees of freedom of in- Grasp Thumb Virtual Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Grasp Thumb Virtual Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Name Picture Type Opp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='Type Position Fingers Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' VF of Fingers No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Name Picture Type Opp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='Type Position Fingers Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' VF of Fingers VF1: VF1: 1 VF1: VF1: 1 large VF2: 2-5 VF2: 2-5 Extension VF2: 2-5 VF2: 2-5 Diameter Power Palm Abducted VF3: VF3: Palm 3 18 Type Power Pad Abducted VF3: VF3: 3 VF1: VF1: Palm VF1: VF1: 1 Small 心 VF2: 2-5 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='Index ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='心 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='VF1: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='VF1: 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='Finger ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='VF2: 3-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='VF2: 3-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='Extension ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='Power ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='Palm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='Adducted ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='VF3: 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='VF3: 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='3Sensors 2022,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' x FOR PEER REVIEW 7 of 20 dividual fingers by the analysis of the functional roles of forces being applied in a grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The virtual fingers oppose each other in the grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Virtual fingers are assigned for each grasp in the grasp taxonomy as mentioned in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Our characterization study re- vised the existing virtual fingers allocation and replaced with new as mentioned in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' In this work, the authors aim for designing a three-finger haptic grasping interface for virtual grasping the common objects in everyday life and tools in motor skill profes- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' For the proposed three-finger haptic grasping interface, the thumb assigned as 𝑉𝐹1, Index finger as 𝑉𝐹2 and other three fingers as a single virtual finger 𝑉𝐹3 as ex- plained in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Characterization Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='1 Overview For designing a multi-finger haptic grasping interface, it is quintessential to study the characteristics of the motion and force distribution of fingers in grasping activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This characterization study helped to propose models for multi-finger haptic rendering and grasping haptic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Here the authors conducted experiments for calculating the finger movements, trajectories, positions, orientations on different grasping activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Also tracked the positions and orientations of the skeleton of each finger through motion tracking techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' A characterization study was carried out to compute the models and the design of the multi-finger haptic device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='2 Subjects and Methods Six subjects, 3 males and 3 females between the ages of 23 and 35 years were taking part in this grasping experiment study with 10 common objects as listed in Table 2 with possible grasp patterns and minimum number of virtual fingers to execute the grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Each subject grasps each object for 5 seconds and repeats for 5 trials results total 300 in- stances for the data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This study aimed on the force distribution and orientation of wrist, palm, and finger in all grasping scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The experiment set up consists of a leap motion sensor [61] to track the movement of fingers and a wearable glove with Force Sensitive Resistor (FSR) [62] to measure the force on fingers while grasping different ob- jects and a computer display with the virtual grasping interface as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Selected objects with possible grasp patterns and minimum number of virtual fingers to execute the grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Index Finger Extension Prismatic 4 Finger Prismatic 3 Finger Prismatic 2 Finger n Sphere Tripod Variation [Parallel Extension Small Daimeter Medium Wrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' [Extension Type Adduction Grip sphere 4 Finger Sphere 3 Finger Grasp Pattern [large Diameter [Power Sphere Disk Lateral Tripod Inferier Pincer [Palmar Pinch Power Disk Fixed Hook Distal Type Adducted Light Tool Precision I Precision Tip Pinch padpeno [Lateral Palmar stick Ring min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' of VF|3 3 2 3 3 3 3 3 3 2 3 3 3 3 3 3 2 3 3 2 3 3 3 2 2 3 3 3 3 3 3 2 3 2 Objects Ball Pen Marker Pen Cube box Toy Wheel 1 Cup Plastic bottle Tennis ball Credit card Scissors ScrewdriverSensors 2022, 22, x FOR PEER REVIEW 8 of 20 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Experimental setup for characterization study: (a) Motion tracking setup, (b) Force track- ing setup for hand fingers during grasping and (c) 3d printed mounting module for FSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The Leap Motion sensor is a small USB peripheral device which is placed on the ta- ble surface and connected to computer interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Subjects grasped the 10 objects in the hemispherical workspace area of Leap motion sensor and traced the position and orien- tation parameters of user’s hand with an average accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='7 mm [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The experi- ment set up for tracking grasping forces in fingers as shown in Figure 2(b), users wore a glove with FSR to measure force and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' But the sensing ability of FSR is depend- ent on its contact area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' To overcome this, a 3d printed mounting module placed on the sensing area of the FSR in the fingertips of gloves as shown in Figure 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The values from FSR processed into multiple linear areas using an Arduino and measured the ex- erted forces in Newton(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Experiment procedure in one trial consist of pick the object, grasp for 5 seconds, and place back the object and repeat for five trials for each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' In each trial subject’s grasp parameters such as the position, speed, orientation, and force are tracked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Further analysis of these primary data leads to the modelling of multi-finger grasping haptics interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content="3 Results and Discussions The 30 primary parameters were tracked and used for calculating the user's hand grasp movements and forces." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Here the x-axis is defined as forward-backward, the y-axis as right-left, and the z-axis as up-down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Roll (γ) is taken to be about the x-axis, pitch (β) about the y-axis and yaw (α) about the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Direction, Trajectory, and Rotation of hand in grasping (a)Motiontrackingsetup (b)Forcetrackingsetup (c)3dprintedmountingmoduleforFSR(a)Directionofhand (b)Trajectoryofhand (c)Rotationofhand 100 Direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='4 Trajectory Yaw Direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='y O Position Pitch Direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='z 50 Position (cm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='6 Angle (degree) Roll axis 0 N-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='8 & 1 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='5 1 100 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='2 1 150 0 500 1000 1500 Yaxis 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='2 0 500 1000 1500 Time (cs) Xaxis Time (cs) (d)Boxplotofyawrotationofhand (e)Boxplotofpitchrotationofhand (f)Boxplotofroll rotationofhand 十 20 5 Angle (degree) 0 Angle (degree) 40 5 60 10 Angle 20 80 15 100 0 20 120 140 1 1 1Sensors 2022, 22, x FOR PEER REVIEW 9 of 20 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Plots for the movement of hand: (a) Direction of hand, (b) Trajectory of hand, (c) Rotation of hand, (d) Box plot of yaw rotation of the hand, (e) Box plot of pitch rotation, (f) Box plot of roll rotation of the hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The direction, trajectory, and rotation of hand in grasping (based on the experiment data set) is plotted as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The direction of the Hand in the 3D co-ordinate axes is shown in Figure 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The x, y and z components of hand direction are spanning from -3 mm to 1 mm, -2 mm to 8 mm, and -10 mm to -6 mm respectively with standard deviation (SD) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='11, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='20 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The trajectory of hand movement direction is plot- ted in Figure 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The movement in the y-direction is more than x and z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The ellipse region in the plot represents the trajectory of hand exactly in the grasping time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Figure 3(c) plots the rotation of hand in the coordinate frame during grasping scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The span of the roll is more than yaw and pitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The boxplot of hand rotation clearly ex- plains the distribution of the yaw, pitch, and roll of hand in grasping exercises shown in Figure 3(d)-(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The minimum angle of hand yaw is -22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='61° and the maximum is -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='86° with an average of -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='57° and SD of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='7°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Inter Quartile Range (IQR) of hand yaw is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='18°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The minimum angle of hand pitch is -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='06° and the maximum is 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='04° with an average of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='04° and SD of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='09°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Inter Quartile Range (IQR) of hand pitch is 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='21°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The minimum angle of hand roll is -136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='69° and the maximum is -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='90° with an average of -53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='68° and SD of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='29°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Inter Quartile Range (IQR) of hand roll is 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='14°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Position and trajectory of wrist and palm in grasping The wrist is acted as the basement for most of the grasping scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The infor- mation about the movement of the wrist is helpful for characterizing the basement of multi-finger gripper modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The Position and trajectory of the wrist in grasping dur- ing the experiments were tracked and plotted as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Here the position of wrist n the 3D coordinate frame shows the span of position in x, y, and z-axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' It is clear from the plot that the span of the position of the wrist in grasping is average of 50 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Also, the plot of the trajectory of wrist shows a minimum workspace of movement for the wrist in grasping is in shape of a square with each side 50 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The thick ellipse re- gion represents the data exactly during the grasping scenarios after noise filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Scat- ter plots in Figure 4 gives a clearer picture of the position of the wrist in the 3d space and 2D planes (x-y, y-z, and x-z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Plots for the movement of the wrist: (a) Position of the wrist, (b) Trajectory and scatter plot of wrist, (c) Position of the wrist in the x-y plane, (d) Position of the wrist in y-z plane, and (e) Position of the wrist in x-z plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' (a)Positionofwrist (b)Trajectoryandscatterplotofwrist 200 Position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='x Trajectory Position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='y 200 Position 150 Position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='z (mm) sixe 150 Position 100 N100 50 50 120 100 80 0 60 200 600 800 1000 1200 80 0 400 40 60 Time (cs) Yaxis 20 Xaxis (c)Position ofwrist inX-Yplane (d)PositionofwristinY-Zplane (e)PositionofwristinX-Zplane 120 160 160 O 110 00 O 140 140 100 120 O 120 axis 90 8 N N 80 O 80 80 O 70 60 60 O 60 40 40 30 40 50 60 70 80 60 70 80 90 100 110 120 30 40 50 60 70 80 Xaxis Yaxis XaxisSensors 2022,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' x FOR PEER REVIEW 10 of 20 Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' to the wrist,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' the palm is also an important part of better grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' In some of the grasping types, the palm acts as an additional supportive area for the successful grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' So, it is important to study the movement factors of palm while grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This will help to model the minimal number of virtual fingers (VF) for a general haptic grasp- ing interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The position and trajectory of the palm are plotted in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' It shows that there is not much deflection in the position of palm while grasping time in three ax- es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The trajectory of the palm shows the workspace of palm during the grasping scenari- os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' After filtering out the noise in the dataset, the maximum span in the 3D space for a grasping activity is 30 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Position and trajectory of palm: (a) Position of palm, (b) Trajectory and scatter plot of palm, (c) Position of palm in the x-y plane, (d) Position of palm in the y-z plane, and (e) Position of palm in the x-z plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Angle of grasp The angle of grasp is one of the common measurements used to describe grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Grasp angle describes the angle of the hand in grasping in relation to the Wrist position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The grasp angle influences comfort and easiness in grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' A little bend in the wrist helps to maintain the grasp angle suitable for comfortable grasping of objects to get a proper grip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This will aid controlling the gripped objects easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Figure 6 shows the vari- ous analysis plots of grasp angle tracked during the experimental scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' (a)Positionofpalm 200 (b)Trajectoryandscatterplotofpalm Position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='x Position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='y Trajectory 150 Position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='z 200 Position (ww) 100 100 axis Position 50 N 0 100 0 200 150 50 60 50 100 30 40 0 200 400 600 800 1000 1200 50 10 20 Time (cs) Yaxis Xaxis (c)Positionofpalm inX-Yplane (d)PositionofpalminY-Zplane (e)Positionofpalm inX-Zplane 180 150 150 160 100 100 axis 80 8 50 50 120 N N O 05 O O 0 0 100 00 O 80 50 50 10 20 30 40 50 60 80 100 120 140 160 180 10 20 30 40 50 60 Xaxis Yaxis XaxisSensors 2022,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' x FOR PEER REVIEW 11 of 20 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Plots for the angle of grasp: (a) Angle of grasp, (b) Box plot of grasp angle, (c) Grasp an- gle with a normal distribution, (d) Probability plot for normal distribution, and (e) Quantile- Quantile plot of grasp angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The angle of grasp is spanning from minimum 0°to maximum of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='12° with an av- erage value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='33° and SD of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='10 as shown in Figure 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The grasp angle data are charted as a box and whisker plot as shown in Figure 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This will help to show the shape of the distribution, its central value, and its variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The first quartile of the grasp angle values lies between 0° to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='41°, second lies between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='41° to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='03°, third lies between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='03° to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='44° and the final lies between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='44° to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='14°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 75% of the grasp angle is below 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='44°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' So, most of the grasp types can perform comfortably with a maximum angle of grasp 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='44°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The ideal maximum angle of grasp for the proposed gripper module should be between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='5° to 4°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Figure 6(c) plots a histogram of grasp angle in data using the number of bins equal to the square root of the number of elements in data and fits a normal density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The bell curve fits the normal distribution with an SD of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 68% of the data falls with- in one SD of the mean 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='33°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The standard deviation controls the spread of the distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Here the larger standard deviation indicates that the data is spread out around the mean and the normal distribution is flat and wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Figure 6(d) draws a normal probabil- ity plot, comparing the distribution of the grasp angle data to the normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The plot includes a reference line helped to judge whether the data follow a normal dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The plot shows that the normal line fit the data except the tails because of the outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Figure 6(e) displays a quantile-quantile plot of the sample quantiles of grasp angles versus theoretical quantiles from a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The plot is close to linear in the IQR, so the distribution of grasp angle is normal during the grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' In the dura- tion of not grasping the plot shows the distribution is not normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='Sphere of grasp The spherical grip is the most used grasp in everyday life [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' It is important to an- alyze the sphere of grasp in common grasping scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Here the authors tracked the center and radius of the sphere of grasp in all the grasping scenarios in the experimental procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Figure 7 shows the various plots related to center and radius of the sphere of grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Through the experiment, the position of the center of the sphere is spanning from minimum (-147 mm, 48 mm, -48 mm) to maximum (181 mm, 259 mm, 85 mm) in x, y, and z-axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' After filtering out the non-grasping samples, the span of sphere grasp is re- duced from minimum (-19 mm, 42 mm, -3 mm) to maximum (68 mm, 150 mm, 78 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The scattering and trajectory of the sphere of grasp shown in Figure 7(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' (a) Angle of grasp (b)Boxplotofgraspangle (c)Graspanglewithnormaldistribution 4 200 Grasp angle 3 Normal distribution 150 degr 100 Angle 50 0 0 0 0 200 400 600 800 1000 1200 2 0 2 4 6 Time (cs) Angle (degree) (d)Probabilityplotfornormaldistribution (e)Quantile-Quantileplotofgraspangle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='9999 8 6 Data Quantiles of Input Sample Duration of not grasping 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='995 Normal 6 Inter Quartile Range %% 十 Data Probability 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='75 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='005 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='0001 0 1 2 3 4 2 0 2 4 Angle (degree) StandardNormalQuantilesSensors 2022, 22, x FOR PEER REVIEW 12 of 20 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Center and radius plots for sphere of grasp: (a) Position of center of sphere of grasp, (b) Trajectory and scatter plot of center of sphere, (c) Radius of sphere of grasp, (d) Histogram of sphere radius with normal distribution, and (e) Box plot of sphere radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The volume of grasp can be represented by analyzing the radius of the sphere of grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Figure 7(c)-(e) shows the plots for the radius of the sphere of grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' In the whole data tracked during the experiment, the radius of the sphere of grasp spanning from minimum 30 mm to 188 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' In the box plot showing in Figure 7(e) 75% of the data is less than 10 mm and 50% of data is 64 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Again, after filtering out the noises the radi- us of the sphere of grasp is spanning from 31 mm to 50 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' So, the authors targeting to model the grasping module with a radius of the sphere of grasp is 50 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='Distance of Pinch Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Plots for the distance of pinch: (a) Distance of pinch, (b) Box plot of pinch distance, (c) Pinch distance with a normal distribution, (d) Probability plot for normal distribution, and (e) Quantile-Quantile plot of pinch distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The pinch distance is the distance between two fingers in grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Pinch gestures are common for touch screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The experiment interface tracked the pinch distance (a)Positionofcenterofsphereofgrasp (b)Trajectoryandscatterplotofcenterofsphere 300 Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='x 100 Trajectory Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='y Position 200 Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='(mm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='axis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='Position ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='200 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='Radius ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='Time (cs) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='Radius (mm)(a) Distance of pinch ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='Normal distribution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='(ww) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='Distance (mm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='Distance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='1200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='time (cs) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='Distance (mm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='(d)Probability plotfornormal distribution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='(e)Quantile-Quantileplotofpinchdistance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='9999 300 Data Duration of not grasping Quantiles of Input Sample X Normal Inter Quartile Range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='995 200 Data %% Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='75 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='25 0 % 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='005 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='0001 200 0 50 100 150 200 4 2 0 2 4 Pinch distance (mm) StandardNormalQuantilesSensors 2022, 22, x FOR PEER REVIEW 13 of 20 throughout the experiment and various plots are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The maximum pinch distance traced in the experiment setup is 110 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 75% of data is less than 90 mm and 50% is less than 70 mm as per the boxplot showed in Figure 8(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The normal proba- bility plot is shown in Figure 8(d) compared the distribution of the pinch distance to the normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The reference normal line fits the data in a range of 20 mm – 100 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Figure 8(e) displays a quantile-quantile plot of the sample quantiles of grasp angles versus theoretical quantiles from a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This quantile-quantile plot is close to linear in the IQR, so the distribution of pinch distance is normal during the grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' In the duration of not grasping the plot shows the distribution is not normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' So, the ideal values for the radius of the sphere of grasp for our proposed model are from 20 mm to 100 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='Finger motion parameters When a person lifts any object, his fingers align in a particular way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This must be re- flected by the device so that the user is at ease when he uses the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The experiment setup traced the length and width of fingers and angular values between fingers in grasping objects differ in size and dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' These angular values measured between fingers must be replicated by the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Unless the user feels comfortable when using the device, its intended purpose cannot be met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' A base point was marked at the center of the outside of the palm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' A line was drawn from the base point to the base of the middle finger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This was the baseline at 0 degrees as seen in Figure 9(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Lines were also drawn to the base of the index and the thumb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The angles between the thumb and the index finger and the angles between the middle and the index fingers were measured during the grasping scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Length, width, and angle of fingers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The average length and width of fingers of the subjects took part in the experiment shown in Figure 9(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The average length and width of Thumb are 50 mm and 18 mm, the index finger is 57 mm and 17 mm, the middle finger is 67 mm and 17 mm, ring finger is 61 mm and 16 mm, and little finger is 48 mm and 14 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This will help the authors to model the wearable multi-finger grasping interface to fit the user’s fingers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The meas- ured angles were tabulated and plotted as shown in Figure 9(c), proves that angular measurements between fingers in grasping and lifting of objects does not depend on gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The angle between thumb and index finger is in range of 50-60degree,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' index and middle finger is in range of 20-30 degree and thumb and middle finger is in range of 70- (a)Length,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='widthandangleoffingers (b)AverageLengthandwidthoffingersofsubjects ODegrees Length Width FingerLength 70 Length & Width (mm) 60 30Degrees FingerWidth 30 20 10 Thumb Thumb Index Middle Ring Little Range (c)Anglesbetweenfingers 100 90 80 70 60 90Degrees 50 40 30 20 10 0 Average Maximun Minimum Average Maximum Minimum Angle Angle Men Women middle andindex IndexandThumb mMiddleandThumbSensors 2022,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' x FOR PEER REVIEW 14 of 20 90 degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This provided the data on how far apart the finger holders must be placed when designing the prototype for comfortable grasping and lifting of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Position of movement of fingers The motion parameters of fingers in grasping are so important to characterize and model the multi-finger grasping module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Here The authors traced all the five-finger’s movement and force distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The movement of the tip of five fingers of the hand in all the grasping scenarios during the experiment was plotted in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The move- ment of the fingertip is minimal in both x and z-axes is around -50 mm to +50 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The movement in x and z-axes are varied for fingers, especially it is very less in case of thumb, ring finger and little finger in the range of -20 mm to +20 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' For index and middle finger, it is an almost same range of -50 mm to +50 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The movement in the y- axis is more compared to the other axes is around 50 mm to 200 mm for all fingers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Span of the trajectory of fingers and (b) Workspace triangle of three Virtual Fingers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The span of the trajectory of all five-finger movement in space during the grasping experiment is plotted together in Figure 11(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This plot gave a clear idea about the ac- tive participation of all the fingers during grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Thumb is more active in grasping and gradually decreasing towards little finger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The thump, index and middle fingers more actively participate in the grasping scenarios than the ring and little fingers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Also, (a)Position of thumb (b)Position of indexfinger (c) Position of middle finger 300 200 200 Position (mm) 200 Positions (mm) Position (mm) 100 100 100 100 100 100 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 Samples Samples Samples (d) Position of ring finger (e) Position of little finger 150 150 Tip position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='x Tip position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='y Position (mm) 100 Position (mm) 100 Tip position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='z 50 50 0 50 50 0 20 40 60 80 0 20 40 60 80 Samples Samples(a)Spanoftrajectoryoffingermovement (b)WorkspacetriangleofthreeVirtualFingers Thumb Thumb 100 Index finger 100 Index finger Middle finger Middle finger Ring finger 80 Littlefinger axis 0 60 N 100 40 220 axis 20 200 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 180 20 160 40 axis 140 120 60 250 100 200 150 80 100 100 60~ 60 80 50 50 20 40 Yaxis 0 40 20 0 0 50 Xaxis 60 40 XaxisSensors 2022, 22, x FOR PEER REVIEW 15 of 20 the workspace of the ring and little finger aligns with the middle finger as shown in Fig- ure 11(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' So, authors proposed to group middle, ring, and little fingers to one Virtual Finger (VF) for multi-finger grasping interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Thumb and index fingers can act as oth- er two Virtual Fingers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The workspace triangle for three VF is shown in Figure 11(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='7 Grasping forces on Fingers The force values obtained through FSR sensors during the grasping exercises were tabulated and the average force was calculated for each person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' As shown in Figure 12, forces are not more than 10N was experienced by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' It can also be seen that the thumb experienced the maximum force, whereas the middle and the index finger expe- rienced somewhat similar forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Average grasping force values on each finger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Computational Model for Tripod Haptic Grasp The multi-finger perception studies confirmed the hypothesis that the minimal con- figuration that allows most of the grasps is the three-finger tripod grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Based on the human grasp analysis, a conceptual, computational model is presented here for the three-finger tripod haptic grasping interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Previous researchers worked on force models [65] and virtual linkages [66] for multi-grasp manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The concept of the virtual finger [15] has been postulated as an abstract representa- tion through which the human brain plans are grasping tasks [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The virtual finger is a functional unit of several fingers work together comprised of at least one real physical finger (which may include the palm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This effectively reduces the many degrees of the human hand to those that are deemed necessary to perform the grasping task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This con- cept replaces the analysis of the mechanical degrees of freedom of individual fingers by the analysis of the functional roles of forces being applied in a grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Here the concept of the virtual finger was implemented to reduce the realistic five-finger grasping to virtual three-finger grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The characterization study revised the existing virtual fingers al- location and replaced with new tripod virtual fingers allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Average Force (N) on Each Finger 9 8 6 5 4 3 2 1 0 7 2 3 4 5 6 7 8 10 Thumb indlex MiddleSensors 2022, 22, x FOR PEER REVIEW 16 of 20 Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Haptic computation model for three-finger tripod haptic grasping interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The haptic computation model is shown in Figure 13 was implemented to create the suitable force feedback for the tripod haptic grasping interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Each finger holder is connected to each slider and these sliders are responsible for creating forces to each fin- ger attachment point on the grasping interface through the closed-loop belt system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Three proxies and Haptic Interactive Points (HIP) are assigned for three fingers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Based on the position information received from each slider, the position of proxy and HIP are updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Collision detection algorithms detected the collision of each fingertip with the virtual object, these collision points act as simple virtual walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Then applying the God object algorithms to these three proxies and calculated the resultant forces as shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Each actuator in the slider generates forces and these forces are transferred to the fingertip through the attached finger holder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' In this proposed model for multi-finger tripod haptic grasping, the thumb is assigned as 𝑉𝐹1, Index finger as 𝑉𝐹2 and other three fingers as a single virtual finger 𝑉𝐹3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The proposed model including object and virtual fingers in the virtual interface measures collision and progress of interactions in terms of applying forces and movements of the fingers and computes force feedback to be pro- vided by the haptic interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' It covers the dependence of the force feedback and the ef- fect of finger motions on the tripod grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' A basic grasping touch in virtual reality is de- fined as a touch that provides vertical force feedbacks on the contact surface coincident with the reverse direction of the fingers in the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Forces and moments from the individual virtual fingers are considered for the ren- dering of resultant forces for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Three proxies and Haptic Interactive Points (HIP) are assigned for three virtual fingers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Based on the position information received from each 𝑉𝐹𝑖, the position of proxy and HIP are updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' When collision detection al- gorithms detected the collision of each 𝑉𝐹𝑖 with the virtual object, these collision points act as simple virtual walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The authors assume that only forces act at the grasp points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Let 𝑢𝑖𝑗 is the unit vector along the virtual finger i to j, 𝑣𝑖 be the vector from the object ref- erence point o to the virtual finger grasp point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Let 𝐹𝑖 be the force exerted on objects through each virtual finger 𝑉𝐹𝑖 and F be the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Based on the object shape, size, and stiffness the applying forces on objects by the virtual fingers are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 𝐹𝑖 = 𝑘 𝑋𝑖 (1) where i = 1,2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 𝐹 = [𝐹1 𝐹2 𝐹3]𝑇 (2) Slider3 Slider2 R2=-k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='X2 R3=-k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='X3 X3 X2 VirtualFinger3 F3 VirtualFinger2 F2 HIP Proxy F1 X1 VirtualFinger1 R1=-k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='X1 Slider1Sensors 2022, 22, x FOR PEER REVIEW 17 of 20 Grasp perception rate depends on the force applied by the fingers 𝐹𝑖, the area of contact between the fingers and the object A, the distance between the proxy and HIP X and the change in position of the HIP ΔX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Mathematically, the perception rate is 𝑃 = 𝑘 𝛥𝑋 𝐹𝑖 𝐴 (3) where k is a constant that depends on the material of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Let 𝑅𝑖 be the result- ant force exerted on each virtual finger 𝑉𝐹𝑖 and R be the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Then applying the God object algorithms [67] to three proxies and calculated the resultant forces 𝑅𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 𝑅 = [𝑅1 𝑅2 𝑅3]𝑇 (4) The relationship between the applied forces F and resultant forces R would be giv- en by 𝑅 = 𝑢𝐹 (5) were 𝑢 = [ 𝑢11 𝑢12 𝑢13 𝑢21 𝑢22 𝑢23 𝑢31 𝑢32 𝑢33 ] (6) The relationships between the resultant force R, the resultant moment m, and the applied forces F are given by 𝐹 = 𝑣 [𝑅 𝑚]𝑇 (7) were 𝑣 = [𝑣1 𝑣2 𝑣3]𝑇 (8) and 𝑚 = [𝑚1 𝑚2 𝑚3]𝑇 (9) Force feedback on the grasping scenarios along virtual fingers has two components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' the force feedback resisting the grasping motion and a frictional force (𝐹𝑟), represented by 𝑅 = 𝑢𝐹 + 𝐹𝑟 (10) An experiment was done to evaluate the rendered force to the fingers through the virtual fingers using the proposed tripod haptic grasp model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The results plotted as shown in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Initially, the grasping forces in the real grasping cases were tracked using the FSR and plotted as shown in Figure 14(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' In case of the real grasping scenari- os, the forces exerted by the thumb is in the range of 6-8N which is greater than the force exerted by the index and middle finger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The force exerted by the index and middle fin- gers in in the range of 2-5N and 3-5N respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The same experiment was carried out to measure the rendered forces in haptic grasping interfaces during virtual grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Force is measured using the FSR sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The FSR was placed inside the finger holder of the gripper at the location of the fingertip contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The sensing part of FSR facing the flat area of the holder and the user keeps a finger on top of the FSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The motors were actuat- ed by the haptic rendering model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The tracked rendered forces were plotted in Figure 14(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This graph shows that the range of rendered forces in virtual grasping is almost in the same range of forces in real grasping scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The range of force rendered at the Thumb, index and middle finger holders is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='7-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='5N, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='6-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='4N and 3-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='5N respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This demonstrates that the three-finger haptic grasping interfaces able to provide realis- tic grasping haptic feedbacks to the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Also, the experiment setup was traced the peak force that can provide the gripper device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' As plotted in Figure 14(c), the device at- tained a peak force of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='4N, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='1N, and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='2N at Thumb, index, and middle finger hold- ers respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' It was found that the interface can give approximately 10N force which is greater than the force when lifting objects as found out in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Finally, the Fig- Sensors 2022, 22, x FOR PEER REVIEW 18 of 20 ure 14(d) shows the performance index of three-finger haptic grasping interface in haptic feedback and finger manipulability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Evaluation results of the haptic grasping interface: (a) Exerted force tracked in the real grasping scenarios, (b) Force rendered in haptic grasping interface during virtual grasping, (c) Peak force generated in the haptic grasping interface, and (d) Performance index of the haptic grasping interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Conclusions The aim of this research was to come up with an analysis, model, and design of a three-finger haptic interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' A detailed literature review was carried out regards the anatomy, kinematics, and dynamics of hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Also, a focused survey on the different human grasps, prehension patterns and grasp taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' As part of this work, authors carried out characterization studies in most of the major aspects of human grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The position, orientation, and forces of hand, wrist, palm, and fingers were analyzed, and the results were discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The characterization studies confirmed the hypothesis that the minimal configuration that allows most of the grasps in grasp taxonomy is the three fin- gers grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' This detailed characterization study leads to the design of a three-finger haptic grasping interface as an extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' As a future work, I am planning to work on the multi-Finger grasping interface for bimanual scenarios which provides the users a complete immersed grasping manipula- tion in the virtual and remote environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Funding: This research received no external funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Institutional Review Board Statement: Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Informed Consent Statement: Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Data Availability Statement: Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Acknowledgments: I wholeheartedly thank the wonderful team at AMMACHI Labs for their support, encouragement and for providing constructive criticism and valuable inputs during the various stages of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Conflicts of Interest: The author declares no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The funders had no role in the design of the study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' in the collection, analyses, or interpretation of data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' in the writing of the man- uscript;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' or in the decision to publish the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Wolff, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' The human hand (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfQfbq/content/2301.00049v1.pdf'} +page_content=' Routledge, 2015 2.' metadata={'source': 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Taghavi, Kh. +Saaidi,∗ and Z. Ossoulian +Department of Physics, Faculty of Science, University of Kurdistan, Sanandaj, Iran. +(Dated: January 9, 2023) +We study the possibility of having HDE as the source of inflation in the frame of f(R, T) gravity +theory. The length scale of the energy density is taken as GO cutoff which is a combination of the +Hubble parameter and its time derivative. It is found that for specific ranges of the free parameters +of the model, the scalar spectral index and the tensor-to-scalar ratio stand in good agreement with +data. Also, by reconstructing the potential in terms of the scalar field the validity of the swampland +criteria is considered. The results indicate that to have scalar field below the Planck mass, the +parameter λ should be at least of the order of O(102). The same order of λ is also required to keep +the gradient of the potential greater than one and satisfy the second conjecture of the swampland +criteria. +I. +INTRODUCTION +The modified versions of Einstein’s gravity have received tremendous attention and have been used for studying +different cosmological and astrophysical phenomena. One of these modified gravity theories is f(R, T) gravity which +was introduced in [1], where R is the Ricci scalar and T is the trace of the energy-momentum tensor. The theory +replaces the Ricci scalar in Einstein-Hilbert action with an arbitrary function of R and T, indicating a non-minimal +coupling between the geometric and matter parts. Different topics including dark energy[2], dark matter [3], worm- +holes [4], gravitational waves [5], inflation [6–8], have been studied in the frame of f(R, T) gravity. +One of the challenges in cosmological studies is understanding the true nature of dark energy. It is assumed to be +the main reason for having an accelerated expansion phase for the current universe which is supported by much +data. [9–14]. Different candidates for dark energy have been introduced such as cosmological constant, quintessence, +tachyon, and so on (refer to [15] for a review of different models of dark energy). Another candidate is holographic +dark energy (HDE) which recently becomes one of the most favorite ones [16–18]. The idea of HDE stands on the +holographic principle [19] which states that the number of degree of freedom of a physical system is scaled with its +bounding area rather than its volume [19–22]. It was first put forth by Cohen and colleagues [23] that in quantum +field theory, due to the limit set by the formation of black holes, a short distant cutoff is related to a long distant +cutoff. It suggests that the total energy in a region of size L should not exceed the mass of a black hole of the same +size. Taking this assumption the HDE density was introduced as ρ = 3c2M 2 +p/L2 [18], where Mp = 8πG is the reduced +Planck mass and c2 is a dimensionless positive constant. The infrared cutoff length L, which is our main concern in +HDE, is usually selected as Hubble length, particle horizon, future event horizon, Ricci scalar curvature, and GO1 +cutoff [24, 25]. The HDE mostly has been used for studying the late time evolution of the universe as a possible +reason for the current accelerated expansion [18, 26–29]. However, recently, the application of HDE as the origin of +the early expansion of the universe, i.e. as a source of inflation, is rising [30–32]. +The universe is assumed to go through another phase of accelerated expansion at very early times, known as the +inflationary phase. The scenario has received a tremendous amount of interest and has been modified on different +levels [33–66]. In a common procedure, it is driven by a scalar field that stands on top of its potential and slowly rolls +down toward the minimum of the potential [67–74]. Due to the slowly varying, the scalar field gets a kinetic energy +density which is ignorable compared to the potential energy density. The scenario has been investigated for different +types of the potential, mostly inspired from particle physics, in different gravity frames including f(R, T) gravity. A +different approach, which has received interest recently, is to assumed that the dark energy which drives inflation is +HDE [30–32]. Different type of the entropies with different length scales were studied in this matter however, up to +our knowledge, no such formalism has been considered in f(R, T) gravity theory. Here, we are going to consider the +HDE inflation in the frame of f(R, T) gravity theory. In this regard, the energy density is assumed to be originated +from Tsallis entropy [75–78] addressed as Tsallis HDE2 (THDE). Since, during the slow-roll inflation, the total energy +density is approximated by the potential term, we take the potential as the THDE with GO cutoff as the length +∗ ksaaidi@uok.ac.ir +1 stand for Granda and Oliveros +2 It was shown by Tsallis and Cirto [79] that the entropy of a black hole should be generalized to S = γAδ where A stands for the area of +the black hole, δ is the Tsallis parameter (or nonextensive parameter), and γ is an unknown parameter. Besides, the quantum gravity +suggests a power-law function of the entropy inspired by Tsallis entropy [80, 81] +arXiv:2301.02631v1 [gr-qc] 5 Jan 2023 + +2 +scale. Then, the potential is determined in terms of the Hubble parameter instead of the usual approach where it +is specified in terms of the scalar field. Applying the observational data and Python coding, the ranges of the free +parameters of the model are determined. Using the result, the potential in terms of the scalar field is reconstructed +and we find our way to examine the validity of the swampland criteria3. +The paper is organized as follows: the f(R, T) gravity and its main dynamical equation will be presented in Sec.II in +brief. In Sec.III, the scenario of the slow-roll inflation is introduced, and by applying the slow-roll approximations, the +dynamical equations get simplified. Then, in Sec.IV, we take the potential equal to a THDE density with GO cutoff +as the length scale. The perturbation parameters are estimated and by comparing with data, the free parameters of +the model are determined. Then, in Sec.V, the potential is reconstructed versus the scalar field and the validity of +the swampland criteria is studied. Finally, the results are summarized in Sec.VI. +II. +f(R, T) GRAVITY THEORY +The f(R, T) gravity is a modified theory of gravity where the Ricci scalar, R, in the Einstein-Hilbert action is +replaced by an arbitrary function of R and T, where it is the trace of the energy-momentum tensor. The general form +of the action is given by [1] +S = +1 +2κ2 +� +d4x√−g +� +f(R, T) + Lm +� +(1) +where f(R, T) is an arbitrary function of R and T, the determinant of the metric gµν is given by g, the Lagrangian +of the matter fields is indicated by Lm, and κ is related to the Newtonian gravitational constant as κ2 = 8πG. +The field equation of the theory is obtained by taking variation of the above action with respect to the metric, which +leads to [1] +� +gµν□ − ∇µ∇ν +� +f,R(R, T) + f,RRµν − 1 +2gµνf(R, T) = κ2Tµν − f,T (R, T) +� +Tµν + Θµν +� +, +(2) +where the tensor Θµν is defined as +Θµν = gαβ δTαβ +δgµν . +(3) +In the case of a perfect fluid with energy density ρ, pressure p, and four velocity uµ, the energy-momentum tensor is +read as +Tµν = (ρ + p)uµuν − pgµν +(4) +and the tensor Θµν is obtained from the definition (3) as +Θ = −Tµν − pgµν. +(5) +Assuming a spatially flat FLRW metric and by taking f(R, T) = R + κ2λT, where λ is a constant, the Friedmann +equations are acquired as +H2 = κ2 +3 +��3 +2λ + 1 +� +ρ − λ +2 p +� +, +(6) +−3H2 − 2 ˙H = κ2 +� +−λ +2 ρ + +�3 +2λ + 1 +� +p +� +, +(7) +Substituting Eq.(6) in (7), one arrives at +− 2 ˙H = κ2 (1 + λ) (ρ + p). +(8) +The conservation equation is achieved by taking the time derivative of Eq.(6) and using Eq.(8) as +˙ρ + 3H(ρ + p) = −3λ +2 +˙ρ + −λ +2 ˙p − 3Hλ(ρ + p) +(9) +It is seen that the conservation equation is modified due to the term λT that implies a non-minimal coupling between +matter and curvature. However, it comes back to the standard conservation equation by putting λ = 0. +3 The swampland criteria, proposed in [82–84], is originated from string theory and it is a mechanism to separate the consistent low-energy +effective field theory (EFT) from the inconsistent ones. Since inflation occurs at a low-energy scale, then, it is expected to be described +by a consistent low-energy EFT. + +3 +III. +SLOW-ROLL INFLATION +Inflation is usually provided by an scalar field, known as inflaton, which is the dominant component of the universe +at the time. The Lagrangian of the scalar field is read as +Lφ = 1 +2∂µφ∂µφ − V (φ), +(10) +where V (φ) is the potential of the scalar field. The energy density and pressure of the scalar field is given by +ρφ = 1 +2 +˙φ2 + V (φ) +(11) +pφ = 1 +2 +˙φ2 − V (φ). +(12) +By substituting above energy density and pressure in the Friedmann equations (6) and (8), one has +H2 = κ2 +3 +�1 +2 (1 + λ) ˙φ2 + (1 + 2λ) V (φ) +� +(13) +−2 ˙H = κ2(1 + λ) ˙φ2. +(14) +Also, the equation of motion of the scalar field takes the following form +(1 + λ) ¨φ + 3H(1 + λ) ˙φ + (1 + 2λ)V ′(φ) = 0, +(15) +which is obtained by inserting Eq.(11) in Eq.(9). +The conditions required for the slow-roll inflation are +| ˙H| ≪ H2, +˙φ2 ≪ V (φ), +|¨φ| ≪ |H ˙φ| +which is known as the slow-roll conditions. Applying the conditions on the Friedmann equations above, leads one to +H2 = κ2 +3 (1 + 2λ) V (φ), +(16) +−2 ˙H = κ2(1 + λ) ˙φ2, +(17) +3H ˙φ = 1 + 2λ +1 + λ V ′(φ). +(18) +The slow-roll conditions are encoded in the slow-roll parameters. The first slow-roll parameter states that the rate +of the Hubble parameter during a Hubble time will be small, i.e. ϵ = − ˙H/H2. This condition provides a quasi- +de Sitter expansion. A common approach to introduce other slow-roll parameters are through a hierarchy, namely +ϵn+1 = ˙ϵn/Hϵ for n ≥ 1. +To proceed further, it is required to determine the potential in terms of the scalar field. There are different types +of the potential used to study the scenario. Chaotic potential, Hilltop potential, natural potential could be addressed +as some of these potential. +In the next section, we are going to follow a different path and propose a suitable function in terms of the Hubble +parameter as the potential. +IV. +THDE AS THE POTENTIAL +The early accelerated expansion of the universe is assumed to be provided by a dynamical scalar field, which is +also a dark energy candidate. The scalar field is taken as the dominant component of the universe which governs the +evolution. On the other hand, the kinetic energy of the scalar field could be ignored compared to the potential part +and the energy density of the scalar field is approximated by the potential part, i.e. ρφ ≈ V (φ). Different types of +the potential in terms of the scalar field have been introduced and studied in the literature. Another competent dark +energy candidate is HDE. After prosperous application of HDE for the late time evolution of the universe, recently +there has been a raising interest for taking it as the source of inflation. Here, we are going to change our prospective +a little and take it as the potential of the scalar field, which is actually the main part of the energy density during + +4 +the inflationary times. +The shape of the HDE depends on the entropy of the system which based on the holographic principle is scaled +with the area rather than the volume[19–22]. Including the quantum correction, the standard entropy is modified +in different ways. One of the modification is achived by following this argument that the thermodynamics of the +gravitational and cosmological systems must be modified to the non-additive entropy. Based on this, Tsallis and +Citro obtained that the entropy of the black hole should be altered to S = γAδ. The resulted HDE density from the +entropy is ρ = Bc2L2δ−4 where B ≡ γ(4π)γ and L is the infrared cutoff. One of the candidate for the cutoff is GO +cutoff which contains the time derivative of the Hubble parameter in addition to the square of the Hubble parameter, +i.e. L−2 = αH2 + β ˙H. +Following the above approach and taking the HDE density as the potential of the scalar field, the Friedmann equation +(16) is rewritten as +H2 = κ2 +3 (1 + 2λ) Bc2� +αH2 + β ˙H +�2−δ. +(19) +Working with the equation, one finds the term ˙H/H2, which is also the definition of the first slow-roll parameter. +Then, we have +ϵ1 = − ˙H +H2 = 1 +β +� +α − AHξ� +(20) +where the defined constants A and ξ respectively are given as +ξ ≡ 2δ − 2 +2 − δ , +A ≡ +� +3M 2 +p +Bc2(1 + 2λ) +� +1 +2−δ +. +To introduce the second slow-roll parameter, the hierarchy procedure is used. Then, the parameter is given by +ϵ2 = +˙ϵ1 +Hϵ1 += ξ +β AHξ. +(21) +Solving the problems of the hot big bang theory requires having enough amount of inflation. which is measured by +the number of e-folds, defined as +N = +� te +t⋆ +H dt = +� He +H⋆ +H +˙H +dt, +(22) +where the subscribes ”e” and ”⋆” respectively indicate that the parameters are estimated at the end of inflation and +the horizon crossing time. Solving the integral and after some manipulation, the Hubble parameter at the time of +horizon crossing is obtained as +AHξ +⋆ = +α(α − β) e +αξN +β +β + (α − β) e +αξN +β +. +(23) +Utilizing the H⋆, the parameters could us estimated at the time of horizon crossing and it will allow us to compare +the model with data. In this regard, we first need to acquired the perturbation parameters. +Following [6–8], the scalar spectral index and the tensor-to-scalar ratio are read as +ns = 1 − +� +2ϵ1 + ϵ2 +� += 1 − 2α +β − ξ − 2 +β +AHξ, +(24) +r = +16 +1 + λ ϵ1 = +16 +β(1 + λ) +� +α − AHξ� +, +(25) +which are related to the slow-roll parameters. Applying Eq.(23), the scalar spectral index and the tensor-to-scalar +ratio are computed at the time of horizon crossing. +Fig.1 illustrates a three dimensional plot of the scalar spectral index versus α and β. It is realized that when the +parameter β decreases there is wider range of the parameter α which takes ns into the data range. The plot gives +some idea about the approximate values of the parameters α and β which we will use later for comparing the model + +5 +FIG. 1. The scalar spectral index is plotted versus the parameters α and β. +(a) +(b) +with data. +By getting some instict from above figure, the tensor-to-scalar ratio is plotted versus the scalar spectral index for +different values of β, in Fig.IV, and δ, as in Fig.IV. The variable parameter in both plot is α which the arrow in the +plots shows the increase direction of α. +To enhance our understanding about the free parameters, Fig.2 portrays a parameteric space of (α, β) so that +for every point in the space, the result about the scalar spectral index and the tensor-to-scalar ratio stand in good +agreement with data. Table.I displays the results of the model for some selected (α, β) points from Fig.2. + +0.9 +0.9 +0.8 +0.8 +0.6 +0.7 +0.5 +0.6 +2.5 +5.0 +7.5 +2.0 +0.4 +-12.3 +0.2 +0.0 +15.0 +0.2 +-17.5 +0.4 +20.0β= - 10 +0.6 +β= - 15 +β= -20 +0.5 +0.4 +0.3 +0.2 +0.1 +0.0 +0.92 +0.93 +0.94 +0.95 +0.960.40 +6= -1.5 +6= -0.5 +0.35 +6= +0.1 +0.30 +0.25 +0.20 +0.15 +0.10 +0.05 +0.940 +0.945 +0.950 +0.955 +0.960 +0.9656 +FIG. 2. The parametric space for the (α, β)) when δ = −1.5, λ = 5.5 and the number of e-folds is N = 65. +α +β +ns +r +0.2500 +−15.0 +0.9608 +0.0595 +0.1500 +−10.0 +0.9619 +0.0662 +0.0800 +−5.0 +0.9613 +0.0621 +0.0220 +−1.5 +0.9622 +0.0676 +0.0078 +−0.5 +0.9616 +0.0637 +0.0055 +−0.3 +0.9596 +0.0533 +0.0016 +−0.1 +0.9613 +0.0621 +TABLE I. The result for the scalar spectal index and tensor-to-scalar ratio for different values of α and β taken from Fig.2 and +number of e-folds N = 65. +V. +SCALAR FIELD, POTENTIAL AND SWAMPLAND CRITERIA +The THDE was introduced as the potential of the scalar field where it is given in terms of the Hubble parameter. +Following the equations, the kinetic terms also could be expressed versus the Hubble parameter, so that +˙φ2 = −2M 2 +p +(1 + λ) +˙H. +(26) +Fig.3 describes the behavior of the potential-kinetic ratio during the inflationary times for different values of the +parameter λ. The ratio decreases by the enhancement of the parameter λ. It is realized that the potential is the +dominant part and it is greater than the kinetic term by the order of O(102). +Taking integral of the above equation, the scalar field is obtained. Then, by combining the definition of the potential +and using parametric plot, the potential is illustrated in terms of the scalar field. The result is depicted in Fig.4 for +different values of the parameter λ. At the onset, the scalar field stands on the top of the potential, and by passing +the time, it increases and rolls down. The potential magnitude reduces by enhancement of the parameter λ. +Besides the observational constraints which we used in the previous section to determine the free parameters of the +model, there are some theoretical constraint which is our interest to be satisfied by an inflationary model. One of +these constraints is the swampland criteria proposed by [82–84]. It contains two conjectures, originated from string +theory, to distinguish the consistent low-energy EFT from the inconsistent ones. The conjectures are +• Distant conjecture: The scalar field excursion in the field space should satisfy the following upper bound +∆φ +Mp +≤ c1, +(27) +where c1 is a constant of the order of one [82–84]. +• de Sitter conjecture: The conjecure is targeting the gradient of the potential and states that it should comply +the following condition [82, 84] +1 +Mp +|V ′| +V +≥ c2 +(28) + +5 +6= -2.5 +0 +-5 +-10 +-15 +-20 +-0.4 +-0.2 +0.0 +0.2 +0.47 +FIG. 3. The ratio of the potential over the kinetic term, i.e. V (φ)/ ˙φ2 for different values of λ. +FIG. 4. The potential is plotted versus the scalar field for different values of the parameter λ. +where c2 is a constant of the order of one. However, some consideration imply that it could be smaller of the +order of O(0.1) [84, 85]. In the refined version of the conjecture, one of the following conditions should be +satisfies [83, 84] +1 +Mp +|V ′| +V +≥ c2, +or +1 +Mp +|V ′′| +V +≥ −c′ +2 +(29) +The first conjecture could be checked out from the Fig.4 which describes the behavior of the potential versus the +scalar field during the inflationary times. It is realized that field excursion decreases by increasing the parameter λ +so that for λ ≳ 102 it will be smaller than one. Then, the model could satisfy the first conjecture. +To consider the validity of the second conjecture, the behavior of the term V ′/V is plotted versus the number of e-folds +in Fig.5 for different values of the parameter λ. At the beginning, the term could be bigger than one and it also gets +larger for higher values of λ. Besides, the term in general increases by approaching to the end of inflation. Therefore, +the second criterion is also satisfied by the model. For both conjectures, the parameter λ plays an important role and + +800 +入= 0.1 +入= 1.0 +入=300.0 +600 +400 +200 +0 +0 +10 +20 +30 +40 +50 +60 +701e-12 +3.0 +入= 400 +入=500 +2.5 +入=600 +2.0 +1.5 +1.0 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.08 +FIG. 5. The behavior of the term V ′/V in the inflationary times is plotted for different values λ. +λ +α +β +ES +|∆φ| +V ′/V +90 +0.0220 +−1.5 +1.84 × 10−3 +1.507 +0.877 +90 +0.0078 +−0.5 +1.81 × 10−3 +1.493 +0.851 +90 +0.0055 +−0.3 +1.73 × 10−3 +1.454 +0.779 +300 +0.0220 +−1.5 +1.36 × 10−3 +0.828 +1.595 +300 +0.0078 +−0.5 +1.34 × 10−3 +0.821 +1.548 +300 +0.0055 +−0.3 +1.28 × 10−3 +0.799 +1.416 +500 +0.0220 +−1.5 +1.20 × 10−3 +0.642 +2.058 +500 +0.0078 +−0.5 +1.18 × 10−3 +0.636 +1.997 +500 +0.0055 +−0.3 +1.13 × 10−3 +0.619 +1.827 +TABLE II. The table presents some results about the energy scale of inflation, and two conjecture of the swampland criteria +for different values of α and β, selected from Fig.2, and λ. The other constant are picked as δ = −2.5 and N = 65. +due to this parameter the model could satisfy swampland criteria. +The results concerning the energy scale (ES) of +inflation and two conjectures of the swampland criteria are presented in Table.II for different values of α and β picked +from Fig.2. It is found that the energy scale of inflation is of the order of 10−3Mp stating that inflation occurs at +energy scale below the Planck scale. Moreover, the role of the parameter λ is clarified when we are considering the +validity of the swampland criteria. It is realized that for values of λ ≲ 300 the field distant becomes larger than one +and the potential gradient less than one. Then, to guarantee the validity of swampland criteria, the parameter λ is +required to approximately be λ ≳ 300. +VI. +CONCLUSION +f(R, T) gravity theory, known as an alternative theory of gravity where matter and curvature have a non-minimal +coupling, was utilized to study the scenario of slow-roll inflation. The common procedure in studying inflation is to +assume that it is driven by a scalar field that stands first at the top of its potential, and then slowly rolls down. Due +to the slow rolling, the kinetic energy of the scalar field is ignorable compared to the potential part, and a quasi-de +Sitter expansion is provided. The potential plays an important role in inflationary times, however, there is no specific +choice for it. Then, different choices for the potential have been considered and some of them provide an agreement +between the model and data. +Instead of introducing a function of the scalar field for the potential, here, the potential was assumed to be described +by HDE built from Tsallis entropy. The IR cutoff was taken as the GO cutoff which in addition to the square of the +Hubble parameter contains the time derivative of the parameter as well. Following this proposal, the perturbation +parameters were obtained at the time of the horizon crossing. Comparing the model with data, the free parameters + +35 +入= 10 +入=100 +30 +入=600 +25 +20 +15 +10 +5 +0 +0 +10 +20 +30 +40 +50 +60 +N9 +of the model were determined and we could illustrate a range for the free parameter so that for every point within +the range, the model completely agrees with the data. +Then, we tried to reconstruct the potential in terms of the scalar field. 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Phys. 66, 1800052 (2018), arXiv:1807.05445 +[hep-th]. + diff --git a/8NE0T4oBgHgl3EQfwQHW/content/tmp_files/load_file.txt b/8NE0T4oBgHgl3EQfwQHW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1a6ef051fdc5859a0b87a2ba20eaa8bf3458369d --- /dev/null +++ b/8NE0T4oBgHgl3EQfwQHW/content/tmp_files/load_file.txt @@ -0,0 +1,893 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf,len=892 +page_content='Holographic inflation in f(R, T) gravity S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Taghavi, Kh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Saaidi,∗ and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Ossoulian Department of Physics, Faculty of Science, University of Kurdistan, Sanandaj, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' (Dated: January 9, 2023) We study the possibility of having HDE as the source of inflation in the frame of f(R, T) gravity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The length scale of the energy density is taken as GO cutoff which is a combination of the Hubble parameter and its time derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' It is found that for specific ranges of the free parameters of the model, the scalar spectral index and the tensor-to-scalar ratio stand in good agreement with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Also, by reconstructing the potential in terms of the scalar field the validity of the swampland criteria is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The results indicate that to have scalar field below the Planck mass, the parameter λ should be at least of the order of O(102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The same order of λ is also required to keep the gradient of the potential greater than one and satisfy the second conjecture of the swampland criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' INTRODUCTION The modified versions of Einstein’s gravity have received tremendous attention and have been used for studying different cosmological and astrophysical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' One of these modified gravity theories is f(R, T) gravity which was introduced in [1], where R is the Ricci scalar and T is the trace of the energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The theory replaces the Ricci scalar in Einstein-Hilbert action with an arbitrary function of R and T, indicating a non-minimal coupling between the geometric and matter parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Different topics including dark energy[2], dark matter [3], worm- holes [4], gravitational waves [5], inflation [6–8], have been studied in the frame of f(R, T) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' One of the challenges in cosmological studies is understanding the true nature of dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' It is assumed to be the main reason for having an accelerated expansion phase for the current universe which is supported by much data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' [9–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Different candidates for dark energy have been introduced such as cosmological constant, quintessence, tachyon, and so on (refer to [15] for a review of different models of dark energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Another candidate is holographic dark energy (HDE) which recently becomes one of the most favorite ones [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The idea of HDE stands on the holographic principle [19] which states that the number of degree of freedom of a physical system is scaled with its bounding area rather than its volume [19–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' It was first put forth by Cohen and colleagues [23] that in quantum field theory, due to the limit set by the formation of black holes, a short distant cutoff is related to a long distant cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' It suggests that the total energy in a region of size L should not exceed the mass of a black hole of the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Taking this assumption the HDE density was introduced as ρ = 3c2M 2 p/L2 [18], where Mp = 8πG is the reduced Planck mass and c2 is a dimensionless positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The infrared cutoff length L, which is our main concern in HDE, is usually selected as Hubble length, particle horizon, future event horizon, Ricci scalar curvature, and GO1 cutoff [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The HDE mostly has been used for studying the late time evolution of the universe as a possible reason for the current accelerated expansion [18, 26–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' However, recently, the application of HDE as the origin of the early expansion of the universe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' as a source of inflation, is rising [30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The universe is assumed to go through another phase of accelerated expansion at very early times, known as the inflationary phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The scenario has received a tremendous amount of interest and has been modified on different levels [33–66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' In a common procedure, it is driven by a scalar field that stands on top of its potential and slowly rolls down toward the minimum of the potential [67–74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Due to the slowly varying, the scalar field gets a kinetic energy density which is ignorable compared to the potential energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The scenario has been investigated for different types of the potential, mostly inspired from particle physics, in different gravity frames including f(R, T) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' A different approach, which has received interest recently, is to assumed that the dark energy which drives inflation is HDE [30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Different type of the entropies with different length scales were studied in this matter however, up to our knowledge, no such formalism has been considered in f(R, T) gravity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Here, we are going to consider the HDE inflation in the frame of f(R, T) gravity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' In this regard, the energy density is assumed to be originated from Tsallis entropy [75–78] addressed as Tsallis HDE2 (THDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Since, during the slow-roll inflation, the total energy density is approximated by the potential term, we take the potential as the THDE with GO cutoff as the length ∗ ksaaidi@uok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='ir 1 stand for Granda and Oliveros 2 It was shown by Tsallis and Cirto [79] that the entropy of a black hole should be generalized to S = γAδ where A stands for the area of the black hole, δ is the Tsallis parameter (or nonextensive parameter), and γ is an unknown parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Besides, the quantum gravity suggests a power-law function of the entropy inspired by Tsallis entropy [80, 81] arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='02631v1 [gr-qc] 5 Jan 2023 2 scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Then, the potential is determined in terms of the Hubble parameter instead of the usual approach where it is specified in terms of the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Applying the observational data and Python coding, the ranges of the free parameters of the model are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Using the result, the potential in terms of the scalar field is reconstructed and we find our way to examine the validity of the swampland criteria3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The paper is organized as follows: the f(R, T) gravity and its main dynamical equation will be presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='II in brief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='III, the scenario of the slow-roll inflation is introduced, and by applying the slow-roll approximations, the dynamical equations get simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Then, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='IV, we take the potential equal to a THDE density with GO cutoff as the length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The perturbation parameters are estimated and by comparing with data, the free parameters of the model are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Then, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='V, the potential is reconstructed versus the scalar field and the validity of the swampland criteria is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Finally, the results are summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' f(R, T) GRAVITY THEORY The f(R, T) gravity is a modified theory of gravity where the Ricci scalar, R, in the Einstein-Hilbert action is replaced by an arbitrary function of R and T, where it is the trace of the energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The general form of the action is given by [1] S = 1 2κ2 � d4x√−g � f(R, T) + Lm � (1) where f(R, T) is an arbitrary function of R and T, the determinant of the metric gµν is given by g, the Lagrangian of the matter fields is indicated by Lm, and κ is related to the Newtonian gravitational constant as κ2 = 8πG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The field equation of the theory is obtained by taking variation of the above action with respect to the metric, which leads to [1] � gµν□ − ∇µ∇ν � f,R(R, T) + f,RRµν − 1 2gµνf(R, T) = κ2Tµν − f,T (R, T) � Tµν + Θµν � , (2) where the tensor Θµν is defined as Θµν = gαβ δTαβ δgµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' (3) In the case of a perfect fluid with energy density ρ, pressure p, and four velocity uµ, the energy-momentum tensor is read as Tµν = (ρ + p)uµuν − pgµν (4) and the tensor Θµν is obtained from the definition (3) as Θ = −Tµν − pgµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' (5) Assuming a spatially flat FLRW metric and by taking f(R, T) = R + κ2λT, where λ is a constant, the Friedmann equations are acquired as H2 = κ2 3 ��3 2λ + 1 � ρ − λ 2 p � , (6) −3H2 − 2 ˙H = κ2 � −λ 2 ρ + �3 2λ + 1 � p � , (7) Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' (6) in (7), one arrives at − 2 ˙H = κ2 (1 + λ) (ρ + p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' (8) The conservation equation is achieved by taking the time derivative of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' (6) and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' (8) as ˙ρ + 3H(ρ + p) = −3λ 2 ˙ρ + −λ 2 ˙p − 3Hλ(ρ + p) (9) It is seen that the conservation equation is modified due to the term λT that implies a non-minimal coupling between matter and curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' However, it comes back to the standard conservation equation by putting λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' 3 The swampland criteria, proposed in [82–84], is originated from string theory and it is a mechanism to separate the consistent low-energy effective field theory (EFT) from the inconsistent ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Since inflation occurs at a low-energy scale, then, it is expected to be described by a consistent low-energy EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' 3 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' SLOW-ROLL INFLATION Inflation is usually provided by an scalar field, known as inflaton, which is the dominant component of the universe at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The Lagrangian of the scalar field is read as Lφ = 1 2∂µφ∂µφ − V (φ), (10) where V (φ) is the potential of the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The energy density and pressure of the scalar field is given by ρφ = 1 2 ˙φ2 + V (φ) (11) pφ = 1 2 ˙φ2 − V (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' (12) By substituting above energy density and pressure in the Friedmann equations (6) and (8), one has H2 = κ2 3 �1 2 (1 + λ) ˙φ2 + (1 + 2λ) V (φ) � (13) −2 ˙H = κ2(1 + λ) ˙φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' (14) Also, the equation of motion of the scalar field takes the following form (1 + λ) ¨φ + 3H(1 + λ) ˙φ + (1 + 2λ)V ′(φ) = 0, (15) which is obtained by inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' (11) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The conditions required for the slow-roll inflation are | ˙H| ≪ H2, ˙φ2 ≪ V (φ), |¨φ| ≪ |H ˙φ| which is known as the slow-roll conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Applying the conditions on the Friedmann equations above, leads one to H2 = κ2 3 (1 + 2λ) V (φ), (16) −2 ˙H = κ2(1 + λ) ˙φ2, (17) 3H ˙φ = 1 + 2λ 1 + λ V ′(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' (18) The slow-roll conditions are encoded in the slow-roll parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The first slow-roll parameter states that the rate of the Hubble parameter during a Hubble time will be small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' ϵ = − ˙H/H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' This condition provides a quasi- de Sitter expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' A common approach to introduce other slow-roll parameters are through a hierarchy, namely ϵn+1 = ˙ϵn/Hϵ for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' To proceed further, it is required to determine the potential in terms of the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' There are different types of the potential used to study the scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Chaotic potential, Hilltop potential, natural potential could be addressed as some of these potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' In the next section, we are going to follow a different path and propose a suitable function in terms of the Hubble parameter as the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' THDE AS THE POTENTIAL The early accelerated expansion of the universe is assumed to be provided by a dynamical scalar field, which is also a dark energy candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The scalar field is taken as the dominant component of the universe which governs the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' On the other hand, the kinetic energy of the scalar field could be ignored compared to the potential part and the energy density of the scalar field is approximated by the potential part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' ρφ ≈ V (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Different types of the potential in terms of the scalar field have been introduced and studied in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Another competent dark energy candidate is HDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' After prosperous application of HDE for the late time evolution of the universe, recently there has been a raising interest for taking it as the source of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Here, we are going to change our prospective a little and take it as the potential of the scalar field, which is actually the main part of the energy density during 4 the inflationary times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The shape of the HDE depends on the entropy of the system which based on the holographic principle is scaled with the area rather than the volume[19–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Including the quantum correction, the standard entropy is modified in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' One of the modification is achived by following this argument that the thermodynamics of the gravitational and cosmological systems must be modified to the non-additive entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Based on this, Tsallis and Citro obtained that the entropy of the black hole should be altered to S = γAδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The resulted HDE density from the entropy is ρ = Bc2L2δ−4 where B ≡ γ(4π)γ and L is the infrared cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' One of the candidate for the cutoff is GO cutoff which contains the time derivative of the Hubble parameter in addition to the square of the Hubble parameter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' L−2 = αH2 + β ˙H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Following the above approach and taking the HDE density as the potential of the scalar field, the Friedmann equation (16) is rewritten as H2 = κ2 3 (1 + 2λ) Bc2� αH2 + β ˙H �2−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' (19) Working with the equation, one finds the term ˙H/H2, which is also the definition of the first slow-roll parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Then, we have ϵ1 = − ˙H H2 = 1 β � α − AHξ� (20) where the defined constants A and ξ respectively are given as ξ ≡ 2δ − 2 2 − δ , A ≡ � 3M 2 p Bc2(1 + 2λ) � 1 2−δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' To introduce the second slow-roll parameter, the hierarchy procedure is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Then, the parameter is given by ϵ2 = ˙ϵ1 Hϵ1 = ξ β AHξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' (21) Solving the problems of the hot big bang theory requires having enough amount of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' which is measured by the number of e-folds, defined as N = � te t⋆ H dt = � He H⋆ H ˙H dt, (22) where the subscribes ”e” and ”⋆” respectively indicate that the parameters are estimated at the end of inflation and the horizon crossing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Solving the integral and after some manipulation, the Hubble parameter at the time of horizon crossing is obtained as AHξ ⋆ = α(α − β) e αξN β β + (α − β) e αξN β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' (23) Utilizing the H⋆, the parameters could us estimated at the time of horizon crossing and it will allow us to compare the model with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' In this regard, we first need to acquired the perturbation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Following [6–8], the scalar spectral index and the tensor-to-scalar ratio are read as ns = 1 − � 2ϵ1 + ϵ2 � = 1 − 2α β − ξ − 2 β AHξ, (24) r = 16 1 + λ ϵ1 = 16 β(1 + λ) � α − AHξ� , (25) which are related to the slow-roll parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Applying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' (23), the scalar spectral index and the tensor-to-scalar ratio are computed at the time of horizon crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='1 illustrates a three dimensional plot of the scalar spectral index versus α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' It is realized that when the parameter β decreases there is wider range of the parameter α which takes ns into the data range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The plot gives some idea about the approximate values of the parameters α and β which we will use later for comparing the model 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The scalar spectral index is plotted versus the parameters α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' (a) (b) with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' By getting some instict from above figure, the tensor-to-scalar ratio is plotted versus the scalar spectral index for different values of β, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='IV, and δ, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The variable parameter in both plot is α which the arrow in the plots shows the increase direction of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' To enhance our understanding about the free parameters, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='2 portrays a parameteric space of (α, β) so that for every point in the space, the result about the scalar spectral index and the tensor-to-scalar ratio stand in good agreement with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='I displays the results of the model for some selected (α, β) points from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='8 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='40 6= -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='5 6= -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='35 6= +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='940 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='950 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='955 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='960 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='9656 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The parametric space for the (α, β)) when δ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='5, λ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='5 and the number of e-folds is N = 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' α β ns r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='2500 −15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='9608 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0595 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='1500 −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='9619 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0662 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0800 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='9613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0621 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0220 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='9622 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0676 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0078 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='9616 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0637 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0055 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='9596 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0533 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0016 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='9613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0621 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The result for the scalar spectal index and tensor-to-scalar ratio for different values of α and β taken from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='2 and number of e-folds N = 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' SCALAR FIELD, POTENTIAL AND SWAMPLAND CRITERIA The THDE was introduced as the potential of the scalar field where it is given in terms of the Hubble parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Following the equations, the kinetic terms also could be expressed versus the Hubble parameter, so that ˙φ2 = −2M 2 p (1 + λ) ˙H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' (26) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='3 describes the behavior of the potential-kinetic ratio during the inflationary times for different values of the parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The ratio decreases by the enhancement of the parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' It is realized that the potential is the dominant part and it is greater than the kinetic term by the order of O(102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Taking integral of the above equation, the scalar field is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Then, by combining the definition of the potential and using parametric plot, the potential is illustrated in terms of the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The result is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='4 for different values of the parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' At the onset, the scalar field stands on the top of the potential, and by passing the time, it increases and rolls down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The potential magnitude reduces by enhancement of the parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Besides the observational constraints which we used in the previous section to determine the free parameters of the model, there are some theoretical constraint which is our interest to be satisfied by an inflationary model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' One of these constraints is the swampland criteria proposed by [82–84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' It contains two conjectures, originated from string theory, to distinguish the consistent low-energy EFT from the inconsistent ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The conjectures are Distant conjecture: The scalar field excursion in the field space should satisfy the following upper bound ∆φ Mp ≤ c1, (27) where c1 is a constant of the order of one [82–84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' de Sitter conjecture: The conjecure is targeting the gradient of the potential and states that it should comply the following condition [82, 84] 1 Mp |V ′| V ≥ c2 (28) 5 6= -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='5 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='47 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The ratio of the potential over the kinetic term, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' V (φ)/ ˙φ2 for different values of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The potential is plotted versus the scalar field for different values of the parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' where c2 is a constant of the order of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' However, some consideration imply that it could be smaller of the order of O(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='1) [84, 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' In the refined version of the conjecture, one of the following conditions should be satisfies [83, 84] 1 Mp |V ′| V ≥ c2, or 1 Mp |V ′′| V ≥ −c′ 2 (29) The first conjecture could be checked out from the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='4 which describes the behavior of the potential versus the scalar field during the inflationary times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' It is realized that field excursion decreases by increasing the parameter λ so that for λ ≳ 102 it will be smaller than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Then, the model could satisfy the first conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' To consider the validity of the second conjecture, the behavior of the term V ′/V is plotted versus the number of e-folds in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='5 for different values of the parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' At the beginning, the term could be bigger than one and it also gets larger for higher values of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Besides, the term in general increases by approaching to the end of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Therefore, the second criterion is also satisfied by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' For both conjectures, the parameter λ plays an important role and 800 入= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='1 入= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0 入=300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0 600 400 200 0 0 10 20 30 40 50 60 701e-12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0 入= 400 入=500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='5 入=600 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='08 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The behavior of the term V ′/V in the inflationary times is plotted for different values λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' λ α β ES |∆φ| V ′/V 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0220 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='84 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='507 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='877 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0078 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='81 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='493 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='851 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0055 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='73 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='454 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='779 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0220 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='36 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='828 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='595 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0078 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='34 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='821 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='548 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0055 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='28 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='799 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='416 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0220 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='20 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='642 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='058 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0078 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='18 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='636 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='997 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='0055 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='13 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='619 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='827 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The table presents some results about the energy scale of inflation, and two conjecture of the swampland criteria for different values of α and β, selected from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='2, and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The other constant are picked as δ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='5 and N = 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' due to this parameter the model could satisfy swampland criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The results concerning the energy scale (ES) of inflation and two conjectures of the swampland criteria are presented in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='II for different values of α and β picked from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' It is found that the energy scale of inflation is of the order of 10−3Mp stating that inflation occurs at energy scale below the Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Moreover, the role of the parameter λ is clarified when we are considering the validity of the swampland criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' It is realized that for values of λ ≲ 300 the field distant becomes larger than one and the potential gradient less than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Then, to guarantee the validity of swampland criteria, the parameter λ is required to approximately be λ ≳ 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' CONCLUSION f(R, T) gravity theory, known as an alternative theory of gravity where matter and curvature have a non-minimal coupling, was utilized to study the scenario of slow-roll inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The common procedure in studying inflation is to assume that it is driven by a scalar field that stands first at the top of its potential, and then slowly rolls down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Due to the slow rolling, the kinetic energy of the scalar field is ignorable compared to the potential part, and a quasi-de Sitter expansion is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The potential plays an important role in inflationary times, however, there is no specific choice for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Then, different choices for the potential have been considered and some of them provide an agreement between the model and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Instead of introducing a function of the scalar field for the potential, here, the potential was assumed to be described by HDE built from Tsallis entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The IR cutoff was taken as the GO cutoff which in addition to the square of the Hubble parameter contains the time derivative of the parameter as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Following this proposal, the perturbation parameters were obtained at the time of the horizon crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Comparing the model with data, the free parameters 35 入= 10 入=100 30 入=600 25 20 15 10 5 0 0 10 20 30 40 50 60 N9 of the model were determined and we could illustrate a range for the free parameter so that for every point within the range, the model completely agrees with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Then, we tried to reconstruct the potential in terms of the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' It was found that the field range and the magnitude of the potential during inflation depend on the value of λ so that for smaller λ there is a wider field range and higher values for the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Finally, the validity of the swampland criteria was investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' The result clarified that to satisfy both conjectures the parameter λ could at least be of the order of O(102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Harko, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Lobo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Nojiri, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Odintsov, f(R, T) gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' D 84, 024020 (2011), arXiv:1104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='2669 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Bhatti and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Yousaf, Dynamical variables and evolution of the universe, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' D 26, 1750029 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Zaregonbadi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Farhoudi, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Riazi, Dark Matter From f(R,T) Gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE0T4oBgHgl3EQfwQHW/content/2301.02631v1.pdf'} +page_content=' Rev.' metadata={'source': 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Tokyo, Tokyo, 113-0033, Japan +2Astronomical Institute, Graduate School of Science, Tohoku University, Sendai, 980-8578, Japan +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +The assembly of supermassive black holes poses a challenge primarily because of observed quasars at high redshift, but +additionally because of the current lack of observations of intermediate mass black holes. One plausible scenario for creating +supermassive black holes is direct collapse triggered by the merger of two gas rich galaxies. This scenario allows the creation +of supermassive stars with up to solar metallicity, where the enhanced metallicity is enabled by extremely rapid accretion. We +investigate the behavior of metal enriched supermassive protostars which collapse due to the general relativistic radial instability. +These stars are rich in both hydrogen and metals and thus may explode due to the CNO cycle (carbon-nitrogen-oxygen) and the +rp process (rapid proton capture). We perform a suite of 1D general relativistic hydrodynamical simulations coupled to a 153 +isotope nuclear network with the effects of neutrino cooling. We determine the mass and metallicity ranges for an explosion. We +then post process using a 514 isotope network which captures the full rp process. We present nucleosynthesis and lightcurves for +selected models. These events are characterized by enhanced nitrogen, suppressed light elements (8 ≥ A ≥ 14), and low mass p +nuclides and they are visible to JWST and other near infrared surveys as decades-long transients. Finally, we provide an estimate +for the number of currently ongoing explosions in the Universe. +Key words: gravitation — (stars:) supernovae: general — nuclear reactions, nucleosynthesis, abundances +1 INTRODUCTION +The study of supermassive stars has arisen from two peculiarities of +the black hole population in the Universe. The first is that supermas- +sive black holes (SMBHs) exist soon after the big bang (Mortlock +et al. 2011; Wu et al. 2015; Bañados et al. 2018; Matsuoka et al. 2019; +Wang et al. 2021). Our current understanding of cosmology requires +that the SMBHs did not exist at the time of the big bang, implying +that they were created in the intervening period. The second pecu- +liarity is that the black hole mass function seems to be bimodal, with +a noticeable lack of intermediate mass black holes (having masses +in between solar mass black holes and SMBHs), although this may +be due to observational bias (Wrobel et al. 2016; Baumgardt 2017; +Kızıltan et al. 2017). If the bimodality is not due to observational bias, +it stands in opposition to the distributions of other self gravitating +objects (stars and galaxies), which have smooth mass functions. +The direct collapse black hole (DCBH) scenario was proposed to +resolve the first of these peculiarities (Bromm & Loeb 2003), and +is sometimes invoked to explain the second (e.g. Banik et al. 2019). +The scenario involves a gas cloud forming a single supermassive star +instead of many individual stars. This can occur in the presence of +local Lyman Werner radiation (Dijkstra et al. 2008; Agarwal et al. +2012; Latif et al. 2014b) or baryon dark matter supersonic streaming +(Latif et al. 2014a; Schauer et al. 2017; Hirano et al. 2017). The +★ E-mail: chrisnagele.astro@gmail.com +resultant supermassive star may be detectable directly (Surace et al. +2018, 2019; Vikaeus et al. 2022), via a general relativistic instability +supernova (GRSN, Chen et al. 2014; Whalen et al. 2013c; Nagele +et al. 2020; Moriya et al. 2021; Nagele et al. 2022b,a), by the obser- +vation of gravitational waves (Shibata et al. 2016; Li et al. 2018), or +as an ultra long gamma ray burst (Sun et al. 2017). +In this paper, however, we consider a slightly different scenario +where the supermassive star formation is triggered by a merger of +two gas rich galaxies (for a review, see Mayer & Bonoli 2019). +The phenomenon of nuclear gaseous disks forming via multi-scale +inflows was first investigated in the context of the M-sigma relation +as a means of providing a source of dynamical friction for a SMBH +binary in order to assist with its eventual merger (Kazantzidis et al. +2005; Mayer et al. 2007). Since then, it has been shown that not only +can this disk influence the behavior of existing SMBHs, but it can +also collapse under its own gravity to form a new black hole (Mayer +et al. 2010). Recently, Zwick et al. (2022) calculated observables +from DCBHs resulting from galaxy mergers. The crucial element of +this scenario as it pertains to the current study is that the scenario +is agnostic to the metallicity of the interstellar medium (ISM). This +means that supermassive stars will form out of metal enriched gas +(Mayer et al. 2015). +But what does it mean to have a metal enriched supermassive star? +These objects radiate at very nearly their Eddington limit and mass +loss rates from our local Universe (Vink et al. 2011) suggest such +objects would not survive for long. The situation is further compli- +© 2022 The Authors +arXiv:2301.01941v1 [astro-ph.HE] 5 Jan 2023 + +2 +C. Nagele et al. +cated by the fact that these metal enriched stars may be accreting +matter to replenish that lost to line driven winds. In this paper, we +sidestep this difficulty by considering only metal enriched supermas- +sive protostars which are massive enough to collapse via the general +relativistic (GR) radial instability (Chandrasekhar 1964) before they +reach the main sequence, thus avoiding any line driven mass loss. +The behavior of such metal enriched supermassive protostars has +been investigated previously (Fuller et al. 1986; Montero et al. 2012). +In particular, it was shown that if the protostars collapse due to the GR +radial instability, then this collapse can cause an explosion powered +by the CNO cycle (which we will term a proton rich or pr-GRSN, to +differentiate it from an 𝛼 process driven GRSN). pr-GRSNe were first +investigated in Fuller et al. (1986). They used a 1D post Newtonian +(PN) code with a 10 isotope nuclear reaction network and found +several exploding models spanning the mass range 5 × 105 − 106 +M⊙. The metallicity floor for the lowest mass model was 5 × 10−3. +Subsequently, Montero et al. (2012) used a 2D BSSN code with +parameterized heating rates to investigate models with similar mass, +and they were able to include the effects of rotation. For their non +rotating models, they found explosions with the same masses as +Fuller et al. (1986), but with slightly higher metallicity. +We use a GR 1D hydrodynamics code coupled to a 153 isotope +network. The large network allows us to more accurately follow the +dynamics of the explosion at higher temperatures, and we thus find +a lower metallicity floor than in previous works. After running our +simulations, we post process the hydrodynamical trajectories with a +514 isotope network designed to fully follow the rp-process on the +proton rich side. Contrary to the conclusions of previous works, we +find that the rp-process can play a critical role in the explosion. +In Sec. 2 we outline our numerical procedures for stellar evolu- +tion, hydrodynamics, post processing, and lightcurves. In Sec. 3.1, +we present the results of the stellar evolution simulations. In Sec. +3.2, we present the results of our hydrodynamical simulations and +post processing, for a fiducial model, as well as for varying mass and +metallicity. In Sec. 3.3 we present the results of our lightcurve calcu- +lations and an estimate of pr-GRSN density. In Sec. 3.4, we discuss +various feedback induced by a pr-GRSN. Finally, we conclude with +a discussion in Sec. 4. +2 METHODS +In this section, we first describe our initial models and stellar evo- +lution code, then provide details of our GR hydrodynamical code, +after which we detail the open source code SNEC, which is used to +calculate lightcurves. +2.1 Stellar evolution +The HOSHI code (Takahashi et al. 2016, 2018, 2019; Yoshida et al. +2019) is a 1D stellar evolution code which solves the stellar structure +and hydrodynamical equations using a Henyey type implicit method. +Nagele et al. (2020) introduced the first order PN correction to the +hydrostatic terms. The PN approximation is extremely accurate for +SMSs in hydrostatic equilibrium because the effects of GR are mi- +nor. These minor effects must be included, however, because SMSs +are radiation dominated and therefore close to instability. Once the +evolution of the star becomes dynamical, HOSHI’s lack of a shock +capture scheme and the PN dynamical corrections neccesitate the +use of another code. HOSHI includes a nuclear reaction network (52 +isotopes), neutrino cooling, mass loss, and rotation. The equation +of state includes contributions from photons, averaged nuclei, elec- +trons, and positrons. HOSHI uses the Rosseland mean opacity of the +OPAL project (Iglesias & Rogers 1996) and solves the Saha equation +to determine the ionization of hydrogen, helium, carbon, nitrogen, +and oxygen. +In this paper, 𝑀 is the total mass, 𝑅 the radius, 𝑇 the temperature, +and 𝜌𝑏 the baryonic density where quantities with 𝑐 subscripts show- +ing the central values. 𝑠𝑟 is the entropy due to radiation at a given +mass (Shapiro & Teukolsky 1983) +𝑠𝑟 = 0.942 +� 𝑀 +M⊙ +�1/2 +. +(1) +Finally, X is the mass fraction of a specified element. +To assist with analysis, we define various global energy quantities. +The internal energy is +𝐸int = +∫ 𝑀 +0 +𝜖 𝑑𝑚𝑟, +(2) +where 𝑚𝑟 is the mass coordinate and 𝜖 is the specific energy. The +gravitational energy is +𝐸grav = − +∫ 𝑀 +0 +𝑔effective 𝑟 𝑑𝑚𝑟, +(3) +where 𝑔effective is the local gravity with the 1st order PN correction +to the static terms (Nagele et al. 2022b). The accuracy of this ap- +proximation degrades with increasing density and velocity, neither +of which are particularly concerning for our purposes. The kinetic +energy is +𝐸kin = +∫ 𝑀 +0 +𝑣2 +2 𝑑𝑚𝑟, +(4) +where 𝑣 is the radial velocity. The binding energy of the star is +the negative of the thermal and gravitational energies (so that a +more tightly bound star has higher 𝐸bind), while the total energy +additionally includes kinetic energy: +𝐸bind = −(𝐸int + 𝐸grav) +(5) +𝐸tot = 𝐸int + 𝐸grav + 𝐸kin. +(6) +As in our previous works, we define the explosion energy as the +total energy at shock breakout. For HYDnuc, we also report the +integration over energy generation due to the nuclear network and +neutrino cooling (dots indicate time derivatives): +𝐸nuc(𝑡) = +∫ 𝑡 +0 +∫ 𝑀 +0 +� +𝜖nuc 𝑑𝑚𝑟 𝑑𝑡 +(7) +𝐸𝜈(𝑡) = +∫ 𝑡 +0 +∫ 𝑀 +0 +�𝜖𝜈 𝑑𝑚𝑟 𝑑𝑡. +(8) +A summary of the results of the HOSHI simulations can be found in +Table 2. +We initiate the HOSHI code in a high entropy state, which imme- +diately relaxes towards a constant entropy radiation dominated state, +though the protostar does not reach all the way to the radiation value +due to the small contribution of gas pressure. This configuration +could represent one of two physically realizable scenarios. Either, +it could be a supermassive protostar which has finished accreting +due to depletion of accretion material, which then contracts to the +radiation dominated constant entropy pre-ZAMS (zero age main se- +quence) state, or it could represent the convective core of a currently +MNRAS 000, 1–16 (2022) + +CNO-rp driven GR instability supernovae. +3 +Table 1. Summary table for the nuclear networks. Entries show the range in A for the specified element. +Element +52 +153 +216 +514 +Element (ctd.) +52 +153 +216 +514 +N +1 +1 +1 +1 +V +47 +45-51 +45-51 +42-53 +P +1-3 +1-3 +1-3 +1-3 +Cr +48 +47-54 +47-54 +44-55 +He +3-4 +3-4 +3-4 +2-4 +Mn +51 +49-55 +49-55 +46-57 +Li +6-7 +6-7 +6-7 +4-7 +Fe +52-56 +51-58 +51-58 +48-60 +Be +7-9 +7-9 +7-9 +5-9 +Co +55-56 +53-59 +53-59 +50-61 +B +8-11 +8-11 +8-11 +6-11 +Ni +56 +55-62 +55-62 +52-66 +C +12-13 +12-13 +12-13 +8-14 +Cu +— +57-63 +57-63 +54-68 +N +13-15 +13-15 +13-15 +10-16 +Zn +— +60-64 +60-64 +56-71 +O +14-18 +14-18 +14-18 +12-18 +Ga +— +— +63-69 +58-73 +F +17-19 +17-19 +16-19 +14-20 +Ge +— +— +65-70 +60-75 +Ne +18-20 +18-22 +18-22 +16-22 +As +— +— +68-75 +62-76 +Na +23 +21-23 +20-23 +18-24 +Se +— +— +72-76 +64-81 +Mg +24 +22-26 +22-26 +20-26 +Br +— +— +74-79 +66-82 +Al +27 +25-27 +24-27 +22-28 +Kr +— +— +76-80 +68-86 +Si +28 +26-32 +26-30 +24-30 +Rb +— +— +78-85 +70-87 +P +31 +29-33 +28-31 +26-32 +Sr +— +— +82-86 +72-89 +S +32 +30-36 +30-36 +28-37 +Y +— +— +84-89 +74-91 +Cl +35 +33-37 +32-37 +30-39 +Zr +— +— +86-90 +76-95 +Ar +36 +34-40 +34-40 +32-43 +Nb +— +— +— +78-96 +K +39 +37-41 +36-41 +34-45 +Mo +— +— +— +80-98 +Ca +40 +38-43 +38-43 +36-48 +Tc +— +— +— +82-98 +Sc +43 +41-45 +41-45 +38-49 +Ru +— +— +— +84-99 +Ti +44 +43-48 +43-48 +40-51 +accreting supermassive star. In the latter case, the accretion envelope +could in theory effect the stability of the system, but such effects are +thought to be small (Haemmerlé 2020). We expect that the envelope +will have no effect on the dynamical behavior of the protostar once +the GR instability is reached besides increasing the overall gravity +(see e.g. Fig. 14 of Nagele et al. 2022b). +We determine the stability of the protostar in HOSHI by solving +the pulsation equation for a hydrostatic, spherically symmetric object +in general relativity (Chandrasekhar 1964): +𝑒−2𝑎−𝑏 d +d𝑟 +� +𝑒3𝑎+𝑏Γ1 +𝑃 +𝑟2 +d +d𝑟 (𝑒−𝑎𝑟2𝜉) +� +−4 +𝑟 +d𝑃 +d𝑟 𝜉+𝑒−2𝑎+2𝑏𝜔2(𝑃+𝜌𝑐2)𝜉 +−8𝜋𝐺 +𝑐4 𝑒2𝑏𝑃(𝑃 + 𝜌𝑐2)𝜉 − +1 +𝑃 + 𝜌𝑐2 +� d𝑃 +d𝑟 +�2 +𝜉 = 0, +(9) +where 𝑎, 𝑏 are the metric coefficients as defined in Haemmerlé +(2021a), 𝑟 is the radius, 𝑃 the pressure, Γ1 the local adiabatic in- +dex at constant entropy (𝑠), 𝜌 = 𝜌𝑏(1 + 𝜖) the relativistic density, +and 𝜖 the specific internal energy (we absorb rest mass due to mass +excess of nuclei into this energy). +The star, or in this case, protostar, is unstable if there exists a trial +function 𝜉(𝑟) ∝ 𝑒𝑖𝜔𝑡 with 𝜔2 < 0, representing a perturbation which +will grow exponentially. There are two main approaches to solving +this equation, either by assuming a nearly linear trial function 𝜉 ∝ 𝑟𝑒𝑎 +(Haemmerlé 2021a), or by iteratively solving for the fundamental +mode of the normal mode decomposition of perturbations to Eq. 9 +(Nagele et al. 2022b). Here we adopt the latter approach, as in our +previous paper, but this choice should have a minimal bearing on the +results. +Even though we will eventually simulate the dynamics of metal +enriched supermassive protostars, we only consider metal free proto- +stars in HOSHI. This is because we are only interested in protostars +which become unstable before the onset of nuclear burning, so metal- +licity, will not effect the protostellar structure except as caused by +Figure 1. Radial (upper) and velocity (lower) snapshots of the fiducial model +at three timesteps, the initial time, the time when the central temperature is +largest, and shock breakout. +changes to the opacity. We consider the following range of masses: +7 − 10 × 104 M⊙ in steps of 104 M⊙ and 1 − 3 × 105 M⊙ in steps of +5 × 104 M⊙. +2.2 Hydrodynamics +HYDnuc is a 1D Lagrangian GR hydrodynamics code which uses +a Roe-type approximate linearized Riemann solver (Yamada 1997; +Takahashi et al. 2016; Nagele et al. 2020). It includes all of the +physics from HOSHI except for convection and ionization. In this +paper, we use a 153 isotope network (Table 1). The 153 isotope +MNRAS 000, 1–16 (2022) + +le13 +initial +1.5 +max T. +shock breakout +r [cm] +1.0 +0.5 +0.0 +6 +4 +2 +0 +-2 +0 +20000 +40000 +60000 +80000 +100000 +mr[M。]4 +C. Nagele et al. +Figure 2. Nucleosynthesis in the post processed 514 isotope network for the fiducial model, 𝑀 = 105 M⊙, 𝑍 = 𝑍⊙. Upper panels — isotope mass fractions for +five snapshots: the initial time, when the temperature rises to 3/4 of the eventual maximum, the maximum temperature, the final time step, and 1012 seconds +later. Lower panel — total energy compared to 𝐸nuc in both the 153 and 514 isotope calculations and compared to 𝐸nuc + 𝐸𝜈 for the 153 isotope calculation. +network (Takahashi et al. 2018) covers the proton rich side (p side) +at a depth of 3-8 isotopes up to zinc. Reaction rates for all networks +are taken from JINA REACLIB (Cyburt et al. 2010). +We use the same scheme to transport our models from HOSHI to +HYDnuc as in Nagele et al. (2022b) which is based on the frequency +function defined in Takahashi et al. (2019). We set the chemical +composition to be constant throughout the star, as a fraction of the +solar composition (Asplund et al. 2009). Due to the exploratory +nature of this study and the requirement of a larger nuclear network, +we use slightly less optimal numerical parameters than in Nagele +et al. (2022b), specifically 255 mesh points and V = 10−4 (maximum +allowed fractional variation of independent variables per timestep). +The effect of these changes is to underestimate the energy generated +by nuclear burning (see Fig. 6 of Nagele et al. 2022b). As in our +previous works, we terminate the simulations when convergence +issues arise due to large radius (𝑟 ∼ 1015 cm). +2.3 Post-processing +After performing the HYDnuc simulations, we post process the hy- +drodynamical trajectories using a 514 isotope network (Table 1) +designed to follow the rp process up to ruthenium. All isotopes on +the p side are covered up to the line with slope 1 and y intercept 5. +Even though this post processing is less computationally expensive, +the network is too large to solve the composition at every timestep of +HYDnuc. We choose to solve the composition with a frequency of +100−1 timesteps−1, and have checked that a) the convergence of 𝐸nuc +as the frequency increases and b) that 𝐸nuc with frequency 100−1 +agrees with 𝐸nuc with frequency 10−1 to within 0.1%. At the end +of the HYDnuc simulation, the post-processed composition contains +many radioactive isotopes. We then fix the temperature and density +and continue to post-process for an additional 1012 seconds while +logarithmically increasing the timestep. 1012 seconds is enough for +most, but not all (e.g. 26Al) radioactive isotopes to decay. +2.4 Lightcurves +The SuperNova Explosion Code (SNEC) is an open source, 1D La- +grangian, radiation hydrodynamics code designed to compute su- +pernova lightcurves (Morozova et al. 2015). It includes artificial +Figure 3. Nucleosynthetic yields (𝑡 = 𝑡final + 1012 s) from the 514 isotope +network for the 𝑀 = 105 M⊙ for the indicated metallicities. +viscosity, an equation of state with a Saha solver for ionization of +hydrogen and helium, and equilibrium flux-limited photon diffusion +using OPAL. As in (Nagele et al. 2022a), we port our HYDnuc mod- +els to SNEC slightly before shock breakout and for the outer layer +of the SNEC model, we use the HOSHI progenitor which enables +increased surface resolution necessary for properly following the +lightcurve. We terminate the SNEC simulations after 109 seconds at +which point any plateau phase in the luminosity has finished. +We then use the effective temperature to construct blackbody spec- +tral energy distributions and assume a standard ΛCDM cosmology. +With these components, we calculate apparent magnitudes for the +passbands of various telescopes at a given redshift. In this study, we +do not take extinction into account. +MNRAS 000, 1–16 (2022) + +t= tinitial +Tc= gTc,max +Te= Tc,max +t = tfinal +t= trinal + 1012 s +0 +20 +-2 +pnumber +-4 +15 +Log +-6 +10 +-8 +10 +15 +20 +25 +10 +15 +20 +25 +10 +15 +20 +25 +10 +15 +20 +25 +10 +15 +20 +25 +-10 +nnumber +1e55 +2 +Etot (153 isotopes) +Etot(t = 0) + Enuc (153) +Etot(t = 0) + Enuc + Ev (153) +Etot(t = 0) + Enuc (514 isotopes) +0 +0 +20000 +40000 +60000 +80000 +100000 +t [s]Z/Z。=0.12254 +4 +Z/Z。=0.1226 +Z/Z。=0.123 +Z/Z。= 0.13 +Sc +3 +Z/Z。= 0.2 +Z/Z。 = 1.0 +2 +[X/Fe] +N +Mo +Ne +0 +-1 +C +Ar +Co +-2 +Mg +5 +10 +15 +20 +25 +30 +35 +40 +45 +Z (proton number)CNO-rp driven GR instability supernovae. +5 +Table 2. Summary of the state of the stellar evolution simulations at the GR instability. The columns are, total mass, radius, central temperature, baryonic density +and entropy, central entropy as a fraction of radiative entropy (for a star of that mass), hydrogen mass fraction and binding energy. +M [105 M⊙] +R [1013 cm] +𝑇𝑐 [108 K] +𝜌𝑏,𝑐 [g/cc] +𝑠𝑐 [𝑘𝑏/baryon] +𝑠𝑐/𝑠𝑟 − 1 +X(1H) +𝐸bind [1054 ergs] +0.7 +264.8 +1.554 +1.881 +265.3 +0.06454 +0.4979 +1.694 +0.9 +32.24 +1.652 +1.968 +306.2 +0.08347 +0.7406 +2.702 +1.0 +2.061 +1.28 +0.8681 +322.4 +0.08224 +0.7599 +3.466 +1.5 +2.94 +0.7642 +0.1508 +390.4 +0.07005 +0.7599 +3.321 +2.0 +3.272 +0.7884 +0.1429 +447.6 +0.06238 +0.7599 +4.144 +2.5 +4.17 +0.6803 +0.08181 +499 +0.05935 +0.7599 +4.487 +3.0 +5.562 +0.5569 +0.04093 +544.7 +0.05577 +0.7599 +4.765 +Figure 4. Monotonic dependence of maximum temperature on metallicity +for the 𝑀 = 105 M⊙ model. +Figure 5. Central temperature evolution of the 𝑀 = 105 M⊙ for the indicated +metallicities. These trajectories were used for the post processing. +3 RESULTS +In this section, we first describe the results of the HOSHI code, after +which we provide details of the nucleosynthesis before moving on to +a discussion of the lightcurves. +3.1 Stellar Evolution +The lowest mass model which we consider in this study (7×104 M⊙) +becomes unstable only after entering the ZAMS phase, with the +instability occurring after roughly half a million years. This model has +a large radius and low proton mass fraction at the onset of instability +(Table 2). Because of its relatively long lifetime, enhanced metallicity +would likely have caused significant mass loss, thus prolonging the +period before the GR instability occurred (less mass means less +gravity). Then, there is the effect of accretion to consider, which, if +present, may be able to restore some of the lost mass. We do not +posses the proper tools to model these physical processes and so we +do not perform any hydrodynamical simulations with this model. +On the other hand, models with mass of 105 M⊙ or greater (9 × +104 M⊙ is a marginal case, which we include), collapse due to the +GR radial instability before any nuclear burning has taken place. For +increasing mass, these models have higher entropy, and are getting +more and more radiation dominated (Table 2, 6th column). This +corresponds to a decrease in the central temperature and density as +well as an increase in the binding energy. +The upper limit for stability in our study (7 × 104 M⊙) is slightly +lower than that found in Fuller et al. (1986) (105 M⊙), which is +likely due to small deviation from the GR pressure gradient in the +PN code which they employed. Indeed, as was shown in Nagele et al. +(2022b), the use of the baryonic density instead of the relativistic +density in the PN correction changes the stellar structure. Our results +are more massive than the cores of the accreting SMSs in Haemmerlé +(2020) (Fig. 8), which is to be expected because we do not consider +the gravity of the envelope. It should be noted that this is not a +completely faithful comparison because the accreting SMSs will +begin nuclear burning before they collapse due to GR (Hosokawa +et al. 2013; Umeda et al. 2016; Haemmerlé et al. 2018). For the high +accretion rates implied by the galaxy merger formation scenario, +however, there is little chance that the hydrogen reservoir will be +depleted before the GR instability, in which case the pr-GRSN would +still occur. As we will show, the less massive the progenitor, the easier +it is for the star to explode (Sec. 3.2). +MNRAS 000, 1–16 (2022) + +1e8 +7 +6 +max T. [K] +5 +4 +3 +10~5 +10-4 +10-3 +10-2 +10-1 +100 +Z/Zo-0.122531e8 +8 +Z/Z。=0.12254 +7 +Z/Z。=0.1226 +Z/Z。=0.123 +Z/Z。=0.13 +6 +Z/Z。=0.2 +Z/Z。=1.0 +5 +3 +2 +1 +0 +20000 +40000 +60000 +80000 +100000 +time [s]6 +C. Nagele et al. +Figure 6. Same as Fig. 2 but for 𝑀 = 105 M⊙, 𝑍 = 0.12254 𝑍⊙, which is the lowest metallicity explosion for 𝑀 = 105 M⊙. +Figure 7. Upper panel — same as upper panel Fig. 6 but showing the final composition at different meshes. Lower panel — mean molecular weight 𝜇 as a +function of mass coordinate. +3.2 Nucleosynthesis +3.2.1 Fiducial model +We take the M=105 M⊙, 𝑍 = Z⊙ as our fiducial model. This mass +is near the lower end of our mass range and the metallicity is well +above the threshold required for an explosion. In the hydrodynamical +simulation, the star begins in a high entropy, but relatively compact +state, due to the star not yet having settled into the main sequence. +At the start of the simulation, the CNO cycle immediately turns on, +but is not strong enough at those temperatures (𝑇𝑐 ≈ 108 K, Fig +5) to arrest the collapse. The star continues to collapse for 50000 +seconds before reaching a temperature of 𝑇𝑐 ≈ 2.3 × 108 K, at which +point the CNO cycle produces enough energy to reverse the motion +of the star. The CNO cycle continues to produce energy, even as the +temperature drops below 𝑇𝑐 ≈ 108 K, but the bulk of the energy +production occurs with the central temperature 𝑇𝑐 > 1.5 × 108 K +(Fig. 2). The star is completely unbound and explodes with energy +𝐸exp = 1.81 × 1055 [ergs]. It’s final radial and velocity profiles as a +function of mass coordinate can be seen in Fig. 1. +After finishing the HYDnuc simulation, we post process the sim- +ulation results as described in Sec. 2.3 with a 514 isotope network +designed to follow the rp process. Fig. 2 shows the isotope mass frac- +tion of the entire star in the 514 network at several time snapshots. In +the fiducial model, the energy generation comes almost entirely from +the CNO cycle, but proton captures on light elements do occur near +the maximum temperature (Fig. 2) and this can be seen in the nucle- +osynthetic yields (Fig. 3). The yields for the fiducial model (Z=Z⊙) +are characterized by two features. The first is enhanced nitrogen and +suppressed carbon and oxygen due to the CNO cycle, which produces +roughly equal amounts of its eponymous elements. The second is a +broad exchange of light elements (flourine, sodium, magnesium) for +slightly heavier elements (aluminium through chlorine) due to mul- +tiple proton captures which then decay back to stability at higher +mass number than they originated (Fig. 2). The only exception to this +trend appears to be neon which experiences proton captures, but is +replenished from below by oxygen exiting the CNO cycle. +3.2.2 Metallicity dependence +Next we consider the behavior of the M=105 M⊙ model as we vary +the metallicity. For each mass, there exists a threshold metallicity +value below which there are not enough seed metals to fuel an ex- +plosion. As one approaches this threshold, that is as one decreases +the metallicity, the model needs to reach higher temperatures before +the collapse can be reversed (Fig. 4). In addition, the time at which +the model reaches the maximum temperature is pushed further back +with decreasing metal content (Fig. 5). +Fig. 6 shows the isotope mass fraction for several time snapshots, +as in Fig. 2, but for Z= 0.12254 Z⊙ which is the metallicity closest +to the explosion threshold (we will refer to this as the metal-poor +MNRAS 000, 1–16 (2022) + +t= tinitial +Te =gTc,max +Te = Tc,max +t= trinal +t = tinal + 1012 s +40 +-4 +-6 +Log +-8 +X +-10 +10 +12 +10 +20 +30 +40 +50 +20 +30 +40 +50 +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +nnumber +le55 +5.0 +Etot (153isotopes) +Etot(t = 0) + Enuc (153) +Etot(t = 0) + Enuc + Ev (153) +Etot(t = 0) + Enuc (514 isotopes) +0.0 +40000 +50000 +60000 +70000 +80000 +90000 +100000 +t [s]-2 +40 +-4 +number +30 +-6 +Log +20 +-8 ++ +-10 +10 +-12 +10 +20 +30 +40 +50 +10 +20 +30 +40 +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +nnumber +0.63 +0.62 +0 +20000 +40000 +60000 +80000 +100000 +mr[M。]CNO-rp driven GR instability supernovae. +7 +model). We emphasize that this fine tuning is not done in order to +find the precise value of the threshold, but rather to demonstrate +that near the threshold, temperatures high enough to trigger the rp +process can be reached. Indeed, panels 2 and 3 of Fig. 6 show extreme +levels of proton captures extending to technetium. Note that in the +HYDnuc simulations with the 153 isotope network (as opposed to +the post processing with 514), the composition cannot go higher +than zinc (Table 1). In the post processing, the heavy p-side elements +then decay back to stability at much higher mass number than they +originated. Fig. 7 shows the final isotopic mass fraction distribution +at different meshes throughout the star. Unlike in the 𝛼 process driven +GRSN, most of the star undergoes nuclear burning and this is one of +the reasons that these explosions can be so energetic. +Fig. 3 shows the final abundances (relative to iron, relative to solar +values Asplund et al. 2009) from the 514 isotope network for selected +metallicities. As was the case in the fiducial model (Z⊙), there are +two main characteristics, with the first being enhanced nitrogen. +The second characteristic is a bulk transport of light elements to +heavy elements, with lower metallicities experiencing a larger and +wider transport. Note that oxygen is part of this transport, whereas +carbon and nitrogen do not vary significantly with metallicity. For +Z= 0.2 Z⊙, the transport extends up to scandium. For Z= 0.13 Z⊙, +iron peak elements are produced, terminating around gallium. For the +three smallest metallicities, the transport extends well past iron, up +to molybdenum in the extreme case. For the lower metallicity cases, +significant contributions to selected element are in the form of low +mass p-nuclides, which reside away from the line of stability (Fig. 6, +5th panel). Below Z= 1.0 Z⊙, all models show a peak at scandium, +with Z= 0.123 Z⊙ peaking again at proton number 31-33 and the two +lowest metallicity models peaking at proton number 31-36. Another +signature of all models is high Cl/Ar and low Ar/K, both resulting +from the bulk transport not necessarily preferring even elements. In +the high temperature models, cobalt is suppressed because its lightest +stable isotope (59Co) cannot be reached from the p side. +3.2.3 Mass dependence +Next, we vary the mass at solar metallicity. As the mass increases, +the binding energy of the star increases, and more nuclear energy is +required to unbind it. This means that the higher mass models will +reach higher temperatures until eventually (M = 2.5 × 105) they can +no longer explode at solar metallicity. Higher temperatures, in turn, +mean that the yields from the 514 isotope post processing should +show more heavier elements, and this is demonstrated in Fig. 8. We +have already discussed 105 M⊙, and 9 × 104 M⊙ is nearly identical. +The 1.5 × 105 M⊙ model shows the same bulk transport described +in the previous section, but this time extending up to titanium, while +the 2 × 105 M⊙ model (we will refer to this as the massive model) +peaks around titanium, but reaches as high as gallium. +Fig. 9 shows the explosion energy in HYDnuc (153 isotopes) as +a function of mass and metallicity. The explosion energy depends +only on mass, as it is a fixed fraction of the star’s binding energy, +unless the model is sufficiently close to the explosion threshold, as +can be seen with M= 2 × 105 M⊙, Z= 0.9 Z⊙. On the other hand, +the metallicity threshold increases with mass for the reasons stated +above. +We have attempted to run additional simulations using a 216 iso- +tope network (Table 1). However, there is a substantial increase in +computational cost (3469 reactions compared with 2463 for the 153 +isotope network) and the network is still not large enough to follow +the rp process to higher mass (Appendix A). For a rough estimate of +Figure 8. Same as Fig. 3, but for different masses at solar metallicity. +Figure 9. Dependence of explosion energy (denoted by color) on mass and +metallicity. The black crosses are models which failed to explode. The gray +region roughly corresponds to non-explosion. +the energy generation rate of the rp process beyond what we investi- +gate in this paper, see Appendix B. +3.3 Lightcurves +Fig. 10 shows the time evolution of photosphere radius, luminosity +and effective temperature for each of the three named models. In +addition, we have rerun the massive model with a much greater en- +velope resolution in order to more accurately characterize the shock +breakout (Fig. 11), though this increased resolution presents compu- +tational challenges during the plateau phase. The shock breakout is +accompanied by an extremely luminous (1049 ergs/s) burst with high +effective temperature, which is potentially observable as an X-ray +outburst. To date, the one observed X-ray outburst associated with +shock breakout (Soderberg et al. 2008) had an X-ray luminosity six +orders of magnitude lower (1043 ergs/s), implying that shock break- +out of a pr-GRSN would be visible, even at high redshift. However, +the short duration of the event and low rate of pr-GRSNe presents +serious challenges to realizing an observation of this shock breakout. +Besides the high luminosity and temperature at shock breakout, +MNRAS 000, 1–16 (2022) + +Sc +Z/Z。= 9e4 +3 +V +Z/Z=1e5 +Z/Z。=1.5e5 +Z/Z。=2e5 +2 +[X/Fe] +1 +N +Ne +0 +C +-1 +Mg +5 +10 +15 +20 +25 +30 +35 +40 +45 +Z (proton number)AA +1e56 +1.0 +1.2 +0.8 +X +1.0 +X +0.85 +X +[ergs +N +X +0.6 +0.4 +X +0.4 +x +0.2 +0.2 +x +0.0 +100000 +150000 +200000 +250000 +M[M。]8 +C. Nagele et al. +Figure 10. Results of the SNEC simulations for the fiducial model, metal-poor +model, and the massive model. Upper panel — photosphere radius. Middle +panel — bolometric luminosity. Lower panel — effective temperature. The +horizontal axis has been normalized so that shock breakout occurs at 10−3 +days. +there is another difference between the lightcurves of the pr-GRSNe +and the lightcurves of standard GRSNe (𝛼 process, Moriya et al. +2021; Nagele et al. 2022a). This is the longer duration of the plateau +phase which follows hydrogen recombination, nearly an order of +magnitude longer than our previous GRSN models. This longer du- +ration may be due to the increased mass or the increased energy in +comparison with previous GRSN models. During this plateau phase, +the hierarchy of the luminosities follows that of the associated ex- +plosion energies. Throughout this phase, the effective temperature +steadily falls and the photosphere radius steadily rises, the latter of +which is slightly different to the stalled photosphere found in standard +GRSNe. +This effect can be seen in Fig. 12 which shows the time evolution +of JWST NIRCAM wideband filters for the three named models +at redshift five, which is consistent with both ZISM = 0.1 Z⊙ and +ZISM = 1.0 Z⊙ (e.g. Pallottini et al. 2014). The prompt emission is +visible in the bluer filters because of the higher effective temperature, +but as this quantity drops, they rapidly fall away and only the four +reddest filters remain during the plateau phase. +3.3.1 Energy input from radioactive decays +When calculating lightcurves in SNEC for the metal-poor model, +there is an additional consideration, namely that the star is undergo- +ing a significant amount of radioactive decays after the end of the +HYDnuc calculation (see panels 4 and 5 of Fig. 6). In Fig. 13, we +Figure 11. Same as Fig. 10, but showing shock breakout for the massive +model. The simulation in this figure has higher surface resolution than those +in Fig. 10. The horizontal axis has been normalized so that the peak luminosity +occurs at 0 s. +show the rate of change in the nuclear energy as a function of time. +The vertical lines separate regions which are powered by different +decays, and these regions are labeled by the mother nuclei. Swells in +the heating rate can be seen for the decay of 56Ni and 57Co. Despite +the staggering amount of energy produced by these radioactive de- +cays (compare to Fig. 10), we do not think that they will affect the +lightcurve for two reasons. The first is that these decays are occurring +deep within the star and are well inside the photosphere. Later on, +it is conceivable that they could contribute, but we have tested run- +ning SNEC with 56Ni decays turned on and there is no difference in +the lightcurve. The second reason is that the total amount of energy +produced by these decays (∼ 1053 ergs) is order one percent of the +explosion energy. +3.3.2 Multiband lightcurves at different redshifts +In this section, we show the observer time evolution of the four reddest +JWST NIRCAM widebands at a variety of redshifts. Because the +galaxy merger scenario does not depend strongly on redshift (there +is a decrease below redshift two, but only by an order of magnitude, +see Fig. 4 of Bonoli et al. 2014), we show redshifts down to one. Fig. +14 shows this for the fiducial model up to redshift ten, above which +we suspect solar metallicity conditions would be challenging (though +not impossible, see e.g. Fig. 3 of Hartwig et al. 2022) to realize. Fig. +15 is a similar figure, but for the metal poor model, which we show +up to redshift 19, though it would not be detectable by JWST much +above redshift 10. Appendix C shows multiband lightcurves of the +MNRAS 000, 1–16 (2022) + +Photo-sphere radius [cm] +1018 +fiducialmodel +metal-poor model +1017 +massivemodel +1016 +1015 +1014 +1047 +I [ergs/s] +1046 +1045 +Lbol +1044 +1043 +1042 +106 +105 +104 +103 +102 +10-4 +10~3 +10-2 +10-1 +100 +101 +102 +103 +t [days]cm] +2.795 +2.790 +2.785 +2.780 +2.775 +2.770 +1.5 +1.0 +0.5 +2:9 +2.0 +got] +1.5 +1.0 +0.5 +0.0 +-2 +0 +2 +4 +6 +8 +10 +t [s]CNO-rp driven GR instability supernovae. +9 +Figure 12. Rest frame time dependence of JWST NIRCAM magnitudes at +𝑧 = 5 for the fiducial model (upper panel), metal-poor model (middle panel) +and massive model (lower panel). The horizontal dotted line shows a typical +limiting magnitude for JWST (29). +Figure 13. Rate of energy produced by radioactive decays in the metal-poor +model after the end of the HYDnuc simulation. Vertical dotted lines separate +regions where the energy production is dominated by different decays. The +mother nuclei of those decays label each region. +fiducial, metal poor, and massive models for JWST, Euclid, Roman, +and Rubin at various redshifts. +Comparing Figs. 14,15, it is clear that from the point of view of +observation, more metal enrichment is better. Not only is the mass +range wider at higher metallicity (Fig. 9), but also the GRSN would +have occurred at lower redshift, making it easier to observe. +3.3.3 With hylotropic envelope +Up to this point, we have not considered the effect that an accretion +envelope (Hosokawa et al. 2013; Umeda et al. 2016; Haemmerlé +et al. 2019) would have on the lightcurve. While the core of an +accreting supermassive protostar has constant entropy, the envelope +is thought to have entropy increasing as a power law Begelman +(2010); Haemmerlé et al. (2019); Haemmerlé (2020, 2021b) and +this structure has been termed a hylotrope (Gk. hyle, ‘matter’ + +tropos, ‘turn’) in Begelman (2010). Specifically, hylotropes obey the +equation of state +𝑃 = 𝐴𝜌4/3𝑀 𝛼 +(10) +where 𝛼 is often taken to be 2/3 as derived from the homology +scalings. In this scenario, we instead take 𝛼 as a parameter and +enforce the mass radius relation of rapidly accreting supermassive +protostars (Eq. 1 of Hosokawa et al. 2013). This choice is motivated +by the sensitivity of lightcurves to the stellar radius. +We construct the hylotropic envelope in a similar manner to the +integration of an 𝑛 = 3 polytrope, as described in Haemmerlé (2020). +We find that, taking the fiducial model as the core, 𝛼 = 0.90481 +satisfies the mass radius relation. This result is not sensitive to the +matching radius. This hylotropic model has 𝑀𝐻 = 4.58 × 105 M⊙ , +𝑅𝐻 = 1.23 × 1016 [cm], so that 𝑀𝐻 /𝑀 = 4.58 and 𝑅𝐻 /𝑅 = 653. +This result resembles numerical models of accreting supermassive +protostars, although self consistent simulations are needed to verify +the results of this section. +We run SNEC by attaching the final timestep of the HYDnuc sim- +ulation to the hylotropic envelope which was constructed using the +initial profile of the HYDnuc simulation. This results in a discon- +tinuity in density, but such a feature is not unexpected around this +radius (see Fig. 14 of Nagele et al. 2022b). We use the thermal bomb +mode with the final explosion energy of the hylotropic model being +the explosion energy of the fiducial model, as the added gravitational +energy of the envelope could not be overcome otherwise. This ap- +proach is not self consistent as it requires the injection of an additional +∼ 1055 ergs, but larger explosion energies (1056 ergs) are achieved +in the massive model, so our strategy is not unreasonable. The pho- +tosphere remains at the surface of the hylotrope for ten days as the +shock propagates through the envelope, after which the photosphere +expands and the effective temperature drops. From a peak value of +𝐿bol = 6.86 × 1046 [ergs/s], the luminosity then falls nearly mono- +tonically (except for a slight increase at hydrogen recombination) in +behavior reminiscent of the pulsations of large radius supermassive +stars in Nagele et al. (2022a). The luminosity of the hylotropic model +falls at a slower rate than for the fiducial model, passing 𝐿bol = 1043 +[ergs/s] only after 9.95 years in the rest frame. Fig. 16 shows the +magnitudes of the hylotropic model at redshift 1 for JWST, Euclid, +Roman, and Rubin. +3.3.4 Rate estimate +We now turn to the question of how frequently these pr-GRSN occur. +Bonoli et al. (2014) calculated the number density of massive galaxy +MNRAS 000, 1–16 (2022) + +AB mag at z = 5 +24 +F070W +F150W +F356W +F090W +F200W +F444W +fiducial model +26 +F115W +F277W +28 +30 +32 +24 +metal-poor model +26 +28 +30 +32 +24 +model +26 +massive +28 +30 +32 +10~3 +10~2 +10-1 +100 +101 +102 +103 +t [days]52Mn +........... +........... +................................. +1047 +725e +..... +....... +: +... +1046 +oGe +............... +........ +...... +..... +1045 +.... +....... +............ +1044 +56Ni +..... +1043 +....... +........ +1042 +..... +....... +...... +...... +1041 +............. +.............. +....... +.............. +'s5Fe +81kr +1040 +....... +......... +1039 +106 +107 +108 +109 +1010 +1011 +1012 +t [s]10 +C. Nagele et al. +Figure 14. Redshift dependence of the four reddest JWST NIRCAM wide bands for the fiducial model. The horizontal axis shows the observer time. +Figure 15. Same as Fig. 14 but for the metal-poor model. The horizontal axis shows the observer time. +MNRAS 000, 1–16 (2022) + +20 +z= 1 +20 +/magnitude +22 +/magnitude +22 +24 +24 +5 +F200W +26 +77W +26 +28 +2 +28 +10 +30 +30 +20 +20 +magnitude +22 +F444Wmagnitude +22 +24 +24 +F356W +26 +26 +28 +28 +30 +30 +0 +5000 +10000 +15000 +20000 +25000 +0 +5000 +10000 +15000 +20000 +25000 +observertime[days] +observertime[days]20 +z= 1 +20 +/magnitude +z = 3 +22 +magnitude +22 +24 +Z = +24 +26 +11 +F200W +13 +77W +26 +28 +15 +2 +28 +30 +30 +20 +20 +magnitude +22 +F444Wmagnitude +22 +24 +24 +356W +26 +26 +28 +28 +30 +30 +0 +2000 +400060008000100001200014000 +0 +2000400060008000100001200014000 +observertime[days] +observertime[days]CNO-rp driven GR instability supernovae. +11 +Figure 16. Magnitudes of the hylotropic model in JWST (first panel), Euclid +(second panel), Roman (third panel), and Rubin (fourth panel) at redshift 1. +The horizontal axis is observer time. +mergers (𝑀halo > 1011 M⊙) which fulfilled the criteria for merger +induced direct collapse (Fig. 4). For a simple order of magnitude +estimation, we will assume a merger rate of 𝜙 = 10−4 [cMpc−3 +Gyr−1] which is fairly conservative (note that Fig. 4 of Bonoli et al. +(2014) has units of [cMpc−3 0.1 Gyr−1]). As discussed in Bonoli +et al. (2014), relaxing the mass asymmetry condition further would +increase the merger rate by an order of magnitude. Another increase +could be had by relaxing the mass constraint, although the minimum +mass at which the merger scenario creates a direct collapse object +is unknown. For a more recent, empirical, estimate of the galaxy +merger rate, see Fig. 3 of O’Leary et al. (2021). +To determine the number of supermassive protostars produced per +unit time by the galaxy merger scenario, we simply multiply the rate +𝜙 by the volume between redshifts 0.1 and 10 (𝑉𝑧∈(0.1,10)). +𝑁protostars = 𝜙𝑉𝑧∈(0.1,10) ≈ 0.3 yr−1 +(11) +The lower bound of this range was chosen because very nearby GRSN +will be visible to many instruments besides the four discussed in this +paper, while the upper bound is nearing the observing threshold for +JWST (Fig. 14). +Next we introduce 𝑓 , the fraction of protostars which explode as +pr-GRSN. We have no ability to estimate the initial mass function +of these objects and thus no ability to estimate 𝑓 . However, it is +reasonable to assume that the mass function decreases for increasing +mass, as does an estimate of the mass function of supermassive stars +(Toyouchi et al. 2022). If this is the case, then 𝑓 will depend heavily +on the behavior of the protostars less massive than those considered +in this paper (Fig. 9). As we have mentioned previously, however, it is +feasible that many of these less massive protostars will also explode, +so an 𝑓 of order unity is feasible. +𝑁pr−GRSN = 𝑓 𝜙𝑉𝑧∈(0.1,10) yr−1 +(12) +From the rate of pr-GRSN, we would then like to obtain a number +of currently ongoing pr-GRSN in the sky. To do this, we need to +know the expected observer duration for the pr-GRSN (⟨𝑡obs⟩). To +do this we perform a Monte Carlo simulation, randomly selecting +lookback times between redshifts 0.1 and 10 and computing the +observer time during which the GRSN is visible to the JWST F444W +band. Averaging the Monte Carlo draws results in +⟨𝑡obs⟩ = 29.5 yr +(13) +so that the expected number of currently observable pr-GRSN is +𝑁observable = 𝑓 𝜙𝑉𝑧∈(0.1,10) ⟨𝑡obs⟩ ≈ 9f +(14) +and we plot this quantity as a function of 𝑓 in Fig. 17, indicating that +it is reasonable to expect a few GRSNe to be observable right now. +On the one hand, this is a small number because JWST only covers +a tiny section of the sky. In addition, the true value of 𝑓 may be small. +However, we point out that the majority of the Monte Carlo draws +occur in the low redshift Universe where gas should be sufficiently +enriched to trigger an explosion. Thus, the pertinent question be- +comes what is the mass function of the supermassive protostars? On +the other hand, Fig. 17 is showing a large number. Bonoli et al. (2014) +and Mayer et al. (2015) are investigating the formation of the most +massive SMBHs in the Universe. This investigation is well founded +upon observations of high redshift quasars with inferred masses in +excess of 108 M⊙, but it does not consider the vast majority of the +black hole population, which have smaller masses (e.g. Inayoshi et al. +2020). For this reason, the mass and mass ratio constraints which are +applied in Fig. 4 of Bonoli et al. (2014) could be far too stringent, +in which case the population of direct collapse objects, and GRSNe, +would be much greater. +In summary, massive uncertainties exists regarding the frequency +of this scenario, but we have shown that it is at least plausible to +expect pr-GRSNe to be observable in the low redshift Universe, if +the galaxy merger scenario is the dominant source of supermassive +black holes. +3.3.5 Observing strategy +Because of the long duration of the pr-GRSNe, identifying them as +transients may not be straightforward, although detection by multi- +ple instruments would ameliorate this difficulty; as well as JWST, +Appendix C shows the magnitudes of Euclid, Roman, and Rubin at +various redshifts. In the rest of this section, however, we will consider +only the four reddest JWST NIRCAM widebands (see Fig. 12). +We divide the lightcurve up into phases, a rising phase, a falling +phase, and a plateau phase in between the rising and falling phases. +We define the rising phase as lasting until peak magnitude is reached, +which occurs at different times in each band (Fig. 12) and the falling +phase as beginning when the photosphere radius jumps (Fig. 10); +the falling phase will not be visible in all bands, especially at high +redshift. Observations of the rising or falling phases will be easily +identifiable as transients (Fig. 14), but the plateau phase, while not +having constant magnitude, will dim at a rate of the order of a few +magnitudes or less per decade (Fig. 18). This rate of dimming is +roughly constant as the duration of the plateau phase increases (for +MNRAS 000, 1–16 (2022) + +16 +F070W +18 +F090W +F115W +JWST +20 +F150W +F200W +22 +F277W +F356W +F444W +24 +16 +VIS +18 +Y +Euclid +20 +H +22 +24 +16 +F062 +18 +F087 +F106 +Roman +20 +F129 +F158 +22 +F184 +F146 +24 +F213 +16 +Z +18 +Rubin +20 +y +u +22 +6 +24 +2500 +5000 +7500 +10000 +12500 +15000 +17500 +20000 +observertime[days]12 +C. Nagele et al. +Table 3. Various quantities related to chemical, mechanical, and radiative feedback for the post-processed models. The columns are mass, metallicity, the change +in metal content, the change in iron, the change in mean molecular weight, the kinetic energy (HYDnuc), the fade radius (at which the shock becomes subsonic), +the radiated energy (SNEC), the number of ionizing photons, and the number of Lyman Werner photons, where the last two assume blackbody emission. +M [105 M⊙] +𝑍/𝑍⊙ +𝑀𝑍 [M⊙] +𝑀Fe [M⊙] +Δ𝜇 +𝐸kin [1055 ergs] +𝑅fade [Mpc] +𝐸rad [1052 ergs] +𝑁𝛾,ion [1054] +𝑁𝛾,LW [1054] +1.0 +0.12254 +201.1 +24.73 +0.01721 +2.684 +1.423 +2.257 +55.9 +9.65 +1.0 +0.1226 +158.5 +26.25 +0.01663 +2.409 +1.278 +2.285 +25.28 +7.807 +1.0 +0.123 +105.8 +26.91 +0.01614 +2.108 +1.118 +2.32 +91.29 +9.491 +1.0 +0.13 +26.74 +2.669 +0.01591 +1.637 +0.8683 +2.43 +28.47 +5.851 +1.0 +0.2 +9.599 +-0.0008733 +0.01698 +1.45 +0.7692 +2.548 +8.964 +3.215 +1.0 +1.0 +4.933 +-0.002745 +0.02357 +1.593 +0.8446 +2.46 +7.814 +2.287 +0.9 +1.0 +4.509 +-0.002353 +0.02311 +1.278 +0.6777 +2.245 +380.7 +267.2 +1.5 +1.0 +48.43 +-0.004265 +0.05847 +8.055 +4.272 +3.914 +173.7 +22.5 +2.0 +1.0 +239.7 +-1.237 +0.0823 +17.26 +9.15 +5.485 +209.6 +41.23 +Figure 17. Expected number of concurrent GRSNe observable by JWST as +a function of the fraction ( 𝑓 ) of supermassive protostars which explode as +pr-GRSN. +higher redshifts), until the photosphere radius jump is no longer +visible (above 29 magnitude). For these fainter, high redshift, sources, +the plateau phase is shorter and the magnitude changes more quickly. +Fig. 19 shows color-color diagrams for the four reddest JWST +NIRCAM filters. All of the data points, including different models +and redshifts, fall neatly along a single line. The results of a linear fit +to this line are shown in each plot. Based on these features, we propose +that pr-GRSNe candidates are identified as NIRCAM sources falling +within at least the three reddest bands which are consistent with Fig. +19. These candidates may then be confirmed or ruled out with long +cadence observations. GRSNe which have already dropped out of +F270W and F2777W will not be identifiable by their color, and can +only be identified with repeated observation. +On the one hand, the long durations and shallow slopes of these +lightcurves pose a challenge to observing these events as transients. +On the other hand, if they are identified as such, their further clas- +sification as GRSNe will be relatively unambiguous, as there exist +no other events one hundred thousand times more energetic than a +supernova. +Figure 18. Rate of dimming during the plateau phase as a function of plateau +phase duration for the four reddest NIRCAM bands (different panels). The +colors denote redshifts and are shown in steps of 0.25, while the symbols +show different models. +3.4 Feedback +The pr-GRSN likely would have been extremely disruptive to its +host halo. Previously, several studies were conducted on the effect +a standard GRSN would have on the halo, specifically on metal +enrichment, gas evacuation, and star formation (Whalen et al. 2013a; +Johnson et al. 2013; Whalen et al. 2013b). The pr-GRSN is more than +an order of magnitude more energetic than in the standard case, and +MNRAS 000, 1–16 (2022) + +8 +S +4 +2 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ffiducial +Z= 1 +0.4 +massive +Z=2 +metalpoor +Z=3 +0.3 +Z = 4 +F200W +z = 5 +0.2 +Z=6 +Z = 7 +0.1 +z=9 +z = 10 +Plateauphase +0.0 +0.4 +0.3 +F277W +0.2 +0.1 +0.0 +0.4 +0.3 +F356W +0.2 +0.1 +0.0 +0.4 +0.3 +0.2 +0.1 +0.0 +0 +10 +20 +30 +40 +50 +Plateauphaseduration[years]CNO-rp driven GR instability supernovae. +13 +Figure 19. Two color-color diagrams for the four reddest NIRCAM wide- +bands. As in Fig. 18, colors denote redshift and symbols denote models. Also +shown is a linear fit (dotted line) as well as its slope and intercept. +has a completely different chemical signature. Whereas the standard +GRSN is characterized by excess silicon and magnesium, the pr- +GRSN will produce nitrogen from the CNO cycle, plus elements +from wherever the bulk transport lands on the line of stability (Figs. +3,8). +But this chemical enrichment will not be distributed evenly +throughout the halo (at least not at first). Heavier elements are pref- +erentially produced in the center of the protostar where higher tem- +peratures are reached. In the resulting outflow, these regions have +lower velocity, as the ejecta has nearly homologous structure (𝑣 ∝ 𝑟). +Thus, portions of the ejecta with different chemical compositions +will end up in different parts of the halo. We have attempted to vi- +sualize this effect in Fig. 20, which shows the yields for different +mass coordinates on the left, and the velocities of those correspond- +ing coordinates on the right. As an example, in the middle panel +(metal-poor model), consider the ratios of [Fe/Ni] and [C/Fe]. The +former will be constant throughout the halo [Fe/Ni] = -3 (or 0 in the +outermost regions), while the latter will vary [C/Fe] ∈ (-2,0). +Table 3 summarizes various feedback effects for selected models. +The third and fourth columns show the total amounts of metals and +iron synthesized during the explosion. Note that models which do not +reach sufficiently high temperature produce metals that do not extend +up to iron which experiences a net loss. The change in mean molecular +weight is shown in the next column. Next are the enormous effects +of mechanical feedback, specifically the total kinetic energy and the +radius at which the shock speed will become subsonic (𝑅fade, Magg +et al. 2020), though this value does not take the effect of the halo’s +gravity into account. Whalen et al. (2013a) found that the GRSN +ejecta reached a radius of 1 kpc before falling back and igniting a +violent starburst, although that was for a less energetic explosion. +Finally, the radiated energy as well as the number of ionizing (E +> 13.6 eV) and Lyman Werner (13.6 > E > 11.2 eV) photons. The +number of high energy photons is fairly small because the effective +temperature is low after shock breakout. +4 DISCUSSION +As the saying goes, hydrogen is flammable. We have shown that +if supermassive protostars have sufficient seed metals when they +collapse, then they will explode through a combination of the CNO +cycle and rp process. The consequences of this are as follows. +Our results present a challenge to the galaxy merger scenario +in two senses of the word. The first is that if the galaxy mergers +do not happen early enough in the Universe, then when they do +occur, the gas may be sufficiently enriched to trigger a pr-GRSN, +thereby preventing the formation of a SMBH seed. If this scenario +is avoided, then SMBH seeds may form from galaxy mergers and +this may explain the presence of the first quasars. However, as the +Universe continues to enrich itself, galaxy mergers continue to occur, +and if these later mergers produce a supermassive protostar, it will +almost certainly contain the requisite metallicity for an explosion. In +addition, these low redshift pr-GRSN will be easily observable. Thus, +if current and future NIR surveys do not see these objects, then the +galaxy merger scenario must posses a way to suppress massive object +formation at lower redshift. Such a mechanism is naturally built into +other scenarios such as atomic cooling halos by the stipulation that +the gas must be nearly metal free (Z/Z⊙ < 10−3, Chon & Omukai +2020; Hirano et al. 2022). +As we have touched upon, the intermediate mass black hole popu- +lation, or relative lack thereof, also presents a challenge to the study +of black holes. If SMBHs originate primarily from the galaxy merger +scenario, then our results present a natural explanation for the lack +of intermediate mass black holes. This is because metal enriched +supermassive stars and protostars which are not massive enough to +collapse to supermassive black holes (say, 106 M⊙) do not then +collapse to intermediate black holes, but instead either explode in a +pr-GRSN or lose most of their mass due to line driven winds while +on the main sequence. In the latter case, the mass of the final remnant +will be determined roughly by how much helium can be produced +before the envelope is completely lost, although mass loss from the +helium core may also be possible in some situations. +Although observation of the lightcurves from a pr-GRSN carries +our best chance at observation, other pathways exist. We have pro- +duced detailed nucleosynthetic signatures of the pr-GRSN and there +is some possibility that we may be able to find traces of these signa- +tures. A signature common to all of our models is enhanced nitrogen +production. A large population of nitrogen-enhanced mildly metal +poor stars has recently been observed (Fernández-Trincado et al. +2020), but any explanation involving a pr-GRSN seems far-fetched +due to the large amount of ejecta produced by a single event. Another +signature found in many of our models is a dearth of light alpha el- +ements, such as magnesium, which serve as the fuel for the proton +captures. Yoshii et al. (2022) reported a quasar with [Mg/Fe] =-1.11 +± 0.12 at redshift 7.54. They discuss how it is challenging to produce +so much iron so early in the Universe, but the pr-GRSN would nat- +urally explain this phenomenon by producing iron and consuming +MNRAS 000, 1–16 (2022) + +fiducial +2.0 +massive +metalpoor +1.5 +F356W +1.0 +z =1 +F277W- +z = 2 +Z =3 +Z=4 +0.5 +z = 5 +Z=6 +z = 7 +..... linear fit +8=Z +0.0 +m= 1.768 +6=Z +b=0.303 +z = 10 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +F200W-F277W +1.75 +fiducial +massive +1.50 ++metalpoor +1.25 +P +F444W +1.00 +F356W- +0.75 +Z = 1 +Z = 2 +z = 3 +0.50 +Z = 4 +Z = 5 +0.25 +Z = 6 +Z = 7 +0.00 +.....linear fit +Z=8 +m= 1.158 +Z=9 +b = 0.266 +Z = 10 +-0.25 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +F277W-F356W14 +C. Nagele et al. +Figure 20. Elemental yields at selected mass coordinates and velocity profiles for the three indicated models. Left column — elemental abundance for mass +coordinates (not the average between mass coordinates) at multiples of 0.2 times the total mass. Right column — velocity profiles at a comparable timestep +near the end of the HYDnuc simulation for each of the three models. The velocity profiles are nearly homologous. Colored circles indicate the mass coordinate +shown in the left column. +magnesium simultaneously. At this redshift, the ISM metallicity can +be greater than the explosion threshold (Pallottini et al. 2014). +We will now summarize some assumptions and shortcoming of the +current study. First is our decision to ignore mass loss and accretion +during the stellar evolution calculation, and thus limit ourselves to +protostars greater than ∼ 105 M⊙. Because only a small mass fraction +of protons is required to trigger the pr-GRSN, if a metal enriched +supermassive star were to collapse via the GR radial instability at +any point during the hydrogen burning phase (besides the very end), +then it would likely explode. Similarly, if the GR instability triggers +during the helium burning phase, a metal enriched version of the 𝛼 +process GRSN is plausible. A logical next step would be to apply our +methods to realistic models of metal enriched accreting supermassive +stars (e.g. Haemmerlé et al. 2019) which would also allow us to +correctly account for the accretion envelope. The lightcurve of the +pr-GRSNe is likely sensitive to the size of the accretion envelope +(Fig. 16). +The second major shortcoming of this study is that we do not +have the resources to couple the 514 isotope network to our GR +hydrodynamics code. Thus, all of the nucleosynthesis that we present +is the result of post-processing, which may not be entirely accurate. +In addition, since the energy generation by the 153 isotope network is +smaller than that of 514 (Sec. 2), we likely underestimate the region +MNRAS 000, 1–16 (2022) + +1.75 +8 +6 +1.50 +[X/Fe] +4 +1.25 +P +S +cm/ +1.00 +model +0 +0.75 +fiducial +2 +Mg +m, = 0.2 M +-4 +0.50 +m,=0.4M +-6 +m,=0.6M +0.25 +m,=0.8M +-8 +m,=1.0M +0.00 +8 +1.75 +6 +1.50 +[X/Fe] +Sc +4 +1.25 +2 +N +metal-poormodel +cm/ +1.00 +0 +0.75 +m,=0.0M +-4 +Co +0.50 +Mg +m.= 0.4 M +-6 +m,=0.6M +0.25 +m, = 0.8 M +-8 +m.=10M +0.00 +8 +1.75 +6 +1.50 +[X/Fe] +1.25 +S +cm/ +1.00 +model +0 +60t] +0.75 +massive +-2 +m,=0.0M +> +Mq +m, = 0.2 M +0.50 +m,=0.4M +-6 +m, = 0.6 M +0.25 +m,=0.8M +-8 +m,=1.0M +0.00 +5 +10 +15 +20 +25 +30 +35 +40 +45 +0 +1 +2 +3 +4 +5 +6 +7 +protonnumber +r [1014 cm]CNO-rp driven GR instability supernovae. +15 +covered by the pr-GRSN (Fig. 9). Other caveats such as effects of +rotation and initial conditions are salient, but likely less important. +We have shown that supermassive protostars which encounter the +GR radial instability before hydrogen burning will explode in a pr- +GRSN if the metallicity is high enough. These events will be clearly +visible to NIR surveys at lower redshifts, and marginally visible at +higher ones. They will also leave distinct chemical imprints on their +host halos, as well as massive amounts of radiative and mechanical +feedback. It is likely that current and future surveys with unprece- +dented breadth and depth will be able to constrain the population of +merger induce DCBHs based on the observation or non observation +of pr-GRSNe. +DATA AVAILABILITY +The data underlying this article will be shared on reasonable request +to the corresponding author. +ACKNOWLEDGEMENTS +This study was supported in part by the Grant-in-Aid for the Sci- +entific Research of Japan Society for the Promotion of Science +(JSPS, Nos. JP19K03837, JP20H01905, JP20H00158, JP21H01123, +JP22K20377) and by Grant-in-Aid for Scientific Research on Inno- +vative areas (JP17H06357, JP17H06365) from the Ministry of Edu- +cation, Culture, Sports, Science and Technology (MEXT), Japan. +REFERENCES +Agarwal B., Khochfar S., Johnson J. L., Neistein E., Dalla Vecchia C., Livio +M., 2012, MNRAS, 425, 2854 +Asplund M., Grevesse N., Sauval A. J., Scott P., 2009, ARA&A, 47, 481 +Bañados E., et al., 2018, Nature, 553, 473 +Banik N., Tan J. 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The 216 isotope network is based on the 153 +isotope network, but contains additional heavy elements up to Zr. +We had created this network because we noticed that our collapsing +models were saturating at Zn, the heaviest element in the 153 network. +Fig. 21 shows the isotopic mass fractions from the central mesh of +this simulation as well as the proton mass fraction (bottom panel). +As is apparent, proton captures stall around a central temperature of +Log 𝑇𝑐 = 8.85, as the network runs out of room for additional proton +captures. In addition, the decay times of the isotopes on the boundary +of the network are longer than the dynamical timescale (∼ 500 s). +This causes a buildup of isotopes at certain positions on the edge +of the network. In an extreme case, 66Ge accumulates nearly half a +percent (0.004) of the mass. This isotope has a half life of roughly +8000 s, largely preventing progress past it. +We offer this demonstration as a cautionary tale, that future net- +works likely need to be larger than the ones we have used here. Such +endeavours are beyond our current computational setup. +6 APPENDIX B +If we did have the resources to couple to a larger network, to what +extent would that alter our results? +To provide a simple estimate of the energy generation of the rp +process, we calculate the energy density produced by 25% of the +star undergoing the rp process up to an isotope, 𝐼𝑚. We assume that, +in this region, every element with atomic number greater than 15 is +converted to the most abundant solar isotope with proton number 𝑚 +(𝐼𝑚). Then, the chemical distribution of the inner 25% will change +as follows: +𝑋′ +𝐼𝑗 = +� +𝑋𝐼𝑗 +𝐴(𝐼 𝑗) < 16 or A(Ij) > A(Im) +0 +15 < 𝐴(𝐼 𝑗) < 𝐴(𝐼𝑚) +(15) +𝑋′ +𝑝 = 𝑋𝑝 − +∑︁ +𝐼𝑗 +(𝑋𝐼𝑗 − 𝑋′ +𝐼𝑗 ) 𝐴(𝐼𝑚) − 𝐴(𝐼 𝑗) +𝐴(𝐼 𝑗) +(16) +𝑋′ +𝐼𝑚 = 𝑋𝐼𝑚 + 𝑋′ +𝑝 + +∑︁ +𝐼𝑗 +(𝑋𝐼𝑗 − 𝑋′ +𝐼𝑗 ) +(17) +Fig. 22 shows the resulting 𝐸nuc (in units of ergs/g) as a function of +metallicity and m. Also shown are 𝐸nuc from the exploding models +in Fig. 9. We have selected 25% so that the border between explosion +and collapse will occur at the edge of our network, roughly m = 30. +Note that this is a somewhat ad hoc approach, as we are ignoring +neutrino cooling and inwards ram pressure, and we thus intend for +this figure to be interpreted as an order of magnitude estimate. As +such, it shows that a larger network would allow access to at least +twice as much energy generated by the rp process if it extended up +to technetium, a situation which we have shown to result from post +processing (Fig. 6). Indeed, the post processed 𝐸nuc is roughly twice +that produced by the 153 isotope network. Thus we can say that +our mass and metallicity range is too conservative, but probably the +correct order of magnitude. +7 APPENDIX C +This appendix shows lightcurves of JWST, Euclid, Roman and Rubin +for the fiducial model, at redshifts 0.1, 0.3, 0.5, 1.0, 3.0, 5.0. Rubin +can observe the pr-GRSN below redshift 1, while Euclid and Roman +extend this up to redshift 3. Observation of more distant events will +only be possible with JWST. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–16 (2022) + +CNO-rp driven GR instability supernovae. +17 +Figure 21. Upper panel — same as Fig. 6, but for only the central mesh of the M = 105 M⊙, Z = 0.1 Z⊙ with 216 isotopes. Lower panel — time evolution of +proton mass fraction. +Figure 22. 𝐸nuc (in units of ergs/g) generated by the rp process up to an +isotope with mass proton number m for a given metallicity. The red triangles +correspond to values of 𝐸nuc from the 153 isotope simulations (Fig. 9). The +vertical dashed line shows the edge of the 153 isotope network. +Figure 23. Same as Fig. 16, but for the fiducial model at redshift 0.1. +MNRAS 000, 1–16 (2022) + +Log T。= 8.31 +Log T, = 8.65 +Log Te = 8.84 +Log Te = 8.90 +Log Te= 8.98 +40 +10 +-2 +-4 +Log +-6 +20 +-8 +-10 +10E +-12 +10 +20 +30 +40 +50 +10 +20 +30 +40 +5010 +20 +30 +40 +50 +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +nnumber +0.75 +... +47500 +50000 +52500 +55000 +57500 +60000 +62500 +65000 +t [s] 1e18 +1.0 +1.17 +1.04 +0.8 +0.91 +0.78 +[ergs/g] +0.6 +0.65 +N +0.52 +0.4 +0.39 +0.26 +0.2 +0.13 +0.00 +10 +15 +20 +25 +30 +35 +40 +protonnumberatrpterminus(m)15.0 +F070W +F090W +17.5 +F115W +5 +F150W +20.0 +M +F200W +22.5 +F277W +F356W +25.0 +F444W +15.0 +VIS +Y +17.5 +Euclid +H +20.0 +22.5 +25.0 +15.0 +F062 +F087 +17.5 +F106 +Roman +F129 +20.0 +F158 +22.5 +F184 +F146 +25.0 +F213 +15.0 +Z +17.5 +Rubin +20.0 +y +u +22.5 +b +25.0 +0 +1000 +2000 +3000 +4000 +5000 +observertime[days]18 +C. Nagele et al. +Figure 24. Same as Fig. 16, but for the fiducial model at redshift 0.3. +Figure 25. Same as Fig. 16, but for the fiducial model at redshift 0.5. +Figure 26. Same as Fig. 16, but for the fiducial model at redshift 1. +Figure 27. Same as Fig. 16, but for the fiducial model at redshift 3. +MNRAS 000, 1–16 (2022) + +17.5 +F070W +F090W +20.0 +F115W +JWST +22.5 +F150W +F200W +25.0 +F27XW +F356W +27.5 +F444W +30.0 +17.5 +VS +Y +20.0 +Euclid +22.5 +H +25.0 +27.5 +30.0 +17.5 +F062 +F087 +20.0 +F106 +ue +22.5 +F129 +Rom +F158 +25.0 +F184 +F146 +27.5 +F213 +30.0 +17.5 +20.0 +Rubin +22.5 +y +u +25.0 +6 +27.5 +30.0 +0 +1000 +2000 +3000 +4000 +5000 +observertime[days]17.5 +F070W +F090W +20.0 +F115W +JWST +F150W +22.5 +F200W +25.0 +F277W +F356W +27.5 +F444W +30.0 +17.5 +VS +Y +20.0 +Euclid +22.5 +H +25.0 +27.5 +30.0 +17.5 +F062 +F087 +20.0 +F106 +ue +22.5 +F129 +Rom +F158 +25.0 +F184 +N146 +27.5 +P213 +30.0 +17.5 +Z +r +20.0 +Rubin +22.5 +y +u +25.0 +b +27.5 +30.0 +0 +1000 +2000 +3000 +4000 +5000 +observertime[days]20.0 +F070W +22.5 +F090W +F115W +JWST +25.0 +F15QW +F200W +27.5 +F277W +30.0 +F356W +F444W +32.5 +20.0 +VS +22.5 +Y +Euclid +25.0 +H +27.5 +30.0 +32.5 +20.0 +F062 +22.5 +F087 +F106 +Roman +25.0 +F129 +F158 +27.5 +F184 +30.0 +F146 +F213 +32.5 +20.0 +22.5 +Rubin +25.0 +u +27.5 +6 +30.0 +32.5 +0 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +observertime[days]22.5 +F070W +F090W +25.0 +F115W +JWST +F150W +27.5 +F200W +30.0 +F277WL +F356W +32.5 +F444W +35.0 +22.5 +VIS +Y +25.0 +Euclid +27.5 +H +30.0 +32.5 +35.0 +22.5 +F062 +F087 +25.0 +F106 +Roman +27.5 +F129 +F158 +30.0 +F184 +F146 +32.5 +F213 +35.0 +22.5 +Z +25.0 +r +i +Rubin +27.5 +u +30.0 +6 +32.5 +35.0 +0 +2000 +4000 +6000 +8000 +10000 +observertime[days]CNO-rp driven GR instability supernovae. +19 +Figure 28. Same as Fig. 16, but for the fiducial model at redshift 5. +MNRAS 000, 1–16 (2022) + +22.5 +F070W +F090W +25.0 +F115W +JWST +27.5 +F150W +F200W +30.0 +F277W +F356W +32.5 +F444W +35.0 +22.5 +VIS +Y +25.0 +Euclid +27.5 +H +30.0 +32.5 +35.0 +22.5 +F062 +F087 +25.0 +F106 +Roman +27.5 +F129 +F158 +30.0 +F184 +F146 +32.5 +F213 +35.0 +22.5 +Z +r +25.0 +Rubin +27.5 +u +30.0 +6 +32.5 +35.0 +0 +2000 +4000 +6000 +8000 +10000 +observertime[days] \ No newline at end of file diff --git a/9NAzT4oBgHgl3EQf-_5O/content/tmp_files/load_file.txt b/9NAzT4oBgHgl3EQf-_5O/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..db6a0c2a4dde1a577a56c476400210bb240fea26 --- /dev/null +++ b/9NAzT4oBgHgl3EQf-_5O/content/tmp_files/load_file.txt @@ -0,0 +1,2063 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf,len=2062 +page_content='MNRAS 000, 1–16 (2022) Preprint 6 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 Light-curves and nucleosynthesis of CNO-rp driven general relativistic instability supernovae in metal enriched supermassive protostars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Chris Nagele,1★ Hideyuki Umeda,1 Koh Takahashi, 2 1Department of Astronomy, Graduate School of Science, the University of Tokyo, Tokyo, 113-0033, Japan 2Astronomical Institute, Graduate School of Science, Tohoku University, Sendai, 980-8578, Japan Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' in original form ZZZ ABSTRACT The assembly of supermassive black holes poses a challenge primarily because of observed quasars at high redshift, but additionally because of the current lack of observations of intermediate mass black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' One plausible scenario for creating supermassive black holes is direct collapse triggered by the merger of two gas rich galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This scenario allows the creation of supermassive stars with up to solar metallicity, where the enhanced metallicity is enabled by extremely rapid accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We investigate the behavior of metal enriched supermassive protostars which collapse due to the general relativistic radial instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' These stars are rich in both hydrogen and metals and thus may explode due to the CNO cycle (carbon-nitrogen-oxygen) and the rp process (rapid proton capture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We perform a suite of 1D general relativistic hydrodynamical simulations coupled to a 153 isotope nuclear network with the effects of neutrino cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We determine the mass and metallicity ranges for an explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We then post process using a 514 isotope network which captures the full rp process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We present nucleosynthesis and lightcurves for selected models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' These events are characterized by enhanced nitrogen, suppressed light elements (8 ≥ A ≥ 14), and low mass p nuclides and they are visible to JWST and other near infrared surveys as decades-long transients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Finally, we provide an estimate for the number of currently ongoing explosions in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Key words: gravitation — (stars:) supernovae: general — nuclear reactions, nucleosynthesis, abundances 1 INTRODUCTION The study of supermassive stars has arisen from two peculiarities of the black hole population in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The first is that supermas- sive black holes (SMBHs) exist soon after the big bang (Mortlock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Bañados et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Matsuoka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Our current understanding of cosmology requires that the SMBHs did not exist at the time of the big bang, implying that they were created in the intervening period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The second pecu- liarity is that the black hole mass function seems to be bimodal, with a noticeable lack of intermediate mass black holes (having masses in between solar mass black holes and SMBHs), although this may be due to observational bias (Wrobel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Baumgardt 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Kızıltan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' If the bimodality is not due to observational bias, it stands in opposition to the distributions of other self gravitating objects (stars and galaxies), which have smooth mass functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The direct collapse black hole (DCBH) scenario was proposed to resolve the first of these peculiarities (Bromm & Loeb 2003), and is sometimes invoked to explain the second (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Banik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The scenario involves a gas cloud forming a single supermassive star instead of many individual stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This can occur in the presence of local Lyman Werner radiation (Dijkstra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Latif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2014b) or baryon dark matter supersonic streaming (Latif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2014a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Schauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Hirano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The ★ E-mail: chrisnagele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='astro@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='com resultant supermassive star may be detectable directly (Surace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2018, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Vikaeus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2022), via a general relativistic instability supernova (GRSN, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Whalen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2013c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Moriya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2022b,a), by the obser- vation of gravitational waves (Shibata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2018), or as an ultra long gamma ray burst (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In this paper, however, we consider a slightly different scenario where the supermassive star formation is triggered by a merger of two gas rich galaxies (for a review, see Mayer & Bonoli 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The phenomenon of nuclear gaseous disks forming via multi-scale inflows was first investigated in the context of the M-sigma relation as a means of providing a source of dynamical friction for a SMBH binary in order to assist with its eventual merger (Kazantzidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Since then, it has been shown that not only can this disk influence the behavior of existing SMBHs, but it can also collapse under its own gravity to form a new black hole (Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Recently, Zwick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (2022) calculated observables from DCBHs resulting from galaxy mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The crucial element of this scenario as it pertains to the current study is that the scenario is agnostic to the metallicity of the interstellar medium (ISM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This means that supermassive stars will form out of metal enriched gas (Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' But what does it mean to have a metal enriched supermassive star?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' These objects radiate at very nearly their Eddington limit and mass loss rates from our local Universe (Vink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2011) suggest such objects would not survive for long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The situation is further compli- © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='01941v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='HE] 5 Jan 2023 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' cated by the fact that these metal enriched stars may be accreting matter to replenish that lost to line driven winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In this paper, we sidestep this difficulty by considering only metal enriched supermas- sive protostars which are massive enough to collapse via the general relativistic (GR) radial instability (Chandrasekhar 1964) before they reach the main sequence, thus avoiding any line driven mass loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The behavior of such metal enriched supermassive protostars has been investigated previously (Fuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Montero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In particular, it was shown that if the protostars collapse due to the GR radial instability, then this collapse can cause an explosion powered by the CNO cycle (which we will term a proton rich or pr-GRSN, to differentiate it from an 𝛼 process driven GRSN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' pr-GRSNe were first investigated in Fuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' They used a 1D post Newtonian (PN) code with a 10 isotope nuclear reaction network and found several exploding models spanning the mass range 5 × 105 − 106 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The metallicity floor for the lowest mass model was 5 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Subsequently, Montero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (2012) used a 2D BSSN code with parameterized heating rates to investigate models with similar mass, and they were able to include the effects of rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' For their non rotating models, they found explosions with the same masses as Fuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (1986), but with slightly higher metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We use a GR 1D hydrodynamics code coupled to a 153 isotope network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The large network allows us to more accurately follow the dynamics of the explosion at higher temperatures, and we thus find a lower metallicity floor than in previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' After running our simulations, we post process the hydrodynamical trajectories with a 514 isotope network designed to fully follow the rp-process on the proton rich side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Contrary to the conclusions of previous works, we find that the rp-process can play a critical role in the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2 we outline our numerical procedures for stellar evolu- tion, hydrodynamics, post processing, and lightcurves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1, we present the results of the stellar evolution simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2, we present the results of our hydrodynamical simulations and post processing, for a fiducial model, as well as for varying mass and metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3 we present the results of our lightcurve calcu- lations and an estimate of pr-GRSN density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='4, we discuss various feedback induced by a pr-GRSN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Finally, we conclude with a discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2 METHODS In this section, we first describe our initial models and stellar evo- lution code, then provide details of our GR hydrodynamical code, after which we detail the open source code SNEC, which is used to calculate lightcurves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1 Stellar evolution The HOSHI code (Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2016, 2018, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Yoshida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2019) is a 1D stellar evolution code which solves the stellar structure and hydrodynamical equations using a Henyey type implicit method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (2020) introduced the first order PN correction to the hydrostatic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The PN approximation is extremely accurate for SMSs in hydrostatic equilibrium because the effects of GR are mi- nor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' These minor effects must be included, however, because SMSs are radiation dominated and therefore close to instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Once the evolution of the star becomes dynamical, HOSHI’s lack of a shock capture scheme and the PN dynamical corrections neccesitate the use of another code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' HOSHI includes a nuclear reaction network (52 isotopes), neutrino cooling, mass loss, and rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The equation of state includes contributions from photons, averaged nuclei, elec- trons, and positrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' HOSHI uses the Rosseland mean opacity of the OPAL project (Iglesias & Rogers 1996) and solves the Saha equation to determine the ionization of hydrogen, helium, carbon, nitrogen, and oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In this paper, 𝑀 is the total mass, 𝑅 the radius, 𝑇 the temperature, and 𝜌𝑏 the baryonic density where quantities with 𝑐 subscripts show- ing the central values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 𝑠𝑟 is the entropy due to radiation at a given mass (Shapiro & Teukolsky 1983) 𝑠𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='942 � 𝑀 M⊙ �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (1) Finally, X is the mass fraction of a specified element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' To assist with analysis, we define various global energy quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The internal energy is 𝐸int = ∫ 𝑀 0 𝜖 𝑑𝑚𝑟, (2) where 𝑚𝑟 is the mass coordinate and 𝜖 is the specific energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The gravitational energy is 𝐸grav = − ∫ 𝑀 0 𝑔effective 𝑟 𝑑𝑚𝑟, (3) where 𝑔effective is the local gravity with the 1st order PN correction to the static terms (Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The accuracy of this ap- proximation degrades with increasing density and velocity, neither of which are particularly concerning for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The kinetic energy is 𝐸kin = ∫ 𝑀 0 𝑣2 2 𝑑𝑚𝑟, (4) where 𝑣 is the radial velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The binding energy of the star is the negative of the thermal and gravitational energies (so that a more tightly bound star has higher 𝐸bind), while the total energy additionally includes kinetic energy: 𝐸bind = −(𝐸int + 𝐸grav) (5) 𝐸tot = 𝐸int + 𝐸grav + 𝐸kin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (6) As in our previous works, we define the explosion energy as the total energy at shock breakout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' For HYDnuc, we also report the integration over energy generation due to the nuclear network and neutrino cooling (dots indicate time derivatives): 𝐸nuc(𝑡) = ∫ 𝑡 0 ∫ 𝑀 0 � 𝜖nuc 𝑑𝑚𝑟 𝑑𝑡 (7) 𝐸𝜈(𝑡) = ∫ 𝑡 0 ∫ 𝑀 0 �𝜖𝜈 𝑑𝑚𝑟 𝑑𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (8) A summary of the results of the HOSHI simulations can be found in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We initiate the HOSHI code in a high entropy state, which imme- diately relaxes towards a constant entropy radiation dominated state, though the protostar does not reach all the way to the radiation value due to the small contribution of gas pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This configuration could represent one of two physically realizable scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Either, it could be a supermassive protostar which has finished accreting due to depletion of accretion material, which then contracts to the radiation dominated constant entropy pre-ZAMS (zero age main se- quence) state, or it could represent the convective core of a currently MNRAS 000, 1–16 (2022) CNO-rp driven GR instability supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Summary table for the nuclear networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Entries show the range in A for the specified element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Element 52 153 216 514 Element (ctd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='153 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='216 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='514 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='accreting supermassive star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In the latter case, the accretion envelope could in theory effect the stability of the system, but such effects are thought to be small (Haemmerlé 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We expect that the envelope will have no effect on the dynamical behavior of the protostar once the GR instability is reached besides increasing the overall gravity (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 14 of Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We determine the stability of the protostar in HOSHI by solving the pulsation equation for a hydrostatic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' spherically symmetric object in general relativity (Chandrasekhar 1964): 𝑒−2𝑎−𝑏 d d𝑟 � 𝑒3𝑎+𝑏Γ1 𝑃 𝑟2 d d𝑟 (𝑒−𝑎𝑟2𝜉) � −4 𝑟 d𝑃 d𝑟 𝜉+𝑒−2𝑎+2𝑏𝜔2(𝑃+𝜌𝑐2)𝜉 −8𝜋𝐺 𝑐4 𝑒2𝑏𝑃(𝑃 + 𝜌𝑐2)𝜉 − 1 𝑃 + 𝜌𝑐2 � d𝑃 d𝑟 �2 𝜉 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (9) where 𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 𝑏 are the metric coefficients as defined in Haemmerlé (2021a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 𝑟 is the radius,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 𝑃 the pressure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Γ1 the local adiabatic in- dex at constant entropy (𝑠),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 𝜌 = 𝜌𝑏(1 + 𝜖) the relativistic density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' and 𝜖 the specific internal energy (we absorb rest mass due to mass excess of nuclei into this energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The star, or in this case, protostar, is unstable if there exists a trial function 𝜉(𝑟) ∝ 𝑒𝑖𝜔𝑡 with 𝜔2 < 0, representing a perturbation which will grow exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' There are two main approaches to solving this equation, either by assuming a nearly linear trial function 𝜉 ∝ 𝑟𝑒𝑎 (Haemmerlé 2021a), or by iteratively solving for the fundamental mode of the normal mode decomposition of perturbations to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 9 (Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Here we adopt the latter approach, as in our previous paper, but this choice should have a minimal bearing on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Even though we will eventually simulate the dynamics of metal enriched supermassive protostars, we only consider metal free proto- stars in HOSHI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This is because we are only interested in protostars which become unstable before the onset of nuclear burning, so metal- licity, will not effect the protostellar structure except as caused by Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Radial (upper) and velocity (lower) snapshots of the fiducial model at three timesteps, the initial time, the time when the central temperature is largest, and shock breakout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' changes to the opacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We consider the following range of masses: 7 − 10 × 104 M⊙ in steps of 104 M⊙ and 1 − 3 × 105 M⊙ in steps of 5 × 104 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 Hydrodynamics HYDnuc is a 1D Lagrangian GR hydrodynamics code which uses a Roe-type approximate linearized Riemann solver (Yamada 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' It includes all of the physics from HOSHI except for convection and ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In this paper, we use a 153 isotope network (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The 153 isotope MNRAS 000, 1–16 (2022) le13 initial 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 max T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' shock breakout r [cm] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 6 4 2 0 2 0 20000 40000 60000 80000 100000 mr[M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=']4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Nucleosynthesis in the post processed 514 isotope network for the fiducial model, 𝑀 = 105 M⊙, 𝑍 = 𝑍⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Upper panels — isotope mass fractions for five snapshots: the initial time, when the temperature rises to 3/4 of the eventual maximum, the maximum temperature, the final time step, and 1012 seconds later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Lower panel — total energy compared to 𝐸nuc in both the 153 and 514 isotope calculations and compared to 𝐸nuc + 𝐸𝜈 for the 153 isotope calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' network (Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2018) covers the proton rich side (p side) at a depth of 3-8 isotopes up to zinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Reaction rates for all networks are taken from JINA REACLIB (Cyburt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We use the same scheme to transport our models from HOSHI to HYDnuc as in Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (2022b) which is based on the frequency function defined in Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We set the chemical composition to be constant throughout the star, as a fraction of the solar composition (Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Due to the exploratory nature of this study and the requirement of a larger nuclear network, we use slightly less optimal numerical parameters than in Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (2022b), specifically 255 mesh points and V = 10−4 (maximum allowed fractional variation of independent variables per timestep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The effect of these changes is to underestimate the energy generated by nuclear burning (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 6 of Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' As in our previous works, we terminate the simulations when convergence issues arise due to large radius (𝑟 ∼ 1015 cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3 Post-processing After performing the HYDnuc simulations, we post process the hy- drodynamical trajectories using a 514 isotope network (Table 1) designed to follow the rp process up to ruthenium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' All isotopes on the p side are covered up to the line with slope 1 and y intercept 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Even though this post processing is less computationally expensive, the network is too large to solve the composition at every timestep of HYDnuc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We choose to solve the composition with a frequency of 100−1 timesteps−1, and have checked that a) the convergence of 𝐸nuc as the frequency increases and b) that 𝐸nuc with frequency 100−1 agrees with 𝐸nuc with frequency 10−1 to within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' At the end of the HYDnuc simulation, the post-processed composition contains many radioactive isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We then fix the temperature and density and continue to post-process for an additional 1012 seconds while logarithmically increasing the timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 1012 seconds is enough for most, but not all (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 26Al) radioactive isotopes to decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='4 Lightcurves The SuperNova Explosion Code (SNEC) is an open source, 1D La- grangian, radiation hydrodynamics code designed to compute su- pernova lightcurves (Morozova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' It includes artificial Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Nucleosynthetic yields (𝑡 = 𝑡final + 1012 s) from the 514 isotope network for the 𝑀 = 105 M⊙ for the indicated metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' viscosity, an equation of state with a Saha solver for ionization of hydrogen and helium, and equilibrium flux-limited photon diffusion using OPAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' As in (Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2022a), we port our HYDnuc mod- els to SNEC slightly before shock breakout and for the outer layer of the SNEC model, we use the HOSHI progenitor which enables increased surface resolution necessary for properly following the lightcurve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We terminate the SNEC simulations after 109 seconds at which point any plateau phase in the luminosity has finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We then use the effective temperature to construct blackbody spec- tral energy distributions and assume a standard ΛCDM cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' With these components, we calculate apparent magnitudes for the passbands of various telescopes at a given redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In this study, we do not take extinction into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' MNRAS 000, 1–16 (2022) t= tinitial Tc= gTc,max Te= Tc,max t = tfinal t= trinal + 1012 s 0 20 2 pnumber 4 15 Log 6 10 8 10 15 20 25 10 15 20 25 10 15 20 25 10 15 20 25 10 15 20 25 10 nnumber 1e55 2 Etot (153 isotopes) Etot(t = 0) + Enuc (153) Etot(t = 0) + Enuc + Ev (153) Etot(t = 0) + Enuc (514 isotopes) 0 0 20000 40000 60000 80000 100000 t [s]Z/Z。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='12254 4 Z/Z。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1226 Z/Z。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='123 Z/Z。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='13 Sc 3 Z/Z。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 Z/Z。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 2 [X/Fe] N Mo Ne 0 1 C Ar Co 2 Mg 5 10 15 20 25 30 35 40 45 Z (proton number)CNO-rp driven GR instability supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 5 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Summary of the state of the stellar evolution simulations at the GR instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The columns are, total mass, radius, central temperature, baryonic density and entropy, central entropy as a fraction of radiative entropy (for a star of that mass), hydrogen mass fraction and binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' M [105 M⊙] R [1013 cm] 𝑇𝑐 [108 K] 𝜌𝑏,𝑐 [g/cc] 𝑠𝑐 [𝑘𝑏/baryon] 𝑠𝑐/𝑠𝑟 − 1 X(1H) 𝐸bind [1054 ergs] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='7 264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='554 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='881 265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='06454 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='4979 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='694 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='9 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='652 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='7599 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='765 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Monotonic dependence of maximum temperature on metallicity for the 𝑀 = 105 M⊙ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Central temperature evolution of the 𝑀 = 105 M⊙ for the indicated metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' These trajectories were used for the post processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3 RESULTS In this section, we first describe the results of the HOSHI code, after which we provide details of the nucleosynthesis before moving on to a discussion of the lightcurves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1 Stellar Evolution The lowest mass model which we consider in this study (7×104 M⊙) becomes unstable only after entering the ZAMS phase, with the instability occurring after roughly half a million years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This model has a large radius and low proton mass fraction at the onset of instability (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Because of its relatively long lifetime, enhanced metallicity would likely have caused significant mass loss, thus prolonging the period before the GR instability occurred (less mass means less gravity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Then, there is the effect of accretion to consider, which, if present, may be able to restore some of the lost mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We do not posses the proper tools to model these physical processes and so we do not perform any hydrodynamical simulations with this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' On the other hand, models with mass of 105 M⊙ or greater (9 × 104 M⊙ is a marginal case, which we include), collapse due to the GR radial instability before any nuclear burning has taken place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' For increasing mass, these models have higher entropy, and are getting more and more radiation dominated (Table 2, 6th column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This corresponds to a decrease in the central temperature and density as well as an increase in the binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The upper limit for stability in our study (7 × 104 M⊙) is slightly lower than that found in Fuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (1986) (105 M⊙), which is likely due to small deviation from the GR pressure gradient in the PN code which they employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Indeed, as was shown in Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (2022b), the use of the baryonic density instead of the relativistic density in the PN correction changes the stellar structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Our results are more massive than the cores of the accreting SMSs in Haemmerlé (2020) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 8), which is to be expected because we do not consider the gravity of the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' It should be noted that this is not a completely faithful comparison because the accreting SMSs will begin nuclear burning before they collapse due to GR (Hosokawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Umeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Haemmerlé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' For the high accretion rates implied by the galaxy merger formation scenario, however, there is little chance that the hydrogen reservoir will be depleted before the GR instability, in which case the pr-GRSN would still occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' As we will show, the less massive the progenitor, the easier it is for the star to explode (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' MNRAS 000, 1–16 (2022) 1e8 7 6 max T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' [K] 5 4 3 10~5 10-4 10-3 10-2 10-1 100 Z/Zo-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='122531e8 8 Z/Z。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='12254 7 Z/Z。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1226 Z/Z。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='123 Z/Z。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='13 6 Z/Z。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 Z/Z。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 5 3 2 1 0 20000 40000 60000 80000 100000 time [s]6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2 but for 𝑀 = 105 M⊙, 𝑍 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='12254 𝑍⊙, which is the lowest metallicity explosion for 𝑀 = 105 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Upper panel — same as upper panel Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 6 but showing the final composition at different meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Lower panel — mean molecular weight 𝜇 as a function of mass coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 Nucleosynthesis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1 Fiducial model We take the M=105 M⊙, 𝑍 = Z⊙ as our fiducial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This mass is near the lower end of our mass range and the metallicity is well above the threshold required for an explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In the hydrodynamical simulation, the star begins in a high entropy, but relatively compact state, due to the star not yet having settled into the main sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' At the start of the simulation, the CNO cycle immediately turns on, but is not strong enough at those temperatures (𝑇𝑐 ≈ 108 K, Fig 5) to arrest the collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The star continues to collapse for 50000 seconds before reaching a temperature of 𝑇𝑐 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3 × 108 K, at which point the CNO cycle produces enough energy to reverse the motion of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The CNO cycle continues to produce energy, even as the temperature drops below 𝑇𝑐 ≈ 108 K, but the bulk of the energy production occurs with the central temperature 𝑇𝑐 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 × 108 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The star is completely unbound and explodes with energy 𝐸exp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='81 × 1055 [ergs].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' It’s final radial and velocity profiles as a function of mass coordinate can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' After finishing the HYDnuc simulation, we post process the sim- ulation results as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3 with a 514 isotope network designed to follow the rp process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2 shows the isotope mass frac- tion of the entire star in the 514 network at several time snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In the fiducial model, the energy generation comes almost entirely from the CNO cycle, but proton captures on light elements do occur near the maximum temperature (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2) and this can be seen in the nucle- osynthetic yields (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The yields for the fiducial model (Z=Z⊙) are characterized by two features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The first is enhanced nitrogen and suppressed carbon and oxygen due to the CNO cycle, which produces roughly equal amounts of its eponymous elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The second is a broad exchange of light elements (flourine, sodium, magnesium) for slightly heavier elements (aluminium through chlorine) due to mul- tiple proton captures which then decay back to stability at higher mass number than they originated (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The only exception to this trend appears to be neon which experiences proton captures, but is replenished from below by oxygen exiting the CNO cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 Metallicity dependence Next we consider the behavior of the M=105 M⊙ model as we vary the metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' For each mass, there exists a threshold metallicity value below which there are not enough seed metals to fuel an ex- plosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' As one approaches this threshold, that is as one decreases the metallicity, the model needs to reach higher temperatures before the collapse can be reversed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In addition, the time at which the model reaches the maximum temperature is pushed further back with decreasing metal content (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 6 shows the isotope mass fraction for several time snapshots, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2, but for Z= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='12254 Z⊙ which is the metallicity closest to the explosion threshold (we will refer to this as the metal-poor MNRAS 000, 1–16 (2022) t= tinitial Te =gTc,max Te = Tc,max t= trinal t = tinal + 1012 s 40 4 6 Log 8 X 10 10 12 10 20 30 40 50 20 30 40 50 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 nnumber le55 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 Etot (153isotopes) Etot(t = 0) + Enuc (153) Etot(t = 0) + Enuc + Ev (153) Etot(t = 0) + Enuc (514 isotopes) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 40000 50000 60000 70000 80000 90000 100000 t [s]-2 40 4 number 30 6 Log 20 8 + 10 10 12 10 20 30 40 50 10 20 30 40 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 nnumber 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='62 0 20000 40000 60000 80000 100000 mr[M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=']CNO-rp driven GR instability supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 7 model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We emphasize that this fine tuning is not done in order to find the precise value of the threshold, but rather to demonstrate that near the threshold, temperatures high enough to trigger the rp process can be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Indeed, panels 2 and 3 of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 6 show extreme levels of proton captures extending to technetium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Note that in the HYDnuc simulations with the 153 isotope network (as opposed to the post processing with 514), the composition cannot go higher than zinc (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In the post processing, the heavy p-side elements then decay back to stability at much higher mass number than they originated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 7 shows the final isotopic mass fraction distribution at different meshes throughout the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Unlike in the 𝛼 process driven GRSN, most of the star undergoes nuclear burning and this is one of the reasons that these explosions can be so energetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3 shows the final abundances (relative to iron, relative to solar values Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2009) from the 514 isotope network for selected metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' As was the case in the fiducial model (Z⊙), there are two main characteristics, with the first being enhanced nitrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The second characteristic is a bulk transport of light elements to heavy elements, with lower metallicities experiencing a larger and wider transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Note that oxygen is part of this transport, whereas carbon and nitrogen do not vary significantly with metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' For Z= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 Z⊙, the transport extends up to scandium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' For Z= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='13 Z⊙, iron peak elements are produced, terminating around gallium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' For the three smallest metallicities, the transport extends well past iron, up to molybdenum in the extreme case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' For the lower metallicity cases, significant contributions to selected element are in the form of low mass p-nuclides, which reside away from the line of stability (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 6, 5th panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Below Z= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 Z⊙, all models show a peak at scandium, with Z= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='123 Z⊙ peaking again at proton number 31-33 and the two lowest metallicity models peaking at proton number 31-36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Another signature of all models is high Cl/Ar and low Ar/K, both resulting from the bulk transport not necessarily preferring even elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In the high temperature models, cobalt is suppressed because its lightest stable isotope (59Co) cannot be reached from the p side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3 Mass dependence Next, we vary the mass at solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' As the mass increases, the binding energy of the star increases, and more nuclear energy is required to unbind it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This means that the higher mass models will reach higher temperatures until eventually (M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 × 105) they can no longer explode at solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Higher temperatures, in turn, mean that the yields from the 514 isotope post processing should show more heavier elements, and this is demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We have already discussed 105 M⊙, and 9 × 104 M⊙ is nearly identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 × 105 M⊙ model shows the same bulk transport described in the previous section, but this time extending up to titanium, while the 2 × 105 M⊙ model (we will refer to this as the massive model) peaks around titanium, but reaches as high as gallium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 9 shows the explosion energy in HYDnuc (153 isotopes) as a function of mass and metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The explosion energy depends only on mass, as it is a fixed fraction of the star’s binding energy, unless the model is sufficiently close to the explosion threshold, as can be seen with M= 2 × 105 M⊙, Z= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='9 Z⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' On the other hand, the metallicity threshold increases with mass for the reasons stated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We have attempted to run additional simulations using a 216 iso- tope network (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' However, there is a substantial increase in computational cost (3469 reactions compared with 2463 for the 153 isotope network) and the network is still not large enough to follow the rp process to higher mass (Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' For a rough estimate of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3, but for different masses at solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Dependence of explosion energy (denoted by color) on mass and metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The black crosses are models which failed to explode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The gray region roughly corresponds to non-explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' the energy generation rate of the rp process beyond what we investi- gate in this paper, see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3 Lightcurves Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 10 shows the time evolution of photosphere radius, luminosity and effective temperature for each of the three named models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In addition, we have rerun the massive model with a much greater en- velope resolution in order to more accurately characterize the shock breakout (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 11), though this increased resolution presents compu- tational challenges during the plateau phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The shock breakout is accompanied by an extremely luminous (1049 ergs/s) burst with high effective temperature, which is potentially observable as an X-ray outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' To date, the one observed X-ray outburst associated with shock breakout (Soderberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2008) had an X-ray luminosity six orders of magnitude lower (1043 ergs/s), implying that shock break- out of a pr-GRSN would be visible, even at high redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' However, the short duration of the event and low rate of pr-GRSNe presents serious challenges to realizing an observation of this shock breakout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Besides the high luminosity and temperature at shock breakout, MNRAS 000, 1–16 (2022) Sc Z/Z。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='= 9e4 3 V Z/Z=1e5 Z/Z。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5e5 Z/Z。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='=2e5 2 [X/Fe] 1 N Ne 0 C 1 Mg 5 10 15 20 25 30 35 40 45 Z (proton number)AA 1e56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='8 X 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='85 X [ergs N X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='4 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='4 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 100000 150000 200000 250000 M[M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=']8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Results of the SNEC simulations for the fiducial model, metal-poor model, and the massive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Upper panel — photosphere radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Middle panel — bolometric luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Lower panel — effective temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The horizontal axis has been normalized so that shock breakout occurs at 10−3 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' there is another difference between the lightcurves of the pr-GRSNe and the lightcurves of standard GRSNe (𝛼 process, Moriya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This is the longer duration of the plateau phase which follows hydrogen recombination, nearly an order of magnitude longer than our previous GRSN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This longer du- ration may be due to the increased mass or the increased energy in comparison with previous GRSN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' During this plateau phase, the hierarchy of the luminosities follows that of the associated ex- plosion energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Throughout this phase, the effective temperature steadily falls and the photosphere radius steadily rises, the latter of which is slightly different to the stalled photosphere found in standard GRSNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This effect can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 12 which shows the time evolution of JWST NIRCAM wideband filters for the three named models at redshift five, which is consistent with both ZISM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1 Z⊙ and ZISM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 Z⊙ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Pallottini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The prompt emission is visible in the bluer filters because of the higher effective temperature, but as this quantity drops, they rapidly fall away and only the four reddest filters remain during the plateau phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1 Energy input from radioactive decays When calculating lightcurves in SNEC for the metal-poor model, there is an additional consideration, namely that the star is undergo- ing a significant amount of radioactive decays after the end of the HYDnuc calculation (see panels 4 and 5 of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 13, we Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 10, but showing shock breakout for the massive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The simulation in this figure has higher surface resolution than those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The horizontal axis has been normalized so that the peak luminosity occurs at 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' show the rate of change in the nuclear energy as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The vertical lines separate regions which are powered by different decays, and these regions are labeled by the mother nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Swells in the heating rate can be seen for the decay of 56Ni and 57Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Despite the staggering amount of energy produced by these radioactive de- cays (compare to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 10), we do not think that they will affect the lightcurve for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The first is that these decays are occurring deep within the star and are well inside the photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Later on, it is conceivable that they could contribute, but we have tested run- ning SNEC with 56Ni decays turned on and there is no difference in the lightcurve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The second reason is that the total amount of energy produced by these decays (∼ 1053 ergs) is order one percent of the explosion energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 Multiband lightcurves at different redshifts In this section, we show the observer time evolution of the four reddest JWST NIRCAM widebands at a variety of redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Because the galaxy merger scenario does not depend strongly on redshift (there is a decrease below redshift two, but only by an order of magnitude, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 4 of Bonoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2014), we show redshifts down to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 14 shows this for the fiducial model up to redshift ten, above which we suspect solar metallicity conditions would be challenging (though not impossible, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3 of Hartwig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2022) to realize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 15 is a similar figure, but for the metal poor model, which we show up to redshift 19, though it would not be detectable by JWST much above redshift 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Appendix C shows multiband lightcurves of the MNRAS 000, 1–16 (2022) Photo-sphere radius [cm] 1018 fiducialmodel metal-poor model 1017 massivemodel 1016 1015 1014 1047 I [ergs/s] 1046 1045 Lbol 1044 1043 1042 106 105 104 103 102 10-4 10~3 10-2 10-1 100 101 102 103 t [days]cm] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='795 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='790 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='785 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='780 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='775 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='770 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 2:9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 got] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 2 0 2 4 6 8 10 t [s]CNO-rp driven GR instability supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 9 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Rest frame time dependence of JWST NIRCAM magnitudes at 𝑧 = 5 for the fiducial model (upper panel), metal-poor model (middle panel) and massive model (lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The horizontal dotted line shows a typical limiting magnitude for JWST (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Rate of energy produced by radioactive decays in the metal-poor model after the end of the HYDnuc simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Vertical dotted lines separate regions where the energy production is dominated by different decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The mother nuclei of those decays label each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' fiducial, metal poor, and massive models for JWST, Euclid, Roman, and Rubin at various redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Comparing Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 14,15, it is clear that from the point of view of observation, more metal enrichment is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Not only is the mass range wider at higher metallicity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 9), but also the GRSN would have occurred at lower redshift, making it easier to observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3 With hylotropic envelope Up to this point, we have not considered the effect that an accretion envelope (Hosokawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Umeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Haemmerlé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2019) would have on the lightcurve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' While the core of an accreting supermassive protostar has constant entropy, the envelope is thought to have entropy increasing as a power law Begelman (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Haemmerlé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Haemmerlé (2020, 2021b) and this structure has been termed a hylotrope (Gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' hyle, ‘matter’ + tropos, ‘turn’) in Begelman (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Specifically, hylotropes obey the equation of state 𝑃 = 𝐴𝜌4/3𝑀 𝛼 (10) where 𝛼 is often taken to be 2/3 as derived from the homology scalings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In this scenario, we instead take 𝛼 as a parameter and enforce the mass radius relation of rapidly accreting supermassive protostars (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 1 of Hosokawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This choice is motivated by the sensitivity of lightcurves to the stellar radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We construct the hylotropic envelope in a similar manner to the integration of an 𝑛 = 3 polytrope, as described in Haemmerlé (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We find that, taking the fiducial model as the core, 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='90481 satisfies the mass radius relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This result is not sensitive to the matching radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This hylotropic model has 𝑀𝐻 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='58 × 105 M⊙ , 𝑅𝐻 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='23 × 1016 [cm], so that 𝑀𝐻 /𝑀 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='58 and 𝑅𝐻 /𝑅 = 653.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This result resembles numerical models of accreting supermassive protostars, although self consistent simulations are needed to verify the results of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We run SNEC by attaching the final timestep of the HYDnuc sim- ulation to the hylotropic envelope which was constructed using the initial profile of the HYDnuc simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This results in a discon- tinuity in density, but such a feature is not unexpected around this radius (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 14 of Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We use the thermal bomb mode with the final explosion energy of the hylotropic model being the explosion energy of the fiducial model, as the added gravitational energy of the envelope could not be overcome otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This ap- proach is not self consistent as it requires the injection of an additional ∼ 1055 ergs, but larger explosion energies (1056 ergs) are achieved in the massive model, so our strategy is not unreasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The pho- tosphere remains at the surface of the hylotrope for ten days as the shock propagates through the envelope, after which the photosphere expands and the effective temperature drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' From a peak value of 𝐿bol = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='86 × 1046 [ergs/s], the luminosity then falls nearly mono- tonically (except for a slight increase at hydrogen recombination) in behavior reminiscent of the pulsations of large radius supermassive stars in Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The luminosity of the hylotropic model falls at a slower rate than for the fiducial model, passing 𝐿bol = 1043 [ergs/s] only after 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='95 years in the rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 16 shows the magnitudes of the hylotropic model at redshift 1 for JWST, Euclid, Roman, and Rubin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='4 Rate estimate We now turn to the question of how frequently these pr-GRSN occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Bonoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (2014) calculated the number density of massive galaxy MNRAS 000, 1–16 (2022) AB mag at z = 5 24 F070W F150W F356W F090W F200W F444W fiducial model 26 F115W F277W 28 30 32 24 metal-poor model 26 28 30 32 24 model 26 massive 28 30 32 10~3 10~2 10-1 100 101 102 103 t [days]52Mn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 1039 106 107 108 109 1010 1011 1012 t [s]10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Redshift dependence of the four reddest JWST NIRCAM wide bands for the fiducial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The horizontal axis shows the observer time.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='observertime[days] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='observertime[days]CNO-rp driven GR instability supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 11 Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Magnitudes of the hylotropic model in JWST (first panel), Euclid (second panel), Roman (third panel), and Rubin (fourth panel) at redshift 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The horizontal axis is observer time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' mergers (𝑀halo > 1011 M⊙) which fulfilled the criteria for merger induced direct collapse (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' For a simple order of magnitude estimation, we will assume a merger rate of 𝜙 = 10−4 [cMpc−3 Gyr−1] which is fairly conservative (note that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 4 of Bonoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (2014) has units of [cMpc−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1 Gyr−1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' As discussed in Bonoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (2014), relaxing the mass asymmetry condition further would increase the merger rate by an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Another increase could be had by relaxing the mass constraint, although the minimum mass at which the merger scenario creates a direct collapse object is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' For a more recent, empirical, estimate of the galaxy merger rate, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3 of O’Leary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' To determine the number of supermassive protostars produced per unit time by the galaxy merger scenario, we simply multiply the rate 𝜙 by the volume between redshifts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1 and 10 (𝑉𝑧∈(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1,10)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 𝑁protostars = 𝜙𝑉𝑧∈(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1,10) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3 yr−1 (11) The lower bound of this range was chosen because very nearby GRSN will be visible to many instruments besides the four discussed in this paper, while the upper bound is nearing the observing threshold for JWST (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Next we introduce 𝑓 , the fraction of protostars which explode as pr-GRSN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We have no ability to estimate the initial mass function of these objects and thus no ability to estimate 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' However, it is reasonable to assume that the mass function decreases for increasing mass, as does an estimate of the mass function of supermassive stars (Toyouchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' If this is the case, then 𝑓 will depend heavily on the behavior of the protostars less massive than those considered in this paper (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' As we have mentioned previously, however, it is feasible that many of these less massive protostars will also explode, so an 𝑓 of order unity is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 𝑁pr−GRSN = 𝑓 𝜙𝑉𝑧∈(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1,10) yr−1 (12) From the rate of pr-GRSN, we would then like to obtain a number of currently ongoing pr-GRSN in the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' To do this, we need to know the expected observer duration for the pr-GRSN (⟨𝑡obs⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' To do this we perform a Monte Carlo simulation, randomly selecting lookback times between redshifts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1 and 10 and computing the observer time during which the GRSN is visible to the JWST F444W band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Averaging the Monte Carlo draws results in ⟨𝑡obs⟩ = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 yr (13) so that the expected number of currently observable pr-GRSN is 𝑁observable = 𝑓 𝜙𝑉𝑧∈(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1,10) ⟨𝑡obs⟩ ≈ 9f (14) and we plot this quantity as a function of 𝑓 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 17, indicating that it is reasonable to expect a few GRSNe to be observable right now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' On the one hand, this is a small number because JWST only covers a tiny section of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In addition, the true value of 𝑓 may be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' However, we point out that the majority of the Monte Carlo draws occur in the low redshift Universe where gas should be sufficiently enriched to trigger an explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Thus, the pertinent question be- comes what is the mass function of the supermassive protostars?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' On the other hand, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 17 is showing a large number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Bonoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (2014) and Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (2015) are investigating the formation of the most massive SMBHs in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This investigation is well founded upon observations of high redshift quasars with inferred masses in excess of 108 M⊙, but it does not consider the vast majority of the black hole population, which have smaller masses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Inayoshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' For this reason, the mass and mass ratio constraints which are applied in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 4 of Bonoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (2014) could be far too stringent, in which case the population of direct collapse objects, and GRSNe, would be much greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In summary, massive uncertainties exists regarding the frequency of this scenario, but we have shown that it is at least plausible to expect pr-GRSNe to be observable in the low redshift Universe, if the galaxy merger scenario is the dominant source of supermassive black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 Observing strategy Because of the long duration of the pr-GRSNe, identifying them as transients may not be straightforward, although detection by multi- ple instruments would ameliorate this difficulty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' as well as JWST, Appendix C shows the magnitudes of Euclid, Roman, and Rubin at various redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In the rest of this section, however, we will consider only the four reddest JWST NIRCAM widebands (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We divide the lightcurve up into phases, a rising phase, a falling phase, and a plateau phase in between the rising and falling phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We define the rising phase as lasting until peak magnitude is reached, which occurs at different times in each band (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 12) and the falling phase as beginning when the photosphere radius jumps (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' the falling phase will not be visible in all bands, especially at high redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Observations of the rising or falling phases will be easily identifiable as transients (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 14), but the plateau phase, while not having constant magnitude, will dim at a rate of the order of a few magnitudes or less per decade (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This rate of dimming is roughly constant as the duration of the plateau phase increases (for MNRAS 000, 1–16 (2022) 16 F070W 18 F090W F115W JWST 20 F150W F200W 22 F277W F356W F444W 24 16 VIS 18 Y Euclid 20 H 22 24 16 F062 18 F087 F106 Roman 20 F129 F158 22 F184 F146 24 F213 16 Z 18 Rubin 20 y u 22 6 24 2500 5000 7500 10000 12500 15000 17500 20000 observertime[days]12 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Various quantities related to chemical, mechanical, and radiative feedback for the post-processed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The columns are mass, metallicity, the change in metal content, the change in iron, the change in mean molecular weight, the kinetic energy (HYDnuc), the fade radius (at which the shock becomes subsonic), the radiated energy (SNEC), the number of ionizing photons, and the number of Lyman Werner photons, where the last two assume blackbody emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' M [105 M⊙] 𝑍/𝑍⊙ 𝑀𝑍 [M⊙] 𝑀Fe [M⊙] Δ𝜇 𝐸kin [1055 ergs] 𝑅fade [Mpc] 𝐸rad [1052 ergs] 𝑁𝛾,ion [1054] 𝑁𝛾,LW [1054] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='26 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='485 209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='6 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='23 Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Expected number of concurrent GRSNe observable by JWST as a function of the fraction ( 𝑓 ) of supermassive protostars which explode as pr-GRSN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' higher redshifts), until the photosphere radius jump is no longer visible (above 29 magnitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' For these fainter, high redshift, sources, the plateau phase is shorter and the magnitude changes more quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 19 shows color-color diagrams for the four reddest JWST NIRCAM filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' All of the data points, including different models and redshifts, fall neatly along a single line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The results of a linear fit to this line are shown in each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Based on these features, we propose that pr-GRSNe candidates are identified as NIRCAM sources falling within at least the three reddest bands which are consistent with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' These candidates may then be confirmed or ruled out with long cadence observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' GRSNe which have already dropped out of F270W and F2777W will not be identifiable by their color, and can only be identified with repeated observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' On the one hand, the long durations and shallow slopes of these lightcurves pose a challenge to observing these events as transients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' On the other hand, if they are identified as such, their further clas- sification as GRSNe will be relatively unambiguous, as there exist no other events one hundred thousand times more energetic than a supernova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Rate of dimming during the plateau phase as a function of plateau phase duration for the four reddest NIRCAM bands (different panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The colors denote redshifts and are shown in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='25, while the symbols show different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='4 Feedback The pr-GRSN likely would have been extremely disruptive to its host halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Previously, several studies were conducted on the effect a standard GRSN would have on the halo, specifically on metal enrichment, gas evacuation, and star formation (Whalen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2013a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Whalen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2013b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The pr-GRSN is more than an order of magnitude more energetic than in the standard case, and MNRAS 000, 1–16 (2022) 8 S 4 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 ffiducial Z= 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='4 massive Z=2 metalpoor Z=3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3 Z = 4 F200W z = 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 Z=6 Z = 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1 z=9 z = 10 Plateauphase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3 F277W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3 F356W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 0 10 20 30 40 50 Plateauphaseduration[years]CNO-rp driven GR instability supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 13 Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Two color-color diagrams for the four reddest NIRCAM wide- bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 18, colors denote redshift and symbols denote models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Also shown is a linear fit (dotted line) as well as its slope and intercept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' has a completely different chemical signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Whereas the standard GRSN is characterized by excess silicon and magnesium, the pr- GRSN will produce nitrogen from the CNO cycle, plus elements from wherever the bulk transport lands on the line of stability (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 3,8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' But this chemical enrichment will not be distributed evenly throughout the halo (at least not at first).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Heavier elements are pref- erentially produced in the center of the protostar where higher tem- peratures are reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In the resulting outflow, these regions have lower velocity, as the ejecta has nearly homologous structure (𝑣 ∝ 𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Thus, portions of the ejecta with different chemical compositions will end up in different parts of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We have attempted to vi- sualize this effect in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 20, which shows the yields for different mass coordinates on the left, and the velocities of those correspond- ing coordinates on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' As an example, in the middle panel (metal-poor model), consider the ratios of [Fe/Ni] and [C/Fe].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The former will be constant throughout the halo [Fe/Ni] = -3 (or 0 in the outermost regions), while the latter will vary [C/Fe] ∈ (-2,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Table 3 summarizes various feedback effects for selected models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The third and fourth columns show the total amounts of metals and iron synthesized during the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Note that models which do not reach sufficiently high temperature produce metals that do not extend up to iron which experiences a net loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The change in mean molecular weight is shown in the next column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Next are the enormous effects of mechanical feedback, specifically the total kinetic energy and the radius at which the shock speed will become subsonic (𝑅fade, Magg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2020), though this value does not take the effect of the halo’s gravity into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Whalen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (2013a) found that the GRSN ejecta reached a radius of 1 kpc before falling back and igniting a violent starburst, although that was for a less energetic explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Finally, the radiated energy as well as the number of ionizing (E > 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='6 eV) and Lyman Werner (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='6 > E > 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 eV) photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The number of high energy photons is fairly small because the effective temperature is low after shock breakout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 4 DISCUSSION As the saying goes, hydrogen is flammable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We have shown that if supermassive protostars have sufficient seed metals when they collapse, then they will explode through a combination of the CNO cycle and rp process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The consequences of this are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Our results present a challenge to the galaxy merger scenario in two senses of the word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The first is that if the galaxy mergers do not happen early enough in the Universe, then when they do occur, the gas may be sufficiently enriched to trigger a pr-GRSN, thereby preventing the formation of a SMBH seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' If this scenario is avoided, then SMBH seeds may form from galaxy mergers and this may explain the presence of the first quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' However, as the Universe continues to enrich itself, galaxy mergers continue to occur, and if these later mergers produce a supermassive protostar, it will almost certainly contain the requisite metallicity for an explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In addition, these low redshift pr-GRSN will be easily observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Thus, if current and future NIR surveys do not see these objects, then the galaxy merger scenario must posses a way to suppress massive object formation at lower redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Such a mechanism is naturally built into other scenarios such as atomic cooling halos by the stipulation that the gas must be nearly metal free (Z/Z⊙ < 10−3, Chon & Omukai 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Hirano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' As we have touched upon, the intermediate mass black hole popu- lation, or relative lack thereof, also presents a challenge to the study of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' If SMBHs originate primarily from the galaxy merger scenario, then our results present a natural explanation for the lack of intermediate mass black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This is because metal enriched supermassive stars and protostars which are not massive enough to collapse to supermassive black holes (say, 106 M⊙) do not then collapse to intermediate black holes, but instead either explode in a pr-GRSN or lose most of their mass due to line driven winds while on the main sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In the latter case, the mass of the final remnant will be determined roughly by how much helium can be produced before the envelope is completely lost, although mass loss from the helium core may also be possible in some situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Although observation of the lightcurves from a pr-GRSN carries our best chance at observation, other pathways exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We have pro- duced detailed nucleosynthetic signatures of the pr-GRSN and there is some possibility that we may be able to find traces of these signa- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' A signature common to all of our models is enhanced nitrogen production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' A large population of nitrogen-enhanced mildly metal poor stars has recently been observed (Fernández-Trincado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2020), but any explanation involving a pr-GRSN seems far-fetched due to the large amount of ejecta produced by a single event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Another signature found in many of our models is a dearth of light alpha el- ements, such as magnesium, which serve as the fuel for the proton captures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Yoshii et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' (2022) reported a quasar with [Mg/Fe] =-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='12 at redshift 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' They discuss how it is challenging to produce so much iron so early in the Universe, but the pr-GRSN would nat- urally explain this phenomenon by producing iron and consuming MNRAS 000, 1–16 (2022) fiducial 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 massive metalpoor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 F356W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 z =1 F277W- z = 2 Z =3 Z=4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 z = 5 Z=6 z = 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' linear fit 8=Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 m= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='768 6=Z b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='303 z = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 F200W-F277W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='75 fiducial massive 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='50 +metalpoor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='25 P F444W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='00 F356W- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='75 Z = 1 Z = 2 z = 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='50 Z = 4 Z = 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='25 Z = 6 Z = 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='linear fit Z=8 m= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='158 Z=9 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='266 Z = 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 F277W-F356W14 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Elemental yields at selected mass coordinates and velocity profiles for the three indicated models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Left column — elemental abundance for mass coordinates (not the average between mass coordinates) at multiples of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 times the total mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Right column — velocity profiles at a comparable timestep near the end of the HYDnuc simulation for each of the three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The velocity profiles are nearly homologous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Colored circles indicate the mass coordinate shown in the left column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' magnesium simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' At this redshift, the ISM metallicity can be greater than the explosion threshold (Pallottini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We will now summarize some assumptions and shortcoming of the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' First is our decision to ignore mass loss and accretion during the stellar evolution calculation, and thus limit ourselves to protostars greater than ∼ 105 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Because only a small mass fraction of protons is required to trigger the pr-GRSN, if a metal enriched supermassive star were to collapse via the GR radial instability at any point during the hydrogen burning phase (besides the very end), then it would likely explode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Similarly, if the GR instability triggers during the helium burning phase, a metal enriched version of the 𝛼 process GRSN is plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' A logical next step would be to apply our methods to realistic models of metal enriched accreting supermassive stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Haemmerlé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2019) which would also allow us to correctly account for the accretion envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The lightcurve of the pr-GRSNe is likely sensitive to the size of the accretion envelope (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The second major shortcoming of this study is that we do not have the resources to couple the 514 isotope network to our GR hydrodynamics code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Thus, all of the nucleosynthesis that we present is the result of post-processing, which may not be entirely accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In addition, since the energy generation by the 153 isotope network is smaller than that of 514 (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 2), we likely underestimate the region MNRAS 000, 1–16 (2022) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='75 8 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='50 [X/Fe] 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='25 P S cm/ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='00 model 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='75 fiducial 2 Mg m, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 M 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='50 m,=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='4M 6 m,=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='6M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='25 m,=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} 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caveats such as effects of rotation and initial conditions are salient, but likely less important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We have shown that supermassive protostars which encounter the GR radial instability before hydrogen burning will explode in a pr- GRSN if the metallicity is high enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' These events will be clearly visible to NIR surveys at lower redshifts, and marginally visible at higher ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' They will also leave distinct chemical imprints on their host halos, as well as massive amounts of radiative and mechanical feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' It is likely that current and future surveys with unprece- dented breadth and depth will be able to constrain the population of merger induce DCBHs based on the observation or non observation of pr-GRSNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' DATA AVAILABILITY The data underlying this article will be shared on reasonable request to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This study was supported in part by the Grant-in-Aid for the Sci- entific Research of Japan Society for the Promotion of Science (JSPS, Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=', Haemmerlé L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=', Klessen R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='02358 MNRAS 000, 1–16 (2022) 16 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 5 APPENDIX A This appendix shows the results of the 216 isotope simulation for M = 105 M⊙, Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1 Z⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The 216 isotope network is based on the 153 isotope network, but contains additional heavy elements up to Zr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We had created this network because we noticed that our collapsing models were saturating at Zn, the heaviest element in the 153 network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 21 shows the isotopic mass fractions from the central mesh of this simulation as well as the proton mass fraction (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' As is apparent, proton captures stall around a central temperature of Log 𝑇𝑐 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='85, as the network runs out of room for additional proton captures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In addition, the decay times of the isotopes on the boundary of the network are longer than the dynamical timescale (∼ 500 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This causes a buildup of isotopes at certain positions on the edge of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' In an extreme case, 66Ge accumulates nearly half a percent (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='004) of the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This isotope has a half life of roughly 8000 s, largely preventing progress past it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We offer this demonstration as a cautionary tale, that future net- works likely need to be larger than the ones we have used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Such endeavours are beyond our current computational setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 6 APPENDIX B If we did have the resources to couple to a larger network, to what extent would that alter our results?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' To provide a simple estimate of the energy generation of the rp process, we calculate the energy density produced by 25% of the star undergoing the rp process up to an isotope, 𝐼𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We assume that, in this region, every element with atomic number greater than 15 is converted to the most abundant solar isotope with proton number 𝑚 (𝐼𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Then, the chemical distribution of the inner 25% will change as follows: 𝑋′ 𝐼𝑗 = � 𝑋𝐼𝑗 𝐴(𝐼 𝑗) < 16 or A(Ij) > A(Im) 0 15 < 𝐴(𝐼 𝑗) < 𝐴(𝐼𝑚) (15) 𝑋′ 𝑝 = 𝑋𝑝 − ∑︁ 𝐼𝑗 (𝑋𝐼𝑗 − 𝑋′ 𝐼𝑗 ) 𝐴(𝐼𝑚) − 𝐴(𝐼 𝑗) 𝐴(𝐼 𝑗) (16) 𝑋′ 𝐼𝑚 = 𝑋𝐼𝑚 + 𝑋′ 𝑝 + ∑︁ 𝐼𝑗 (𝑋𝐼𝑗 − 𝑋′ 𝐼𝑗 ) (17) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 22 shows the resulting 𝐸nuc (in units of ergs/g) as a function of metallicity and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Also shown are 𝐸nuc from the exploding models in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' We have selected 25% so that the border between explosion and collapse will occur at the edge of our network, roughly m = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Note that this is a somewhat ad hoc approach, as we are ignoring neutrino cooling and inwards ram pressure, and we thus intend for this figure to be interpreted as an order of magnitude estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' As such, it shows that a larger network would allow access to at least twice as much energy generated by the rp process if it extended up to technetium, a situation which we have shown to result from post processing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Indeed, the post processed 𝐸nuc is roughly twice that produced by the 153 isotope network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Thus we can say that our mass and metallicity range is too conservative, but probably the correct order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 7 APPENDIX C This appendix shows lightcurves of JWST, Euclid, Roman and Rubin for the fiducial model, at redshifts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Rubin can observe the pr-GRSN below redshift 1, while Euclid and Roman extend this up to redshift 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Observation of more distant events will only be possible with JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' MNRAS 000, 1–16 (2022) CNO-rp driven GR instability supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 17 Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Upper panel — same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 6, but for only the central mesh of the M = 105 M⊙, Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1 Z⊙ with 216 isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Lower panel — time evolution of proton mass fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 𝐸nuc (in units of ergs/g) generated by the rp process up to an isotope with mass proton number m for a given metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The red triangles correspond to values of 𝐸nuc from the 153 isotope simulations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' The vertical dashed line shows the edge of the 153 isotope network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 16, but for the fiducial model at redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' MNRAS 000, 1–16 (2022) Log T。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='= 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='31 Log T, = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='65 Log Te = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='84 Log Te = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='90 Log Te= 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='98 40 10 2 4 Log 6 20 8 10 10E 12 10 20 30 40 50 10 20 30 40 5010 20 30 40 50 10 20 30 40 50 10 20 30 40 50 nnumber 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='75 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 47500 50000 52500 55000 57500 60000 62500 65000 t [s] 1e18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='78 [ergs/g] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='65 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='00 10 15 20 25 30 35 40 protonnumberatrpterminus(m)15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 F070W F090W 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 F115W 5 F150W 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 M F200W 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 F277W F356W 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 F444W 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 VIS Y 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 Euclid H 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 F062 F087 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 F106 Roman F129 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 F158 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 F184 F146 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 F213 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 Z 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 Rubin 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 y u 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 b 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 0 1000 2000 3000 4000 5000 observertime[days]18 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Nagele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 16, but for the fiducial model at redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Figure 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 16, but for the fiducial model at redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Figure 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 16, but for the fiducial model at redshift 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Figure 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' 16, but for the fiducial model at redshift 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content=' MNRAS 000, 1–16 (2022) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 F070W F090W 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 F115W JWST 22.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 F062 F087 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 F106 ue 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='5 F129 Rom F158 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQf-_5O/content/2301.01941v1.pdf'} +page_content='0 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+CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, +Chinese Academy of Sciences, Beijing 100190, China and +School of Physical Sciences, University of Chinese Academy of Sciences, +No. +19A Yuquan Road, Beijing 100049, China +(Dated: January 3, 2023) +The signals from international pulsar timing arrays have presented a hint of gravitational +stochastic background in nHz band frequency. Further confirmation will be based on whether +the signals follow the angular correlation curves formulated by the overlap reduction func- +tions, known as Hellings-Downs curves. This paper investigates the non-linear corrections of +overlap reduction functions in the present of non-Gaussianity, in which the self-interaction +of gravity is first taken into considerations. Based on perturbed Einstein field equations +for the second order metric perturbations, and perturbed geodesic equations to the second +order, we obtain non-linear corrections for the timing residuals of pulsar timing, and theo- +retically study corresponding overlap reduction functions for pulsar timing arrays. There is +order-one correction for the overlap reduction functions from the three-point correlations of +gravitational waves, and thus the shapes of the overlap reduction functions with non-linear +corrections can be distinguished from the Hellings-Downs curves. +I. +INTRODUCTION +The stochastic gravitational wave background (SGWB) might contain lots of information of +the Universe, since it can be originated from inflationary GWs [1–5], produced from early-time +phase transitions [6–8], sourced by cosmic string [9–12], or formed by superpositions of unresolved +individual GW sources such as binary systems [13–17], core-collapse supernovae [18–21], and de- +formed rotating neutron stars [19, 22]. In the 10Hz–1kHz frequency band, the ground-based GW +detectors LIGO/Virgo/KAGRA have presented an upper limits of SGWBs [23, 24]. In nHz fre- +quency band, the international timing pulsar array projects (IPTA) [25–27] found and confirmed +a common spectrum process from the pulsar-timing data sets, and suggested that further evidence +for SGWBs might rely on its angular correlation signature [28–30]. +The angular correlations of output of a pair of GW detectors can give characteristic signature of +∗ zhuqh@itp.ac.cn +arXiv:2301.00311v1 [gr-qc] 1 Jan 2023 + +2 +GWs, which is formulated by overlap reduction functions (ORFs). For GW detector networks made +by pulsar timing arrays (PTAs), the ORFs of GWs are known as Hellings-Downs curve for a pair +of pulsars [31]. Motivated by the observation from IPTA on the angular correlations of SGWBs, +it is necessary to clarify physical causes of deviations of the Hellings-Downs curve. It might come +from the SGWBs beyond isotropy approximation [32, 33], polarized SGWBs [34–36], non-tensor +modes from modified gravity [37–42], or simply a careful calculation on the pulsar terms [43–45]. +Recently, there was also study on the non-linear corrections for PTAs from higher order effect of +gravity [46]. This correction for ORFs was shown to be order one in the present of non-Gaussianity +of the GWs. In this paper, we will extend the study on the non-linear corrections of the ORFs, in +which the self-interaction of gravity is taken into considerations. +Due to the self-interaction of gravity, the linear order of GWs can generate the non-linear one +even in vacuum, which can affect the propagation of light, thereby changing the response of GW +detectors. There was shown to be order-one corrections for the ORFs in the present of the non- +Gaussianity of GWs [46]. To obtain a solid derivation for GW detectors to the non-linear regime, +we utilize perturbed Einstein field equations for evolution of metric perturbations, and perturbed +geodesic equations for evaluating timing residuals of pulsar timing. We compute the ORFs with +the non-linear corrections, and study its shapes. +The rest of the paper is organized as follows. In Sec. II, the evolutions of metric perturbations +to the second order are presented. In Sec. III, based on propagation of light formulated by the +perturbed geodesic equations to the second order, we obtain timing residuals of pulsar timing with +non-linear corrections. In Sec. IV, we compute ORFs with different shape of non-Gaussianity, and +show its deviation from the Hellings-Downs curves. In Sec. V, conclusions and discussions are +summarized. +II. +PROPAGATION OF SECONDARY GRAVITATIONAL WAVE WITHIN PTAS +From the theory of perturbations in general relativity, the first order GWs, the transverse- +traceless modes of metric perturbations, should give rise to secondary effects from the higher order +metric perturbations. Because all the space-time fluctuations can affect propagation of light, there +are inevitably corrections of response of GW detectors in non-linear regime. In this section, to +study the secondary effects of GWs on PTAs, we will firstly show how the non-linear order metric +perturbations freely propagate in vacuum. + +3 +The perturbed metric in Minkowski background is given by [47] +ds2 = −dt2 + +� +δij(1 − ψ(2)) + ∂i∂jE(2) + 1 +2∂iC(2) +j ++ 1 +2∂jC(2) +i ++ h(1) +ij + 1 +2h(2) +ij +� +dxidxj , +(1) +where δij is Kronecker symbol, h(1) +ij is the first order transverse-traceless metric perturbation known +as GWs, and h(2) +ij , C(2) +j , and ψ(2) are the second order tensor, vector, and scalar perturbations, +respectively. Because GW detectors are reckon in the freely-falling frame, we have adopted Syn- +chronous gauge for the perturbed metric. +From Einstein field equations for the first order GWs, the motion of equations reduce to wave +equations, +h(1) +ij +′′ − ∆h(1) +ij = 0 . +(2) +Thus, the solutions can be given by +h(1) +ij = +� +d3k +(2π)3 ¯h(1) +ij,ke−i(kt−k·x) , +(3) +where the h(1) +ij,k is the Fourier mode of ¯h(1) +ij , which contains the physical information before the +GWs reaching the detectors. +Since the GW detectors can response to tensor, vector, and scalar modes of the metric pertur- +bations, we should further consider all the second order metric perturbations. Using the metric in +Eq. (1), the perturbed Einstein field equations in the second order take the form of [48, 49] +h(2) +ij +′′ − ∆h(2) +ij = −Λab +ij Sab , +(4a) +C(2) +j +′′ = −Vab +j Sab , +(4b) +−2ψ(2)′′ + ∆ +� +E(2)′′ + ψ(2)� += −S(Ψ),abSab , +(4c) +E(2)′′ + ψ(2) = −S(E),abSab , +(4d) +where Λab +ij , Vab +j , S(Ψ),ab and S(E),ab are helicity decomposition operators [47], and the source on +the right hand side of Eq. (4) is given by +Sab = −2δcd∂0h(1) +ac ∂0h(1) +bd + 2h(1),cd∂c∂dh(1) +ab − 2h(1),cd∂c∂ah(1) +db +−2h(1),cd∂c∂bh(1) +da − 2∂dh(1) +ac ∂ch(1) +bd + 2δcd∂jh(1) +ad ∂jh(1) +bc + ∂ah(1),cd∂bh(1) +cd + 2h(1),cd∂a∂bh(1) +cd ++δab +�3 +2∂0h(1) +cd ∂0h(1),cd + 2h(1) +cd ∂2 +0h(1),cd − 2h(1) +cd ∆h(1),cd + ∂jh(1) +cd ∂dh(1),cj − 3 +2∂jh(1) +cd ∂jh(1),cd +� +. +(5) + +4 +Differed from Eq. (2), the second metric perturbations are sourced the by the first order GWs. By +making use of Eq. (3) and (4), we obtain the solutions in the form of +ψ(2) (t, x) = +� +d3k +(2π)3 +�� +d3p +(2π)3 +� +ˆF ab +ψ ¯Sab (k, p) e−i(|k−p|+p)t� +eik·x +� +, +(6a) +E(2) (t, x) = +� +d3k +(2π)3 +�� +d3p +(2π)3 +� +ˆF ab +E ¯Sab (k, p) e−i(|k−p|+p)t� +eik·x +� +, +(6b) +C(2) +j +(t, x) = +� +d3k +(2π)3 +�� +d3p +(2π)3 +� +ˆF ab +C,j ¯Sab (k, p) e−i(|k−p|+p)t� +eik·x +� +, +(6c) +h(2) +ij (t, x) = +� +d3k +(2π)3 +�� +d3p +(2π)3 +� +ˆF ab +h,ij ¯Sab (k, p) e−i(|k−p|+p)t� +eik·x +� +, +(6d) +where +ˆF ab +h,ij (k, p) ≡ − +Λab +ij (k) +k2 − (|k − p| + p)2 , +(7a) +ˆF ab +C,j (k, p) ≡ +Vab +j (k) +(|k − p| + p)2 , +(7b) +ˆF ab +ψ (k, p) ≡ −S(Ψ),ab (k) + k2S(E),ab (k) +2 (|k − p| + p)2 +, +(7c) +ˆF ab +E (k, p) ≡ +S(E),ab (k) +(|k − p| + p)2 − S(Ψ),ab (k) + k2S(E),ab (k) +2 (|k − p| + p)4 +. +(7d) +The ¯Sab (k, p) in Eq. (6) is defined with +¯Sij (k, p) = fcdab +ij +(k, p) ¯h(1) +cd,k−p¯h(1) +ab,p , +(8) +where +fbclm +ij +(k, p) = δblδcm (−3δij (p |k − p| − p · (k − p)) + pi(kj − 2pj) + ki(pj − 2kj)) ++2 +� +pb(−δl +iδm +j pc + δcm(δl +jpi + δl +ipj + δij(pl − kl)) ++δb +j +� +δcm � +δl +i (p |k − p| − p · (k − p)) + ki(kl − pl) + pi(pl − kl) +� ++ δl +ipc(km − pm) +� ++δb +i +� +δcm � +δl +j (p |k − p| − p · (k − p)) + kj(kl − pl) + pj(pl − kl) +� ++ δl +jpc(km − pm) +� +−δb +i δc +j(klkm − 2klpm + plpm) +� +. +(9) +The ¯Sab (k, p) is derived from the source in Eq. (5) via +Sab (t, x) = +� +d3k +(2π)3 Sab (t, k, p) eik·x = +� +d3k +(2π)3 ¯Sab (k, p) e−i(|k−p|+p)t+ik·x . +(10) +Note that the Sab (t, k, p) is the Fourier mode of the source, and the ¯Sab (k, p) is not. Here, as +shown in Eqs. (6), the explicit solutions of perturbations ψ(2), E(2), C(2) +j +and h(2) +ij can be obtained +based on the known h(1) +ij in Eq. (3). + +5 +We consider the non-linear corrections for GW detectors originated from the gravity self- +interaction. In this case, all the second order perturbations are generated by the first order one. +In fact, there could be second order perturbations that are independent of the h(1) +ij . Because the +response of the second order perturbations of this type to the GWs detector has nothing different +from the response for the first order GWs, we did not consider this situation in the present study. +III. +PERTURBED GEODESIC EQUATIONS +The space-time fluctuations can affect time of arrivals of radio beams from a pulsar. Thus, via +monitoring pulsar timing, the PTA observations can reflect the space-time fluctuations over the +Universe in principle. To formulate it, and extend it to the non-linear regime, we will calculate the +perturbed geodesic equations to the second order. +Based on geodesic equations in Minkowski background, namely, +P µ∂µP ν = 0 , +(11) +one can obtain the 4-momentum of a light ray, +P µ = P 0(1, −ˆnj) , +(12) +where ˆnj is a constant unit vector, and can be used to locate a pulsar. By making use of above +4-momentums, one can obtain trajectories of light rays from a pulsar to the detectors. The pulsar +emits a radio beam at the event (t − L, Lˆnj), and the beam is detected on the earth at the event +of (t, 0), where the L is distance between the pulsar and the detectors. +From the first order perturbed geodesic equations, +0 = δP µ∂µP ν + P µ∂µδP ν + gνρ +� +∂µh(1) +λρ − 1 +2∂ρh(1) +µλ +� +P µP λ , +(13) +we can evaluate it by using Eq. (12), namely, +(∂0 − ˆn · ∂) +�δP 0 +P 0 +� += −1 +2 ˆnaˆnb∂0h(1) +ab , +(14a) +(∂0 − ˆn · ∂) +�δP j +P 0 +� += δjbˆna +� +∂0h(1) +ab − ˆnc +� +∂ch(1) +ab − 1 +2∂bh(1) +ac +�� +, +(14b) +where δP µ is the first order perturbed 4-momentum. The temporal and spatial components of +the above perturbed geodesic equations take different forms, because the time components of h(1) +µν +in Eq. (13) vanish in the Synchronous gauge. From Eq. (14), the solutions in Fourier space take + +6 +the form of +δP 0 +k +P 0 ≡ K0,ab (k, ˆn) ¯h(1) +ab,ke−ikt = − +1 +2(1 + ˆn · ˆk) +ˆnaˆnb¯h(1) +ab,ke−ikt , +(15a) +δP j +k +P 0 ≡ Kj,ab (k, ˆn) ¯h(1) +ab,ke−ikt = +� +δjbˆna − +ˆkjˆnaˆnb +2(1 + ˆn · ˆk) +� +¯h(1) +ab,ke−ikt , +(15b) +where the k is wave number, and the ˆk describes propagation direction of the first order GWs. +The results shown in Eqs. (15) are the basis of GW detectors, and astrometric detection with Gaia +[31, 50]. +In the non-linear regime, we extend the calculations to the second order perturbed geodesic +equations, namely, +0 = δ2P µ∂µP ν + 2δP µ∂µδP ν + P µ∂µδ2P ν + 2gνρP µδP λ(∂µh(1) +λρ − ∂ρh(1) +µλ + ∂λh(1) +µρ ) ++P µP λ +� +gνρ +� +∂λδg(2) +µρ − 1 +2∂ρδg(2) +µλ +� ++ gνκgρωh(1) +ωκ(∂ρh(1) +µλ − 2∂λh(1) +µρ ) +� +, +(16) +where where δ2P µ is the second order perturbed 4-momentum, and the second order metric per- +turbations in Synchronous gauge is +δg(2) +00 = δg(2) +i0 = 0 , +(17a) +δg(2) +ij = −2δijψ(2) + 2∂i∂jE(2) + ∂iC(2) +j ++ ∂jC(2) +i ++ h(2) +ij . +(17b) +To obtain the timing residuals to the second order, we calculate temporal components of Eq. (16) +by making use of Eqs. (12) and (17), namely, +(∂0 − ˆn · ∂) +�δ2P 0 +P 0 +� += −1 +2 ˆnaˆnb∂0δg(2) +ab − 2 +�δP µ +P 0 +� +∂µ +�δP 0 +P 0 +� ++ 2ˆna +�δP b +P 0 +� +∂0h(1) +ab . +(18) +Differed from the linearized equations in Eq. (14a), the second and the third terms on the right hand +side of above equation should be attributed to the non-linear contributions from perturbed geodesic +equations. It was partly considered in previous study [46], in which the non-linear corrections were +obtained by expanding the proper time to the second order. By making use of Eqs. (15), the +solutions of Eq. (18) can be obtained, +δ2P 0 +k +P 0 += +� +d3q +(2π)3 +� +Fcd,ab (k, q) ¯h(1) +cd,k−q¯h(1) +ab,qe−i(|k−q|+q)t� +, +(19) +where +Fcd,ab (k, q, ˆn) ≡ − +1 +|k − q| + q + ˆn · k +� +�Fcd,ab +L +(k, q) + |k − q| + q +2 +� +∗=ψ,E,C,h +Fcd,ab +∗ +(k, q) +� +� , +(20) + +7 +and +Fcd,ab +L +(k, q) ≡ (|k − q| + q) K0,cd +k−qK0,ab +q +− qjKj,cd +k−qK0,ab +q +, +−(kj − qj)K0,cd +k−qKj,ab +q +− qˆnaKb,cd +k−q − |k − q| ˆncKd,ab +q +, +(21a) +Fcd,ab +ψ +(k, q) ≡ −2 ˆF ij +ψ (k, q) fcdab +ij +(k, q) , +(21b) +Fcd,ab +E +(k, q) ≡ −2(ˆn · k)2 ˆF ij +E (k, q) fcdab +ij +(k, q) , +(21c) +Fcd,ab +C +(k, q) ≡ iˆnj(ˆn · k) ˆF ml +C,j (k, q) fcdab +ml +(k, q) , +(21d) +Fcd,ab +h +(k, q) ≡ ˆniˆnj ˆF ml +h,ij (k, q) fcdab +ml +(k, q) , +(21e) +where Kµ,ab +k +has been defined in Eq. (15) and ˆF ij +ψ (k, q), ˆF ij +E (k, q), ˆF ij +C (k, q) and ˆF ij +h (k, q) have +been given in Eq. (7). The Fcd,ab +L +(k, q) is derived from the second and the third terms of Eq. (18), +and the Fcd,ab +ψ +(k, q), Fcd,ab +E +(k, q), Fcd,ab +C +(k, q), and Fcd,ab +h +(k, q) are obtained by the solving the +motion of equations of second order scalar, vector, and tensor perturbations, respectively. +The difference of the time of arrivals can be quantified by the redshift caused by GWs fluctua- +tions, namely, +z = uµ ˜P µ|obs +uµ ˜P µ|src +, +(22) +where ˜P µ(≡ P µ + δP µ + 1 +2δ2P µ + O(3)) is the total 4-momentum of a radio beam from a pulsar. +For comoving observers in Synchronous gauge uµ = (1, 0, 0, 0), we obtain the redshift and its +fluctuations as follows, +z(0) = 0 , +(23a) +z(1) = δP 0 +obs +P 0 +obs +− δP 0 +src +δP 0src +, +(23b) +z(2) = δ2P 0 +obs +P 0 +obs +− δ2P 0 +src +P 0src +− 2 +�δP 0 +src +P 0src +� �δP 0 +obs +P 0 +obs +− δP 0 +src +δP 0src +� +, +(23c) +where the perturbed 4-momentums can be given by δ(n)P 0/P 0 = +� +d3k +(2π)3 +� +(δ(n)P 0 +k/P 0)eik·x� +, and +the expressions of δ(n)P 0 +k/P 0 have been given in Eqs. (15) and (19). From Eq. (12), we have known +the events of a pulsar emitting a radio beam tsrc = t − L and xj +src = Lˆnj, and the events of the +beam reaching the earth tobs = t and xj +obs = 0. Therefore, the redshifts and its fluctuations in + +8 +Eqs. (23) can be rewritten in the form of +z(1) = +� +d3k +(2π)3 +� +K0,ab (k, ˆn) ¯h(1) +ab,ke−ikt(1 − eikL(1+ˆn·ˆk)) +� +, +(24a) +z(2) = +� d3kd3q +(2π)6 +� +Fcd,ab (k, q, ˆn) ¯h(1) +cd,k−q¯h(1) +ab,qe−i(|k−q|+q)t � +1 − eiL(|k−q|+q+ˆn·k)� ++K0,cd (k, ˆn) K0,ab (q, ˆn) ¯h(1) +cd,k¯h(1) +ab,qe−i(k+q)t +×1 +2 +� +eiqL(1+ˆn·ˆq)(1 − eikL(1+ˆn·ˆk)) + eikL(1+ˆn·ˆk)(1 − eiqL(1+ˆn·ˆq)) +�� +. +(24b) +In practical, the observables are the timing residuals of pulsar timing. It can be obtained by +integration of the redshifts in Eqs. (24) over observation duration t, namely, R(n)(t) = +� t +0 z(n)(¯t)d¯t. +Thus, one can obtain expressions of the timing residuals in the form of +R(1) = +� +d3k +(2π)3 +� 1 +ikK0,ab (k, ˆn) ¯h(1) +ab,k(1 − e−ikt)(1 − eikL(1+ˆn·ˆk)) +� +, +(25) +R(2) = +� d3kd3q +(2π)6 +� +1 +i (|k − q| + q)Fcd,ab (k, q, ˆn) ¯h(1) +cd,k−q¯h(1) +ab,q(1 − e−i(|k−q|+q)t) +� +1 − eiL(|k−q|+q+ˆn·k)� ++ +1 +i(k + q)K0,cd (k, ˆn) K0,ab (q, ˆn) ¯h(1) +cd,k¯h(1) +ab,q(1 − e−i(k+q)t) +×1 +2 +� +eiqL(1+ˆn·ˆq)(1 − eikL(1+ˆn·ˆk)) + eikL(1+ˆn·ˆk)(1 − eiqL(1+ˆn·ˆq)) +�� +. +(26) +Due to kt ≪ 1 for the nHz band PTAs, the timing residuals can be expanded as kt → 0. In +this approximation, the leading-order timing residuals reduce to R(n) = tz(n)|t=0. It indicates that +the corrections of the outputs (timing residuals) of PTAs can be obtained via the correlations of +redshifts, namely, ⟨R(x)R(x′)⟩ = t2⟨z(x)z(x′)⟩. +IV. +SPATIAL CORRELATIONS AND OVERLAP REDUCTION FUNCTIONS +A. +Non-linear correction of the correlations and non-Gaussianity +The timing residuals can reflect physical information of SGWBs in the space. Thus, it also +should be studied statistically, due to stochastic nature of the SGWBs. The spatial correlations of +the timing residuals, as mentioned above, are proportional to spatial correlations of the redshifts +⟨z(x)z(x′)⟩, where the x and x′ can represent the locations of pulsar pairs. For illustration, we let +zα ≡ z(x) and zβ ≡ z(x′) in the rest of the paper. +To calculate the correlations order by order, we expand correlations for the redshifts zα in the +form of +⟨zαzβ⟩ = ⟨z(1) +α z(1) +β ⟩ + 1 +2(⟨z(1) +α z(2) +β ⟩ + ⟨z(2) +α z(1) +β ⟩) + O(4) . +(27) + +9 +For PTAs, the angular correlation derived from ⟨z(1) +α z(1) +β ⟩ is known as Hellings-Downs curve [31]. +In this section, we will extend the spatial correlations to the non-linear regime with ⟨z(1) +α z(2) +β ⟩ and +⟨z(2) +α z(1) +β ⟩. +The purpose of PTAs is extracting the physical information of h(1) +ij based on observation of the +timing residuals. Since z(1) ∝ h(1) and z(2) ∝ (h(1))2 shown in Eq. (24), the ⟨z(1) +α z(1) +β ⟩ and ⟨z(1) +α z(2) +β ⟩ +could encode two-point and three-point correlations of h(1) +ij , respectively. Here, we strict our study +to the isotropic and unpolarized GWs. In this case, the two-point correlations of hλ +k in Fourier +space can be given by +� +h(1),λ1 +k1 +h(1),λ2 +k2 +� += (2π)3δ (k1 + k2) δλ1λ2P(k2) , +(28) +where the P(k) is power spectrum, the λ∗(= +, ×) is polarization index, and the Kronecker symbol +δλ1λ2 indicates that h(1),λ +k +is unpolarized. In the non-linear regime, the three-point correlations in +Fourier space can be given by +� +h(1),λ1 +k1 +h(1),λ2 +k2 +h(1),λ3 +k3 +� += (2π)6δ(3) (k1 + k2 + k3) Bλ1λ2λ3(k1, k2, k3) , +(29) +where the Bλ1λ2λ3(k1, k2, k3) is bi-spectrum. +It does not vanish due to the non-Gaussianity of +SGWBs. For the unpolarized GWs, i) different polarizations of h(1),λ have no correlations, namely, +Bλ1λ2λ3(k1, k2, k3) does not vanish, only if λ1 = λ2 = λ3, and ii) different components of the bi- +spectrum are equally weighted, namely, the B+++(k1, k2, k3) = B×××(k1, k2, k3). To quantifying +shapes of the non-Gaussianity, we follow the parameterization scenario used in Ref. [46]. With all +the above assumptions, the bi-spectrum in Eq. (29) can be given in the form of +Bλ1λ2λ3(k1, k2, k3) = Hλ1λ2λ3(k3)P(k3)δ(k1 − χk3)δ(k2 − ζk3) , +(30) +where we have H+++(k3) = H×××(k3) ≡ κ(k3) due to the unpolarized GWs, the P(k) is the power +spectrum defined in Eq. (28), and the ζ and χ are the dimensionless quantities formulating the +shape of bi-spectrum. In principle, the shape of bi-spectrum quantified by κ, ζ and χ should be +given based on generation mechanism of GWs. Here, we did not involve any physical models for +the generation mechanism, but utilize the non-Gaussianity based on a phenomenological param- +eterization. Because of k1 + k2 + k3 = 0 in Eq. (29), the parameters ζ and χ should satisfy the +relations, +χ + ζ ⩾ 1 , and |χ − ζ| ⩽ 1 . +(31) +In Fig. 1, we show scheme diagram for the shape of the bi-spectrum, in which the ζ and χ are +defined with ζ ≡ BC/AB and χ ≡ CA/AB. +In Fig. 2, we show the parameter space (ζ, χ), + +10 +k3 +k1 +k2 +A +B +C +Figure 1: The scheme diagram of the parameterized bi-spectrum defined in Eq. (30). +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +ζ +χ +ζ  0.60 +χ  0.60 +ζ  0.87 +χ  0.87 +ζ  1.1 +χ  1.1 +ζ  1.4 +χ  1.4 +ζ  1.7 +χ  1.7 +ζ  1.9 +χ  1.9 +ζ  1.0 +χ  1.0 +ζ  1.3 +χ  0.87 +ζ  1.4 +χ  0.87 +ζ  1.9 +χ  1.1 +Figure 2: Left panel: parameter space of (ζ, χ) for the parameterized non-Gaussianity of hλ +ij,k. The +points locate the available domain of the parameters. The dashed curve is formulated by ζ = χ, and the +dotted curve is formulated by 1 +2ζχ +� +1 − +� +ζ2+χ2−1 +2ζχ +�2 += +√ +3 +4 . Right panel: The shape of non-Gaussianity for +selected parameters ζ and χ on the dashed and dotted curves. +where the points locate the domain of available parameters in Eqs. (31). Here, every single point +represents a distinguishable shape of non-Gaussianity. As shown in the right panel of Fig. 2, for +example, the orange points on the dashed line represent isosceles triangles, and the green points +on the dotted curve represent triangles with the same height. +B. +Overlap reduction functions of PTAs in second order +By making use of Eqs. (24a) and (28), the linear-order correlations of the redshifts for a pulsar +pair can be given by +⟨z(1) +α z(1) +β ⟩ = +� k2dk +2π2 P(k) +� dΩ +4π +� +K0,ab (k, ˆnα) K0,cd (k, ˆnβ) eλ +ab(ˆk)eλ +cd(ˆk) +×(1 − eikLα(1+ˆnα·ˆk))(1 − e−ikLβ(1+ˆnβ·ˆk)) , +� +(32) + +11 +where the eλ +cd(ˆk) is polarization tensor for h(1) +ij,k, the θαβ ≡ cos−1(ˆnα · ˆnβ), and the P(k) is the +power spectrum defined in Eq. (28). The ORFs describe angular correlations of outputs of the GW +detectors, and can be obtained via surface integrals over the unit sphere ˆk. Rewriting Eq. (32) in +the form of +⟨z(1) +α z(1) +β ⟩ ≡ +� k2dk +2π2 P(k)Γ(2)(k, θαβ) , +(33) +one can read the ORFs, +Γ(2)(k, θab) = +� dΩ +4π +� +K0,ab (k, ˆnα) K0,cd (k, ˆnβ) eλ +ab(ˆk)eλ +cd(ˆk) +×(1 − eikLα(1+ˆnα·ˆk))(1 − e−ikLβ(1+ˆnβ·ˆk)) +� +. +(34) +Because of the approximation kL ≫ 1 for the known frequency band and arms length of PTAs, +above ORFs would reduce to Hellings-Downs curve [31], namely, +ΓHD(θab) ≡ Γ(2)(k, θab)|kLα≫1,kLβ≫1 = +� dΩ +4π +� +K0,ab (k, ˆnα) K0,cd (k, ˆnβ) eλ +ab(ˆk)eλ +cd(ˆk) +� +. (35) +Here, the oscillation parts in Eq. (34) is suppressed by the factor (kL)−1, and thus can be neglected +for PTAs. Besides, the ORFs without the approximation were also studied [43–45]. +To the second order, we further compute non-linear corrections for the correlations of redshift +in Eq. (24), namely, +⟨z(1) +α z(2) +β ⟩ = +� d3kd3k′d3q′ +(2π)12 +�� +h(1) +k,mlh(1) +k′−q′,cd +∗h(1) +q,ab +∗� +K0,ml (k, ˆnα) Fcd,ab � +k′, q′, ˆnβ +� +×e−ikt(1 − eikLα(1+ˆnα·ˆk))eit(|k′−q′|+q′) � +1 − e−iLβ(|k′−q′|+q′+ˆnβ·k′)� ++1 +2 +� +h(1) +k,mlh(1) +k′,cd +∗h(1) +q,ab +∗� +K0,ml (k, ˆnα) K0,cd � +k′, ˆnβ +� +K0,ab � +q′, ˆnβ +� +e−ikt(1 − eikLα(1+ˆnα·ˆk))ei(k′+q′)t +× +� +e−iq′Lβ(1+ˆnβ·ˆq′)(1 − e−ik′Lβ(1+ˆnβ·ˆk′)) + e−iq′Lβ(1+ˆnβ·ˆq′)(1 − e−ik′Lβ(1+ˆnβ·ˆk′)) +�� +, +(36) +where above three-point correlations of h(1) +ij,k can be written in terms of polarization components +h(1),λ +k +, +� +h(1) +k,cdh(1) +ab,ph(1) +q,ml +� += eλ1 +cd(ˆk)eλ2 +ab(ˆp)eλ3 +ml(ˆq) +� +h(1),λ1 +k +h(1),λ2 +p +h(1),λ3 +q +� +. +(37) + +12 +By making use of Eqs. (29) and (30), the Eq. (34) is evaluated to be +⟨z(1) +α z(2) +β ⟩ + ⟨z(2) +α z(1) +β ⟩ = +� +d3k +(2π)3 +� dφq +2π +� +eλ1 +ml(ˆk)eλ2 +cd( � +k − q)eλ3 +ab(ˆq)Hλ1λ2λ3(k)P(k) +× +�� +K0,ml (k, ˆnα) Fcd,ab (k, q, ˆnβ) (1 − eikLα(1+ˆnα·ˆk))(1 − e−ikLβ(χ+ζ+ˆnβ·ˆk)) ++1 +2K0,ml (k, ˆnα) K0,cd (k − q, ˆnβ) K0,ab (q, ˆnβ) +×e−iζkLβ(1+ˆnβ·ˆq)(1 − eikLα(1+ˆnα·ˆk))(1 − e−iLβ(χk+ˆnβ·(k−q))) ++1 +2K0,ml (k, ˆnα) K0,cd (q, ˆnβ) K0,ab (k − q, ˆnβ) +×e−iLβ(χk+ˆnβ·(k−q))(1 − eikLα(1+ˆnα·ˆk))(1 − e−iζkLβ(1+ˆnβ·ˆq))) +� +e−ikt(1−χ−ζ) ++ +� +Fcd,ab (k, q, ˆnα) K0,ml (k, ˆnβ) (1 − eikLα(ζ+χ+ˆnα·ˆk))(1 − e−ikLβ(1+ˆnβ·ˆk)) ++1 +2Kml (k, ˆnβ) K0,cd (k − q, ˆnα) K0,ab (q, ˆnα) +×eiζkLα(1+ˆnα·ˆq)(1 − eiLα(χk+ˆnα·(k−q)))(1 − e−ikLβ(1+ˆnβ·ˆk)) ++1 +2Kml (k, ˆnβ) K0,cd (q, ˆnα) K0,ab (k − q, ˆnα) +×eiLα(χk+ˆnα·(k−q))(1 − eiζkLα(1+ˆnα·ˆq))(1 − e−ikLβ(1+ˆnβ·ˆk)) +� +eikt(1−χ−ζ)�� +, +(38) +where the momentums are give by +q = +�1 + ζ2 − χ2 +2 +� +k + +� +((ζ + χ)2 − 1)(1 − (ζ − χ)2) +2 +k (cos φqu + sin φqυ) , +(39a) +ˆq = 1 + ζ2 − χ2 +2ζ +ˆk + +� +((ζ + χ)2 − 1)(1 − (ζ − χ)2) +2ζ +(cos φqu + sin φqυ) , +(39b) +� +k − q = χ2 − ζ2 + 1 +2χ +ˆk − +� +((ζ + χ)2 − 1)(1 − (ζ − χ)2) +2χ +(cos φqu + sin φqυ) . +(39c) +Here, the u and υ represent polarization vectors with respect to the ˆk. Using Eq. (39a), one can +verify the relations of q = ζk and |k − q| = χk shown in Fig. 1. From Eq. (38), it is found that the +three-point correlations are proportional to e±ikt(1−χ−ζ). It indicates that the values of correlations +would oscillate with time around zero. In practical, due to kt ≪ 1 for nHz band PTAs, we here +can let e±ikt(1−χ−ζ) ≃ 1. +Similarly, the correlations in Eq. (38) can be rewritten in the form of +⟨z(1) +α z(2) +β ⟩ + ⟨z(2) +α z(1) +β ⟩ = +� k2dk +2π2 P(k)Γ(3)(k, θab) , +(40) + +13 +where the ORFs in the non-linear order are given by +Γ(3)(k, θαβ) = +� dΩ +4π +� dφq +2π +� +eλ1 +ml(ˆk)eλ2 +cd( � +k − q)eλ3 +ab(ˆq)Hλ1λ2λ3(k)P(k) +× +� +K0,ml (k, ˆnα) Fcd,ab (k, q, ˆnβ) (1 − eikLα(1+ˆnα·ˆk))(1 − e−ikLβ(χ+ζ+ˆnβ·ˆk)) ++Fcd,ab (k, q, ˆnα) K0,ml (k, ˆnβ) (1 − eikLα(ζ+χ+ˆnα·ˆk))(1 − e−ikLβ(1+ˆnβ·ˆk)) ++1 +2K0,ml (k, ˆnα) K0,cd (k − q, ˆnβ) K0,ab (q, ˆnβ) +×e−iζkLβ(1+ˆnβ·ˆq)(1 − eikLα(1+ˆnα·ˆk))(1 − e−iLβ(χk+ˆnβ·(k−q))) ++1 +2K0,ml (k, ˆnα) K0,cd (q, ˆnβ) K0,ab (k − q, ˆnβ) +×e−iLβ(χk+ˆnβ·(k−q))(1 − eikLα(1+ˆnα·ˆk))(1 − e−iζkLβ(1+ˆnβ·ˆq))) ++1 +2Kml (k, ˆnβ) K0,cd (k − q, ˆnα) K0,ab (q, ˆnα) +×eiζkLα(1+ˆnα·ˆq)(1 − eiLα(χk+ˆnα·(k−q)))(1 − e−ikLβ(1+ˆnβ·ˆk)) ++1 +2Kml (k, ˆnβ) K0,cd (q, ˆnα) K0,ab (k − q, ˆnα) +×eiLα(χk+ˆnα·(k−q))(1 − eiζkLα(1+ˆnα·ˆq))(1 − e−ikLβ(1+ˆnβ·ˆk)) +�� +. +(41) +The expression of Hλ1λ2λ3(k) has been given in the Eq. (30). Since the oscillation parts in the +integration is suppressed by the factor (kL)−1 for PTAs, we also adopt kLα ≫ 1 and kLβ ≫ 1 for +evaluating Eq. (41). Namely, the ORFs can be simplified in the form of +Γnl(k, θαβ) ≡ Γ(3)(k, θab)|kLα≫1,kLβ≫1 += κ +� dΩ +4π +� dφq +2π +�� +e+ +ml(ˆk)e+ +cd( � +k − q)e+ +ab(ˆq) + e× +ml(ˆk)e× +cd( � +k − q)e× +ab(ˆq) +� +× +� +K0,ml (k, ˆnα) Fcd,ab (k, q, ˆnβ) + Fcd,ab (k, q, ˆnα) K0,ml (k, ˆnβ) +�� +. +(42) +In the following, we would show the results of Γnl(k, θab) with selected parameters ζ and χ. In +Fig. 3, we show the ORFs over the κ as function of parameter (ζ, χ) for given angle θαβ. It is +found that the values of Γln/κ tend to be zero for large ζ and χ, and to be larger in the case of +ζ + χ = 1. In Fig. 4, we show the ORFs for select parameters given in right panel of Fig. 2. It +is confirmed that there is a larger value of ORFs for the parameters ζ, χ → 0. Differed from the +ORFs in the linear order, there are three extreme points in the curves of the non-linear ORFs. In +order to clarify the extreme cases, such as ζ − χ = 1 or ζ + χ = 1, we show the ORFs as function +of θαβ for ζ − χ → 1 and ζ + χ → 1 in Figs. 7 and 6, respectively. From Fig. 7, the values of ORFs +tend to be zero as ζ − χ → 1, and these ORFs have the same zero of the Γ(k, θαβ) with respect +to θαβ. From Fig. 6, the values of ORFs tend to be larger, and the numbers of extreme points are +reduced as ζ + χ → 1, which is different from the results shown in the left panel of Fig. 4 for a +larger ζ + χ. +In the case of ζ + χ = 1, we can further simplify the expression of ORFs as + +14 +Figure 3: Non-linear ORFs over the κ as function of parameter (ζ, χ) for θαβ = 0, π/4, π/2, π. +Γnl(k, θαβ) = κ +� dΩ +4π +�� +e+ +ml(ˆk)e+ +cd(�k)e+ +ab(ˆk) + e× +ml(ˆk)e× +cd(�k)e× +ab(ˆk) +� +× +� +K0,ml (k, ˆnα) Fcd,ab (k, ζk, ˆnβ) + Fcd,ab (k, ζk, ˆnα) K0,ml (k, ˆnβ) +�� +. +(43) +Here, the integration over the angle φq simply gives 2π. In Fig. 7, we show the ORFs in the case +of ζ + χ = 1 for different ζ. It is found that the values of ORFs get smaller as ζ → 0. +In principle, the parameter κ in Eq. (42) depends on the wave number k, and the shape of +non-Gaussianity quantified by ζ and χ. There seems no reason that the non-linear corrections +come from the non-Gaussianity with one of the available parameters (ζ, χ). For a general case, the +total ORFs to the non-linear order should be the sum of all the shape of non-Gaussianity weighted +by parameter κ, namely, +Γ(k, θαβ) = ΓHD(θαβ) + 1 +2 +� +ζ,χ +Γnl(k, θαβ)∆σ , +(44) + +15 +ζ  χ  0.60 +ζ  χ  0.87 +ζ  χ  1.1 +ζ  χ  1.4 +ζ  χ  1.7 +ζ  χ  1.9 +0.0 0.5 1.0 1.5 2.0 2.5 3.0 +-0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +θαβ +Γnlθαβ +ζ  χ  1.0 +ζ  1.3 , χ  0.87 +ζ  1.4 , χ  0.87 +ζ  1.9 , χ  1.1 +0.0 0.5 1.0 1.5 2.0 2.5 3.0 +-0.04 +-0.02 +0.00 +0.02 +0.04 +0.06 +0.08 +θαβ +Γnlθαβ +Figure 4: Non-linear ORFs for selected parameters ζ and χ shown in right panel of Fig. 2, and κ = 1. +Left panel: the non-Gaussianity in the shape of isosceles triangles. Right panel: the non-Gaussianity in the +shape of the triangles with the same height. +ζ  0.500 , χ  1.40 +ζ  0.500 , χ  1.47 +ζ  0.500 , χ  1.49 +0.0 0.5 1.0 1.5 2.0 2.5 3.0 +-0.002 +-0.001 +0.000 +0.001 +0.002 +0.003 +0.004 +0.005 +θαβ +Γnlθαβ +ζ  1.00 , χ  1.90 +ζ  1.00 , χ  1.97 +ζ  1.00 , χ  1.99 +0.0 0.5 1.0 1.5 2.0 2.5 3.0 +-0.0005 +0.0000 +0.0005 +0.0010 +0.0015 +0.0020 +θαβ +Γnlθαβ +ζ  0.500 +χ  1.40 +ζ  0.500 +χ  1.47 +ζ  0.500 +χ  1.49 +ζ  1.00 +χ  1.90 +ζ  1.00 +χ  1.97 +ζ  1.00 +χ  1.99 +Figure 5: Non-linear ORFs for selected parameters ζ − χ → 1, and κ = 1. + +16 +ζ  χ  0.600 +ζ  χ  0.533 +ζ  χ  0.510 +ζ  χ  0.501 +ζ  χ  0.500 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +θαβ +Γnlθαβ +ζ  0.600 +χ  0.600 +ζ  0.533 +χ  0.533 +ζ  0.510 +χ  0.510 +ζ  0.501 +χ  0.501 +ζ  0.500 +χ  0.500 +Figure 6: Non-linear ORFs for selected parameters ζ + χ → 1, and κ = 1. +ζ  χ  0.50 +ζ  0.75 , χ  0.25 +ζ  0.90 , χ  0.10 +ζ  0.99 , χ  0.010 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +θαβ +Γnlθαβ +ζ  0.500 +χ  0.500 +ζ  0.750 +χ  0.250 +ζ  0.900 +χ  0.100 +ζ  0.990 +χ  0.0100 +Figure 7: Non-linear ORFs for selected parameters ζ + χ = 1, ζ → 1, and κ = 1. +where ∆σ is the size of grids of points in the parameter space (ζ, χ). +For example, we have +∆σ = 0.018 for the grids in the left panel of Fig. 2. Here, the Γnl(k, θαβ) is proportional to the +parameter κ(k; ζ, χ) shown in Eq. (42). The dependence of Γ(k, θαβ) on the parameters ζ and + +17 +ΓHD(θαβ) +Γtot(θαβ) +ζ,χ<2,κ=1 +Γtot(θαβ) +ζ,χ<10,κ=1 +Γtot(θαβ) +ζ,χ<2,κ=-1 +Γtot(θαβ) +ζ,χ<10,κ=-1 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +-0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +θαβ +Γθαβ +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +ζ +χ +0 +2 +4 +6 +8 +10 +0 +2 +4 +6 +8 +10 +ζ +χ +Figure 8: Left panel: total ORFs with non-linear corrections. The non-linear ORFs are sum of the +selected parameters κ = ±1 and (ζ, χ) shown with the points in the right panel. Right panel: The sets of +selected parameters in the plots of (ζ, χ). The ζ, χ ∈ (0, 2), and ζ, χ ∈ (0, 10) for the top-right panel, and +bottom-right panel, respectively. +χ should be based on specific physical models, which are not involved in our study. Here, we +phenomenologically show the ORFs in Eq. (44) by letting |κ| ≡ 1 on the left panel of Fig. 8. +Because of the parameter space ζ, χ ∈ (0, ∞) in Eq. (31), it is not practical to consider to all the +shape of non-Gaussianity with |κ| = 1. In principle, if the deviation from Hellings-Downs curve is +completely ascribed to the non-linear corrections of the ORFs, one can fit the κ(k; ζ, χ) with real +data [28]. +V. +CONCLUSIONS AND DISCUSSIONS +In this paper, we extended the study on the non-linear corrections of the ORFs in the present +non-Gaussianity, in which self-interaction of gravity is first taken into considerations. Due to the +self-interaction of gravity, the linear order GWs can generate the non-linear one, which will change +the response of GW detectors. Based on the perturbed Einstein field equations for the second order +metric perturbations, and perturbed geodesic equations to the second order, we obtained non-linear +order timing residuals of pulsar timing, and compute the ORFs with non-linear corrections in the + +18 +PTA frequency band. +We considered the self-interaction of gravity through evaluating Einstein field equations in +vacuum for the second order metric perturbations. Namely, the space-time fluctuations are freely +propagating within the GW detectors described in Einstein’s gravity. +It is suggested that the +influence from secondary effect of GWs on the detectors could be different in the alternative theory +of gravity, or in the present of (dark) matter. +This paper showed that the leading order non-linear corrections for the ORFs come from the +three-point correlations of h(1) +k,ij. It is different from pioneers’ study that the correlations are from +the four-points functions [46]. 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Lett. 119, 261102 (2017), +arXiv:1707.06239 [astro-ph.IM]. + diff --git a/AdAyT4oBgHgl3EQfd_gY/content/tmp_files/load_file.txt b/AdAyT4oBgHgl3EQfd_gY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9f507559899617b29c60fa35ffe30662ff4924b7 --- /dev/null +++ b/AdAyT4oBgHgl3EQfd_gY/content/tmp_files/load_file.txt @@ -0,0 +1,1051 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf,len=1050 +page_content='Non-linear corrections of overlap reduction functions for pulsar timing arrays Qing-Hua Zhu∗ CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China and School of Physical Sciences, University of Chinese Academy of Sciences, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 19A Yuquan Road, Beijing 100049, China (Dated: January 3, 2023) The signals from international pulsar timing arrays have presented a hint of gravitational stochastic background in nHz band frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Further confirmation will be based on whether the signals follow the angular correlation curves formulated by the overlap reduction func- tions, known as Hellings-Downs curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' This paper investigates the non-linear corrections of overlap reduction functions in the present of non-Gaussianity, in which the self-interaction of gravity is first taken into considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Based on perturbed Einstein field equations for the second order metric perturbations, and perturbed geodesic equations to the second order, we obtain non-linear corrections for the timing residuals of pulsar timing, and theo- retically study corresponding overlap reduction functions for pulsar timing arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' There is order-one correction for the overlap reduction functions from the three-point correlations of gravitational waves, and thus the shapes of the overlap reduction functions with non-linear corrections can be distinguished from the Hellings-Downs curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' INTRODUCTION The stochastic gravitational wave background (SGWB) might contain lots of information of the Universe, since it can be originated from inflationary GWs [1–5], produced from early-time phase transitions [6–8], sourced by cosmic string [9–12], or formed by superpositions of unresolved individual GW sources such as binary systems [13–17], core-collapse supernovae [18–21], and de- formed rotating neutron stars [19, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In the 10Hz–1kHz frequency band, the ground-based GW detectors LIGO/Virgo/KAGRA have presented an upper limits of SGWBs [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In nHz fre- quency band, the international timing pulsar array projects (IPTA) [25–27] found and confirmed a common spectrum process from the pulsar-timing data sets, and suggested that further evidence for SGWBs might rely on its angular correlation signature [28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' The angular correlations of output of a pair of GW detectors can give characteristic signature of ∗ zhuqh@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='cn arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='00311v1 [gr-qc] 1 Jan 2023 2 GWs, which is formulated by overlap reduction functions (ORFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' For GW detector networks made by pulsar timing arrays (PTAs), the ORFs of GWs are known as Hellings-Downs curve for a pair of pulsars [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Motivated by the observation from IPTA on the angular correlations of SGWBs, it is necessary to clarify physical causes of deviations of the Hellings-Downs curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' It might come from the SGWBs beyond isotropy approximation [32, 33], polarized SGWBs [34–36], non-tensor modes from modified gravity [37–42], or simply a careful calculation on the pulsar terms [43–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Recently, there was also study on the non-linear corrections for PTAs from higher order effect of gravity [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' This correction for ORFs was shown to be order one in the present of non-Gaussianity of the GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In this paper, we will extend the study on the non-linear corrections of the ORFs, in which the self-interaction of gravity is taken into considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Due to the self-interaction of gravity, the linear order of GWs can generate the non-linear one even in vacuum, which can affect the propagation of light, thereby changing the response of GW detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' There was shown to be order-one corrections for the ORFs in the present of the non- Gaussianity of GWs [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' To obtain a solid derivation for GW detectors to the non-linear regime, we utilize perturbed Einstein field equations for evolution of metric perturbations, and perturbed geodesic equations for evaluating timing residuals of pulsar timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' We compute the ORFs with the non-linear corrections, and study its shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' II, the evolutions of metric perturbations to the second order are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' III, based on propagation of light formulated by the perturbed geodesic equations to the second order, we obtain timing residuals of pulsar timing with non-linear corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' IV, we compute ORFs with different shape of non-Gaussianity, and show its deviation from the Hellings-Downs curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' V, conclusions and discussions are summarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' PROPAGATION OF SECONDARY GRAVITATIONAL WAVE WITHIN PTAS From the theory of perturbations in general relativity, the first order GWs, the transverse- traceless modes of metric perturbations, should give rise to secondary effects from the higher order metric perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Because all the space-time fluctuations can affect propagation of light, there are inevitably corrections of response of GW detectors in non-linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In this section, to study the secondary effects of GWs on PTAs, we will firstly show how the non-linear order metric perturbations freely propagate in vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 3 The perturbed metric in Minkowski background is given by [47] ds2 = −dt2 + � δij(1 − ψ(2)) + ∂i∂jE(2) + 1 2∂iC(2) j + 1 2∂jC(2) i + h(1) ij + 1 2h(2) ij � dxidxj , (1) where δij is Kronecker symbol, h(1) ij is the first order transverse-traceless metric perturbation known as GWs, and h(2) ij , C(2) j , and ψ(2) are the second order tensor, vector, and scalar perturbations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Because GW detectors are reckon in the freely-falling frame, we have adopted Syn- chronous gauge for the perturbed metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' From Einstein field equations for the first order GWs, the motion of equations reduce to wave equations, h(1) ij ′′ − ∆h(1) ij = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (2) Thus, the solutions can be given by h(1) ij = � d3k (2π)3 ¯h(1) ij,ke−i(kt−k·x) , (3) where the h(1) ij,k is the Fourier mode of ¯h(1) ij , which contains the physical information before the GWs reaching the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Since the GW detectors can response to tensor, vector, and scalar modes of the metric pertur- bations, we should further consider all the second order metric perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Using the metric in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (1), the perturbed Einstein field equations in the second order take the form of [48, 49] h(2) ij ′′ − ∆h(2) ij = −Λab ij Sab , (4a) C(2) j ′′ = −Vab j Sab , (4b) −2ψ(2)′′ + ∆ � E(2)′′ + ψ(2)� = −S(Ψ),abSab , (4c) E(2)′′ + ψ(2) = −S(E),abSab , (4d) where Λab ij , Vab j , S(Ψ),ab and S(E),ab are helicity decomposition operators [47], and the source on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (4) is given by Sab = −2δcd∂0h(1) ac ∂0h(1) bd + 2h(1),cd∂c∂dh(1) ab − 2h(1),cd∂c∂ah(1) db −2h(1),cd∂c∂bh(1) da − 2∂dh(1) ac ∂ch(1) bd + 2δcd∂jh(1) ad ∂jh(1) bc + ∂ah(1),cd∂bh(1) cd + 2h(1),cd∂a∂bh(1) cd +δab �3 2∂0h(1) cd ∂0h(1),cd + 2h(1) cd ∂2 0h(1),cd − 2h(1) cd ∆h(1),cd + ∂jh(1) cd ∂dh(1),cj − 3 2∂jh(1) cd ∂jh(1),cd � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (5) 4 Differed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (2), the second metric perturbations are sourced the by the first order GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' By making use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (3) and (4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' we obtain the solutions in the form of ψ(2) (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' x) = � d3k (2π)3 �� d3p (2π)3 � ˆF ab ψ ¯Sab (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' p) e−i(|k−p|+p)t� eik·x � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (6a) E(2) (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' x) = � d3k (2π)3 �� d3p (2π)3 � ˆF ab E ¯Sab (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' p) e−i(|k−p|+p)t� eik·x � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (6b) C(2) j (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' x) = � d3k (2π)3 �� d3p (2π)3 � ˆF ab C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='j ¯Sab (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' p) e−i(|k−p|+p)t� eik·x � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (6c) h(2) ij (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' x) = � d3k (2π)3 �� d3p (2π)3 � ˆF ab h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ij ¯Sab (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' p) e−i(|k−p|+p)t� eik·x � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (6d) where ˆF ab h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ij (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' p) ≡ − Λab ij (k) k2 − (|k − p| + p)2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (7a) ˆF ab C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='j (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' p) ≡ Vab j (k) (|k − p| + p)2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (7b) ˆF ab ψ (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' p) ≡ −S(Ψ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab (k) + k2S(E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab (k) 2 (|k − p| + p)2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (7c) ˆF ab E (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' p) ≡ S(E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab (k) (|k − p| + p)2 − S(Ψ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab (k) + k2S(E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab (k) 2 (|k − p| + p)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (7d) The ¯Sab (k, p) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (6) is defined with ¯Sij (k, p) = fcdab ij (k, p) ¯h(1) cd,k−p¯h(1) ab,p , (8) where fbclm ij (k, p) = δblδcm (−3δij (p |k − p| − p · (k − p)) + pi(kj − 2pj) + ki(pj − 2kj)) +2 � pb(−δl iδm j pc + δcm(δl jpi + δl ipj + δij(pl − kl)) +δb j � δcm � δl i (p |k − p| − p · (k − p)) + ki(kl − pl) + pi(pl − kl) � + δl ipc(km − pm) � +δb i � δcm � δl j (p |k − p| − p · (k − p)) + kj(kl − pl) + pj(pl − kl) � + δl jpc(km − pm) � −δb i δc j(klkm − 2klpm + plpm) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (9) The ¯Sab (k, p) is derived from the source in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (5) via Sab (t, x) = � d3k (2π)3 Sab (t, k, p) eik·x = � d3k (2π)3 ¯Sab (k, p) e−i(|k−p|+p)t+ik·x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (10) Note that the Sab (t, k, p) is the Fourier mode of the source, and the ¯Sab (k, p) is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Here, as shown in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (6), the explicit solutions of perturbations ψ(2), E(2), C(2) j and h(2) ij can be obtained based on the known h(1) ij in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 5 We consider the non-linear corrections for GW detectors originated from the gravity self- interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In this case, all the second order perturbations are generated by the first order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In fact, there could be second order perturbations that are independent of the h(1) ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Because the response of the second order perturbations of this type to the GWs detector has nothing different from the response for the first order GWs, we did not consider this situation in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' PERTURBED GEODESIC EQUATIONS The space-time fluctuations can affect time of arrivals of radio beams from a pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Thus, via monitoring pulsar timing, the PTA observations can reflect the space-time fluctuations over the Universe in principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' To formulate it, and extend it to the non-linear regime, we will calculate the perturbed geodesic equations to the second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Based on geodesic equations in Minkowski background, namely, P µ∂µP ν = 0 , (11) one can obtain the 4-momentum of a light ray, P µ = P 0(1, −ˆnj) , (12) where ˆnj is a constant unit vector, and can be used to locate a pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' By making use of above 4-momentums, one can obtain trajectories of light rays from a pulsar to the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' The pulsar emits a radio beam at the event (t − L, Lˆnj), and the beam is detected on the earth at the event of (t, 0), where the L is distance between the pulsar and the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' From the first order perturbed geodesic equations, 0 = δP µ∂µP ν + P µ∂µδP ν + gνρ � ∂µh(1) λρ − 1 2∂ρh(1) µλ � P µP λ , (13) we can evaluate it by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (12), namely, (∂0 − ˆn · ∂) �δP 0 P 0 � = −1 2 ˆnaˆnb∂0h(1) ab , (14a) (∂0 − ˆn · ∂) �δP j P 0 � = δjbˆna � ∂0h(1) ab − ˆnc � ∂ch(1) ab − 1 2∂bh(1) ac �� , (14b) where δP µ is the first order perturbed 4-momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' The temporal and spatial components of the above perturbed geodesic equations take different forms, because the time components of h(1) µν in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (13) vanish in the Synchronous gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (14), the solutions in Fourier space take 6 the form of δP 0 k P 0 ≡ K0,ab (k, ˆn) ¯h(1) ab,ke−ikt = − 1 2(1 + ˆn · ˆk) ˆnaˆnb¯h(1) ab,ke−ikt , (15a) δP j k P 0 ≡ Kj,ab (k, ˆn) ¯h(1) ab,ke−ikt = � δjbˆna − ˆkjˆnaˆnb 2(1 + ˆn · ˆk) � ¯h(1) ab,ke−ikt , (15b) where the k is wave number, and the ˆk describes propagation direction of the first order GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' The results shown in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (15) are the basis of GW detectors, and astrometric detection with Gaia [31, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In the non-linear regime,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' we extend the calculations to the second order perturbed geodesic equations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 0 = δ2P µ∂µP ν + 2δP µ∂µδP ν + P µ∂µδ2P ν + 2gνρP µδP λ(∂µh(1) λρ − ∂ρh(1) µλ + ∂λh(1) µρ ) +P µP λ � gνρ � ∂λδg(2) µρ − 1 2∂ρδg(2) µλ � + gνκgρωh(1) ωκ(∂ρh(1) µλ − 2∂λh(1) µρ ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (16) where where δ2P µ is the second order perturbed 4-momentum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' and the second order metric per- turbations in Synchronous gauge is δg(2) 00 = δg(2) i0 = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (17a) δg(2) ij = −2δijψ(2) + 2∂i∂jE(2) + ∂iC(2) j + ∂jC(2) i + h(2) ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (17b) To obtain the timing residuals to the second order, we calculate temporal components of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (16) by making use of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (12) and (17), namely, (∂0 − ˆn · ∂) �δ2P 0 P 0 � = −1 2 ˆnaˆnb∂0δg(2) ab − 2 �δP µ P 0 � ∂µ �δP 0 P 0 � + 2ˆna �δP b P 0 � ∂0h(1) ab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (18) Differed from the linearized equations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (14a), the second and the third terms on the right hand side of above equation should be attributed to the non-linear contributions from perturbed geodesic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' It was partly considered in previous study [46], in which the non-linear corrections were obtained by expanding the proper time to the second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' By making use of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (15), the solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (18) can be obtained,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' δ2P 0 k P 0 = � d3q (2π)3 � Fcd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q) ¯h(1) cd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='k−q¯h(1) ab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='qe−i(|k−q|+q)t� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (19) where Fcd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆn) ≡ − 1 |k − q| + q + ˆn · k � �Fcd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab L (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q) + |k − q| + q 2 � ∗=ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='h Fcd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab ∗ (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q) � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (20) 7 and Fcd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab L (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q) ≡ (|k − q| + q) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='cd k−qK0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab q − qjKj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='cd k−qK0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' −(kj − qj)K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='cd k−qKj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab q − qˆnaKb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='cd k−q − |k − q| ˆncKd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (21a) Fcd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab ψ (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q) ≡ −2 ˆF ij ψ (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q) fcdab ij (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (21b) Fcd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab E (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q) ≡ −2(ˆn · k)2 ˆF ij E (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q) fcdab ij (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (21c) Fcd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab C (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q) ≡ iˆnj(ˆn · k) ˆF ml C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='j (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q) fcdab ml (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (21d) Fcd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab h (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q) ≡ ˆniˆnj ˆF ml h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ij (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q) fcdab ml (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (21e) where Kµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab k has been defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (15) and ˆF ij ψ (k, q), ˆF ij E (k, q), ˆF ij C (k, q) and ˆF ij h (k, q) have been given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' The Fcd,ab L (k, q) is derived from the second and the third terms of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (18), and the Fcd,ab ψ (k, q), Fcd,ab E (k, q), Fcd,ab C (k, q), and Fcd,ab h (k, q) are obtained by the solving the motion of equations of second order scalar, vector, and tensor perturbations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' The difference of the time of arrivals can be quantified by the redshift caused by GWs fluctua- tions, namely, z = uµ ˜P µ|obs uµ ˜P µ|src , (22) where ˜P µ(≡ P µ + δP µ + 1 2δ2P µ + O(3)) is the total 4-momentum of a radio beam from a pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' For comoving observers in Synchronous gauge uµ = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' we obtain the redshift and its fluctuations as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' z(0) = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (23a) z(1) = δP 0 obs P 0 obs − δP 0 src δP 0src ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (23b) z(2) = δ2P 0 obs P 0 obs − δ2P 0 src P 0src − 2 �δP 0 src P 0src � �δP 0 obs P 0 obs − δP 0 src δP 0src � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (23c) where the perturbed 4-momentums can be given by δ(n)P 0/P 0 = � d3k (2π)3 � (δ(n)P 0 k/P 0)eik·x� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' and the expressions of δ(n)P 0 k/P 0 have been given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (15) and (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (12), we have known the events of a pulsar emitting a radio beam tsrc = t − L and xj src = Lˆnj, and the events of the beam reaching the earth tobs = t and xj obs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Therefore, the redshifts and its fluctuations in 8 Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (23) can be rewritten in the form of z(1) = � d3k (2π)3 � K0,ab (k, ˆn) ¯h(1) ab,ke−ikt(1 − eikL(1+ˆn·ˆk)) � , (24a) z(2) = � d3kd3q (2π)6 � Fcd,ab (k, q, ˆn) ¯h(1) cd,k−q¯h(1) ab,qe−i(|k−q|+q)t � 1 − eiL(|k−q|+q+ˆn·k)� +K0,cd (k, ˆn) K0,ab (q, ˆn) ¯h(1) cd,k¯h(1) ab,qe−i(k+q)t ×1 2 � eiqL(1+ˆn·ˆq)(1 − eikL(1+ˆn·ˆk)) + eikL(1+ˆn·ˆk)(1 − eiqL(1+ˆn·ˆq)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (24b) In practical, the observables are the timing residuals of pulsar timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' It can be obtained by integration of the redshifts in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (24) over observation duration t, namely, R(n)(t) = � t 0 z(n)(¯t)d¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Thus, one can obtain expressions of the timing residuals in the form of R(1) = � d3k (2π)3 � 1 ikK0,ab (k, ˆn) ¯h(1) ab,k(1 − e−ikt)(1 − eikL(1+ˆn·ˆk)) � , (25) R(2) = � d3kd3q (2π)6 � 1 i (|k − q| + q)Fcd,ab (k, q, ˆn) ¯h(1) cd,k−q¯h(1) ab,q(1 − e−i(|k−q|+q)t) � 1 − eiL(|k−q|+q+ˆn·k)� + 1 i(k + q)K0,cd (k, ˆn) K0,ab (q, ˆn) ¯h(1) cd,k¯h(1) ab,q(1 − e−i(k+q)t) ×1 2 � eiqL(1+ˆn·ˆq)(1 − eikL(1+ˆn·ˆk)) + eikL(1+ˆn·ˆk)(1 − eiqL(1+ˆn·ˆq)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (26) Due to kt ≪ 1 for the nHz band PTAs, the timing residuals can be expanded as kt → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In this approximation, the leading-order timing residuals reduce to R(n) = tz(n)|t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' It indicates that the corrections of the outputs (timing residuals) of PTAs can be obtained via the correlations of redshifts, namely, ⟨R(x)R(x′)⟩ = t2⟨z(x)z(x′)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' SPATIAL CORRELATIONS AND OVERLAP REDUCTION FUNCTIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Non-linear correction of the correlations and non-Gaussianity The timing residuals can reflect physical information of SGWBs in the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Thus, it also should be studied statistically, due to stochastic nature of the SGWBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' The spatial correlations of the timing residuals, as mentioned above, are proportional to spatial correlations of the redshifts ⟨z(x)z(x′)⟩, where the x and x′ can represent the locations of pulsar pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' For illustration, we let zα ≡ z(x) and zβ ≡ z(x′) in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' To calculate the correlations order by order, we expand correlations for the redshifts zα in the form of ⟨zαzβ⟩ = ⟨z(1) α z(1) β ⟩ + 1 2(⟨z(1) α z(2) β ⟩ + ⟨z(2) α z(1) β ⟩) + O(4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (27) 9 For PTAs, the angular correlation derived from ⟨z(1) α z(1) β ⟩ is known as Hellings-Downs curve [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In this section, we will extend the spatial correlations to the non-linear regime with ⟨z(1) α z(2) β ⟩ and ⟨z(2) α z(1) β ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' The purpose of PTAs is extracting the physical information of h(1) ij based on observation of the timing residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Since z(1) ∝ h(1) and z(2) ∝ (h(1))2 shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (24), the ⟨z(1) α z(1) β ⟩ and ⟨z(1) α z(2) β ⟩ could encode two-point and three-point correlations of h(1) ij , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Here, we strict our study to the isotropic and unpolarized GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In this case, the two-point correlations of hλ k in Fourier space can be given by � h(1),λ1 k1 h(1),λ2 k2 � = (2π)3δ (k1 + k2) δλ1λ2P(k2) , (28) where the P(k) is power spectrum, the λ∗(= +, ×) is polarization index, and the Kronecker symbol δλ1λ2 indicates that h(1),λ k is unpolarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In the non-linear regime, the three-point correlations in Fourier space can be given by � h(1),λ1 k1 h(1),λ2 k2 h(1),λ3 k3 � = (2π)6δ(3) (k1 + k2 + k3) Bλ1λ2λ3(k1, k2, k3) , (29) where the Bλ1λ2λ3(k1, k2, k3) is bi-spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' It does not vanish due to the non-Gaussianity of SGWBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' For the unpolarized GWs, i) different polarizations of h(1),λ have no correlations, namely, Bλ1λ2λ3(k1, k2, k3) does not vanish, only if λ1 = λ2 = λ3, and ii) different components of the bi- spectrum are equally weighted, namely, the B+++(k1, k2, k3) = B×××(k1, k2, k3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' To quantifying shapes of the non-Gaussianity, we follow the parameterization scenario used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' With all the above assumptions, the bi-spectrum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (29) can be given in the form of Bλ1λ2λ3(k1, k2, k3) = Hλ1λ2λ3(k3)P(k3)δ(k1 − χk3)δ(k2 − ζk3) , (30) where we have H+++(k3) = H×××(k3) ≡ κ(k3) due to the unpolarized GWs, the P(k) is the power spectrum defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (28), and the ζ and χ are the dimensionless quantities formulating the shape of bi-spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In principle, the shape of bi-spectrum quantified by κ, ζ and χ should be given based on generation mechanism of GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Here, we did not involve any physical models for the generation mechanism, but utilize the non-Gaussianity based on a phenomenological param- eterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Because of k1 + k2 + k3 = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (29), the parameters ζ and χ should satisfy the relations, χ + ζ ⩾ 1 , and |χ − ζ| ⩽ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (31) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 1, we show scheme diagram for the shape of the bi-spectrum, in which the ζ and χ are defined with ζ ≡ BC/AB and χ ≡ CA/AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 2, we show the parameter space (ζ, χ), 10 k3 k1 k2 A B C Figure 1: The scheme diagram of the parameterized bi-spectrum defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 ζ χ ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='60 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='60 ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='87 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='87 ζ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='1 χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='1 ζ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='4 χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='4 ζ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='7 χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='7 ζ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='9 χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='9 ζ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 ζ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='3 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='87 ζ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='4 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='87 ζ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='9 χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='1 Figure 2: Left panel: parameter space of (ζ, χ) for the parameterized non-Gaussianity of hλ ij,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' The points locate the available domain of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' The dashed curve is formulated by ζ = χ, and the dotted curve is formulated by 1 2ζχ � 1 − � ζ2+χ2−1 2ζχ �2 = √ 3 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Right panel: The shape of non-Gaussianity for selected parameters ζ and χ on the dashed and dotted curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' where the points locate the domain of available parameters in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Here, every single point represents a distinguishable shape of non-Gaussianity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' As shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 2, for example, the orange points on the dashed line represent isosceles triangles, and the green points on the dotted curve represent triangles with the same height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Overlap reduction functions of PTAs in second order By making use of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (24a) and (28), the linear-order correlations of the redshifts for a pulsar pair can be given by ⟨z(1) α z(1) β ⟩ = � k2dk 2π2 P(k) � dΩ 4π � K0,ab (k, ˆnα) K0,cd (k, ˆnβ) eλ ab(ˆk)eλ cd(ˆk) ×(1 − eikLα(1+ˆnα·ˆk))(1 − e−ikLβ(1+ˆnβ·ˆk)) , � (32) 11 where the eλ cd(ˆk) is polarization tensor for h(1) ij,k, the θαβ ≡ cos−1(ˆnα · ˆnβ), and the P(k) is the power spectrum defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' The ORFs describe angular correlations of outputs of the GW detectors, and can be obtained via surface integrals over the unit sphere ˆk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Rewriting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (32) in the form of ⟨z(1) α z(1) β ⟩ ≡ � k2dk 2π2 P(k)Γ(2)(k, θαβ) , (33) one can read the ORFs, Γ(2)(k, θab) = � dΩ 4π � K0,ab (k, ˆnα) K0,cd (k, ˆnβ) eλ ab(ˆk)eλ cd(ˆk) ×(1 − eikLα(1+ˆnα·ˆk))(1 − e−ikLβ(1+ˆnβ·ˆk)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (34) Because of the approximation kL ≫ 1 for the known frequency band and arms length of PTAs, above ORFs would reduce to Hellings-Downs curve [31], namely, ΓHD(θab) ≡ Γ(2)(k, θab)|kLα≫1,kLβ≫1 = � dΩ 4π � K0,ab (k, ˆnα) K0,cd (k, ˆnβ) eλ ab(ˆk)eλ cd(ˆk) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (35) Here, the oscillation parts in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (34) is suppressed by the factor (kL)−1, and thus can be neglected for PTAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Besides, the ORFs without the approximation were also studied [43–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' To the second order, we further compute non-linear corrections for the correlations of redshift in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ⟨z(1) α z(2) β ⟩ = � d3kd3k′d3q′ (2π)12 �� h(1) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='mlh(1) k′−q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='cd ∗h(1) q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab ∗� K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ml (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnα) Fcd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab � k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnβ � ×e−ikt(1 − eikLα(1+ˆnα·ˆk))eit(|k′−q′|+q′) � 1 − e−iLβ(|k′−q′|+q′+ˆnβ·k′)� +1 2 � h(1) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='mlh(1) k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='cd ∗h(1) q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab ∗� K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ml (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnα) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='cd � k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnβ � K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab � q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnβ � e−ikt(1 − eikLα(1+ˆnα·ˆk))ei(k′+q′)t × � e−iq′Lβ(1+ˆnβ·ˆq′)(1 − e−ik′Lβ(1+ˆnβ·ˆk′)) + e−iq′Lβ(1+ˆnβ·ˆq′)(1 − e−ik′Lβ(1+ˆnβ·ˆk′)) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (36) where above three-point correlations of h(1) ij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='k can be written in terms of polarization components h(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='λ k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' � h(1) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='cdh(1) ab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ph(1) q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ml � = eλ1 cd(ˆk)eλ2 ab(ˆp)eλ3 ml(ˆq) � h(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='λ1 k h(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='λ2 p h(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='λ3 q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (37) 12 By making use of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (29) and (30), the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (34) is evaluated to be ⟨z(1) α z(2) β ⟩ + ⟨z(2) α z(1) β ⟩ = � d3k (2π)3 � dφq 2π � eλ1 ml(ˆk)eλ2 cd( � k − q)eλ3 ab(ˆq)Hλ1λ2λ3(k)P(k) × �� K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ml (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnα) Fcd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnβ) (1 − eikLα(1+ˆnα·ˆk))(1 − e−ikLβ(χ+ζ+ˆnβ·ˆk)) +1 2K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ml (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnα) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='cd (k − q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnβ) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnβ) ×e−iζkLβ(1+ˆnβ·ˆq)(1 − eikLα(1+ˆnα·ˆk))(1 − e−iLβ(χk+ˆnβ·(k−q))) +1 2K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ml (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnα) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='cd (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnβ) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab (k − q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnβ) ×e−iLβ(χk+ˆnβ·(k−q))(1 − eikLα(1+ˆnα·ˆk))(1 − e−iζkLβ(1+ˆnβ·ˆq))) � e−ikt(1−χ−ζ) + � Fcd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnα) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ml (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnβ) (1 − eikLα(ζ+χ+ˆnα·ˆk))(1 − e−ikLβ(1+ˆnβ·ˆk)) +1 2Kml (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnβ) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='cd (k − q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnα) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnα) ×eiζkLα(1+ˆnα·ˆq)(1 − eiLα(χk+ˆnα·(k−q)))(1 − e−ikLβ(1+ˆnβ·ˆk)) +1 2Kml (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnβ) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='cd (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnα) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab (k − q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnα) ×eiLα(χk+ˆnα·(k−q))(1 − eiζkLα(1+ˆnα·ˆq))(1 − e−ikLβ(1+ˆnβ·ˆk)) � eikt(1−χ−ζ)�� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (38) where the momentums are give by q = �1 + ζ2 − χ2 2 � k + � ((ζ + χ)2 − 1)(1 − (ζ − χ)2) 2 k (cos φqu + sin φqυ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (39a) ˆq = 1 + ζ2 − χ2 2ζ ˆk + � ((ζ + χ)2 − 1)(1 − (ζ − χ)2) 2ζ (cos φqu + sin φqυ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (39b) � k − q = χ2 − ζ2 + 1 2χ ˆk − � ((ζ + χ)2 − 1)(1 − (ζ − χ)2) 2χ (cos φqu + sin φqυ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (39c) Here, the u and υ represent polarization vectors with respect to the ˆk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (39a), one can verify the relations of q = ζk and |k − q| = χk shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (38), it is found that the three-point correlations are proportional to e±ikt(1−χ−ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' It indicates that the values of correlations would oscillate with time around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In practical, due to kt ≪ 1 for nHz band PTAs, we here can let e±ikt(1−χ−ζ) ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Similarly, the correlations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (38) can be rewritten in the form of ⟨z(1) α z(2) β ⟩ + ⟨z(2) α z(1) β ⟩ = � k2dk 2π2 P(k)Γ(3)(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' θab) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (40) 13 where the ORFs in the non-linear order are given by Γ(3)(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' θαβ) = � dΩ 4π � dφq 2π � eλ1 ml(ˆk)eλ2 cd( � k − q)eλ3 ab(ˆq)Hλ1λ2λ3(k)P(k) × � K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ml (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnα) Fcd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnβ) (1 − eikLα(1+ˆnα·ˆk))(1 − e−ikLβ(χ+ζ+ˆnβ·ˆk)) +Fcd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnα) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ml (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnβ) (1 − eikLα(ζ+χ+ˆnα·ˆk))(1 − e−ikLβ(1+ˆnβ·ˆk)) +1 2K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ml (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnα) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='cd (k − q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnβ) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnβ) ×e−iζkLβ(1+ˆnβ·ˆq)(1 − eikLα(1+ˆnα·ˆk))(1 − e−iLβ(χk+ˆnβ·(k−q))) +1 2K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ml (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnα) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='cd (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnβ) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab (k − q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnβ) ×e−iLβ(χk+ˆnβ·(k−q))(1 − eikLα(1+ˆnα·ˆk))(1 − e−iζkLβ(1+ˆnβ·ˆq))) +1 2Kml (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnβ) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='cd (k − q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnα) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnα) ×eiζkLα(1+ˆnα·ˆq)(1 − eiLα(χk+ˆnα·(k−q)))(1 − e−ikLβ(1+ˆnβ·ˆk)) +1 2Kml (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnβ) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='cd (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnα) K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='ab (k − q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ˆnα) ×eiLα(χk+ˆnα·(k−q))(1 − eiζkLα(1+ˆnα·ˆq))(1 − e−ikLβ(1+ˆnβ·ˆk)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (41) The expression of Hλ1λ2λ3(k) has been given in the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Since the oscillation parts in the integration is suppressed by the factor (kL)−1 for PTAs, we also adopt kLα ≫ 1 and kLβ ≫ 1 for evaluating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Namely, the ORFs can be simplified in the form of Γnl(k, θαβ) ≡ Γ(3)(k, θab)|kLα≫1,kLβ≫1 = κ � dΩ 4π � dφq 2π �� e+ ml(ˆk)e+ cd( � k − q)e+ ab(ˆq) + e× ml(ˆk)e× cd( � k − q)e× ab(ˆq) � × � K0,ml (k, ˆnα) Fcd,ab (k, q, ˆnβ) + Fcd,ab (k, q, ˆnα) K0,ml (k, ˆnβ) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (42) In the following, we would show the results of Γnl(k, θab) with selected parameters ζ and χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 3, we show the ORFs over the κ as function of parameter (ζ, χ) for given angle θαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' It is found that the values of Γln/κ tend to be zero for large ζ and χ, and to be larger in the case of ζ + χ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 4, we show the ORFs for select parameters given in right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' It is confirmed that there is a larger value of ORFs for the parameters ζ, χ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Differed from the ORFs in the linear order, there are three extreme points in the curves of the non-linear ORFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In order to clarify the extreme cases, such as ζ − χ = 1 or ζ + χ = 1, we show the ORFs as function of θαβ for ζ − χ → 1 and ζ + χ → 1 in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 7 and 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 7, the values of ORFs tend to be zero as ζ − χ → 1, and these ORFs have the same zero of the Γ(k, θαβ) with respect to θαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 6, the values of ORFs tend to be larger, and the numbers of extreme points are reduced as ζ + χ → 1, which is different from the results shown in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 4 for a larger ζ + χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In the case of ζ + χ = 1, we can further simplify the expression of ORFs as 14 Figure 3: Non-linear ORFs over the κ as function of parameter (ζ, χ) for θαβ = 0, π/4, π/2, π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Γnl(k, θαβ) = κ � dΩ 4π �� e+ ml(ˆk)e+ cd(�k)e+ ab(ˆk) + e× ml(ˆk)e× cd(�k)e× ab(ˆk) � × � K0,ml (k, ˆnα) Fcd,ab (k, ζk, ˆnβ) + Fcd,ab (k, ζk, ˆnα) K0,ml (k, ˆnβ) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (43) Here, the integration over the angle φq simply gives 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 7, we show the ORFs in the case of ζ + χ = 1 for different ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' It is found that the values of ORFs get smaller as ζ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In principle, the parameter κ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (42) depends on the wave number k, and the shape of non-Gaussianity quantified by ζ and χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' There seems no reason that the non-linear corrections come from the non-Gaussianity with one of the available parameters (ζ, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' For a general case, the total ORFs to the non-linear order should be the sum of all the shape of non-Gaussianity weighted by parameter κ, namely, Γ(k, θαβ) = ΓHD(θαβ) + 1 2 � ζ,χ Γnl(k, θαβ)∆σ , (44) 15 ζ \uf7d9 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='60 ζ \uf7d9 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='87 ζ \uf7d9 χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='1 ζ \uf7d9 χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='4 ζ \uf7d9 χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='7 ζ \uf7d9 χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='20 θαβ Γnl\uf000θαβ\uf006 ζ \uf7d9 χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 ζ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='3 , χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='87 ζ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='4 , χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='87 ζ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='9 , χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='08 θαβ Γnl\uf000θαβ\uf006 Figure 4: Non-linear ORFs for selected parameters ζ and χ shown in right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 2, and κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Left panel: the non-Gaussianity in the shape of isosceles triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Right panel: the non-Gaussianity in the shape of the triangles with the same height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='500 , χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='40 ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='500 , χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='47 ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='500 , χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='005 θαβ Γnl\uf000θαβ\uf006 ζ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='00 , χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='90 ζ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='00 , χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='97 ζ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='00 , χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0020 θαβ Γnl\uf000θαβ\uf006 ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='500 χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='40 ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='500 χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='47 ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='500 χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='49 ζ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='00 χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='90 ζ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='00 χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='97 ζ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='00 χ \uf7d9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='99 Figure 5: Non-linear ORFs for selected parameters ζ − χ → 1, and κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 16 ζ \uf7d9 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='600 ζ \uf7d9 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='533 ζ \uf7d9 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='510 ζ \uf7d9 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='501 ζ \uf7d9 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='30 θαβ Γnl\uf000θαβ\uf006 ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='600 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='600 ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='533 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='533 ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='510 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='510 ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='501 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='501 ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='500 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='500 Figure 6: Non-linear ORFs for selected parameters ζ + χ → 1, and κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ζ \uf7d9 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='50 ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='75 , χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='25 ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='90 , χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='10 ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='99 , χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='35 θαβ Γnl\uf000θαβ\uf006 ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='500 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='500 ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='750 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='250 ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='900 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='100 ζ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='990 χ \uf7d9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0100 Figure 7: Non-linear ORFs for selected parameters ζ + χ = 1, ζ → 1, and κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' where ∆σ is the size of grids of points in the parameter space (ζ, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' For example, we have ∆σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='018 for the grids in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Here, the Γnl(k, θαβ) is proportional to the parameter κ(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ζ, χ) shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' The dependence of Γ(k, θαβ) on the parameters ζ and 17 ΓHD(θαβ) Γtot(θαβ) ζ,χ<2,κ=1 Γtot(θαβ) ζ,χ<10,κ=1 Γtot(θαβ) ζ,χ<2,κ=-1 Γtot(θαβ) ζ,χ<10,κ=-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='4 θαβ Γ\uf000θαβ\uf006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='0 ζ χ 0 2 4 6 8 10 0 2 4 6 8 10 ζ χ Figure 8: Left panel: total ORFs with non-linear corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' The non-linear ORFs are sum of the selected parameters κ = ±1 and (ζ, χ) shown with the points in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Right panel: The sets of selected parameters in the plots of (ζ, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' The ζ, χ ∈ (0, 2), and ζ, χ ∈ (0, 10) for the top-right panel, and bottom-right panel, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' χ should be based on specific physical models, which are not involved in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Here, we phenomenologically show the ORFs in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (44) by letting |κ| ≡ 1 on the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Because of the parameter space ζ, χ ∈ (0, ∞) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (31), it is not practical to consider to all the shape of non-Gaussianity with |κ| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' In principle, if the deviation from Hellings-Downs curve is completely ascribed to the non-linear corrections of the ORFs, one can fit the κ(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' ζ, χ) with real data [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' CONCLUSIONS AND DISCUSSIONS In this paper, we extended the study on the non-linear corrections of the ORFs in the present non-Gaussianity, in which self-interaction of gravity is first taken into considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Due to the self-interaction of gravity, the linear order GWs can generate the non-linear one, which will change the response of GW detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Based on the perturbed Einstein field equations for the second order metric perturbations, and perturbed geodesic equations to the second order, we obtained non-linear order timing residuals of pulsar timing, and compute the ORFs with non-linear corrections in the 18 PTA frequency band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' We considered the self-interaction of gravity through evaluating Einstein field equations in vacuum for the second order metric perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Namely, the space-time fluctuations are freely propagating within the GW detectors described in Einstein’s gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' It is suggested that the influence from secondary effect of GWs on the detectors could be different in the alternative theory of gravity, or in the present of (dark) matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' This paper showed that the leading order non-linear corrections for the ORFs come from the three-point correlations of h(1) k,ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' It is different from pioneers’ study that the correlations are from the four-points functions [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' It is because the contributions from three-point correlations in our study are all derived from the self-interaction of gravity shown in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' (21a)–(21e), which was not considered in pioneers’ study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' The author thanks Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Qing-Guo Huang and Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Sai Wang for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Grishchuk, Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content=' Eksp.' metadata={'source': 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arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='06239 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} +page_content='IM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAyT4oBgHgl3EQfd_gY/content/2301.00311v1.pdf'} diff --git a/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf b/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6179d7344c0b74146cebddaff51f2cb18e62274c --- /dev/null +++ b/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9f5c9c9a4571ac4941b2f54ced60874e93e685a02df5ab9225f4ebc80e58c376 +size 2777947 diff --git a/AdE3T4oBgHgl3EQfsgtu/vector_store/index.faiss b/AdE3T4oBgHgl3EQfsgtu/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..700faf988c6da735de66994ea4cf0abb5e2e8c75 --- /dev/null +++ b/AdE3T4oBgHgl3EQfsgtu/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e8a69ef6538b31ae80c8b70f747e0d8f6c936404c05081c73e0dfb5673328df5 +size 1966125 diff --git a/AdE5T4oBgHgl3EQfSg_z/content/tmp_files/2301.05530v1.pdf.txt b/AdE5T4oBgHgl3EQfSg_z/content/tmp_files/2301.05530v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..671e4c70a2bc9397ad162e322abfaa6b3d7c975b --- /dev/null +++ b/AdE5T4oBgHgl3EQfSg_z/content/tmp_files/2301.05530v1.pdf.txt @@ -0,0 +1,1426 @@ +1 +An RTL Implementation of the Data Encryption +Standard (DES) +Ruby Kumari, Student Member, IEEE, Jai Gopal Pandey, Senior Member,IEEE, and Abhijit Karmakar, +Abstract—Data Encryption Standard (DES) is based on the +Feistel block cipher, developed in 1971 by IBM cryptography +researcher Horst Feistel. DES uses 16 rounds of the Feistel +structure. But with the changes in recent years, the internet is +starting to be used more to connect devices to each other. These +devices can range from powerful computing devices, such as +desktop computers and tablets, to resource constrained devices, +When it comes to these constrained devices, using a different key +for each round cryptography algorithms fail to provide necessary +security and performance. +Index Terms—Keywords: Cryptography, DES , SDES, Feistel +block Cipher. +I. INTRODUCTION +T +HIS Security is a prevalent concern in information and +data systems of all types. Historically, military and na- +tional security issues drove the need for secure communica- +tions. Recently, security issues have pervaded the business and +private sectors. E-commerce has driven the need for secure +internet communications. Many businesses have fire- walls to +protect internal corporate information from competitors. In the +private sector, personal privacy is a growing concern. Products +are available to scramble both e-mail and telephone communi- +cations. One means of providing security in communications +is through encryption. By encryption, data is transformed in a +way that it is rendered unrecognizable. Only by decryption can +this data be recovered. Ostensibly, the process of decryption +can only be performed correctly by the intended recipients. +The validity of this assertion determines the “strength” or +“security” of the encryption scheme. Many communications +products incorporate encryption as a feature to provide se- +curity. This application report studies the implementation of +one of the most historically famous and widely implemented +encryption algorithms, the Data Encryption Standard (DES). +The Data Encryption Standard is a symmetric-key block +cipher published by the National Institute of Standards and +Technology (NIST) for the encryption of digital data. DES +is probably one of the best-known cryptographic algorithms +and has been widely used since its introduction in 1976. +Although its short key length of 56 bits makes it too inse- +1 cure for applications, it has been highly influential in the +advancement of cryptography. The DES must be stronger than +the other cryptosystems in security. The goal of this project is +to develop a python code for SDES and DES. Before building +our design, we need an overview of cryptography, followed +by a description of the DES algorithm. +Manuscript received December 19, 2022 +1) Overview of Cryptography: Cryptography is a type of +rule or technique by which private or sensitive information is +secured from the public or other members. It plays a vital role +in preserving data integrity, confidentiality and user privacy. +An encryption algorithm can convert imported essential data +to encrypted data (plaintext into cipher- text). This data +would be of no use to a person that does not possess the +encryption key. The use of Cryptography in passwords is a +very famous example. Cryptography is based on mathematical +theory and some Computer Science principles. There are many +terminologies related to cryptography. Some terms are defined +below. +• Ciphertext: Conversion of plain text into intelligible text +is called ciphertext. +• Cipher: It is a technique of encryption and decryption. +Critical and algorithms play vital role in this technique. +• Symmetric: It is a kind of cryptosystem. It uses same key +for encryption and decryption. It is faster than asymmetric. +• Asymmetric: It is also a kind of cryptosystem. It uses +a public key for the encryption and a private key for the +decryption of any message. +• Cryptanalysis: It studies cracking the encryption of the +algorithms. +A. Symmetric Ciphers Model +Symmetric-key (or private-key) encryption can be simply +illustrated with the schematic shown in Figure 1. +Fig. 1: Symmetric Cryptosystem model. +A symmetric encryption scheme has five main parts, that is, +• Encryption algorithm: The encryption algorithm per- +forms various substitutions and transformations on plain- +text. +• Secret key: The secret key is also input to the encryption +algorithm. The key is a value independent of the plaintext +and the algorithm. The algorithm will produce a different +arXiv:2301.05530v1 [cs.CR] 13 Jan 2023 + +Cryptanalyst +Message +Encryption +Decryption +Sources +Destination +Algorithm +Algorithm +Secure Channel +Key +Sources2 +output depending on the specific key. The exact substi- +tutions and transformations performed by the algorithm +depend on the key. +• Ciphertext: This is the scrambled message produced as +output. It depends on the plaintext and the secret key. +For a given message, two different keys will produce two +different ciphertexts. The ciphertext is an random stream +of data. +• Decryption algorithm: This is essentially the encryption +algorithm run in reverse. It takes the ciphertext and the +secret key and produces the original plaintext. +Alice and Bob want to communicate over an un-secure +channel, but Oscar is trying to read the message. So Alice and +Bob must use a cryptosystem to prevent Oscar from reading +the message. Let us take a closer look at the essential elements +of a symmetric encryption scheme using Figure 1. A source +produces a message in plaintext, X = [X1, X2, . . . , XM]. +The M elements of X are letters in some finite alphabet. +Traditionally, the alphabet usually consisted of t6 capital +letters. Nowadays, the binary alphabet 0, 1 is typically used. +For encryption, a key of the form K = [K1, K2, . . . . . .., KJ] +is generated. If the key is generated at the message source, +then it must also be provided to the destination using some +secure channel. Alternatively, a third party could generate key +and securely deliver it to both source and destination. The +encryption algorithm forms the ciphertext as given in 1. +Y = [Y 1, Y 2, ...., Y N] +(1) +with the message X and the encryption key K as it. We can +write this as given in 2. +Y = E(K, X) +(2) +This notation indicates that Y is produced by using en- +cryption algorithm E as a function of the plaintext X, with +the specific process determined by the value of the key K. +The intended receiver, in possession of the key, can invert the +transformation: X = D(K, Y ). An opponent, observing Y but +not having access to K or X, may attempt to recover X or K +or both X and K. It is assumed that the opponent knows the +encryption (E) and decryption (D) algorithms. If the opponent +is interested in only this particular message, then the focus of +the effort is to recover X by generating a plaintext estimate +X. Often, however, the opponent is interested in being able +to read future messages as well, in which case an attempt is +made to recover K by generating an estimate K. +B. Simplified Data Encryption Standard +The S-DES encryption algorithm takes an 8-bit block of +plaintext and a 10-bit key as input and produces an 8-bit block +of ciphertext as output. The S-DES decryption algorithm takes +an 8- bit block of ciphertext and the same 10-bit key used to +produce that ciphertext as input and produces the original 8-bit +block of plaintext. Simplified DES (SDES) was designed for +educational purposes only, to help students learn about modern +cryptanalytic techniques [1]. SDES has similar properties and +structure as DES but has been simplified to make it much +easier to perform encryption and decryption by hand with +Fig. 2: Simplified DES (SDES) +Pencil and paper. Some people feel that learning SDES gives +insight into DES and other block ciphers, and insight into +various cryptanalytic attacks against them. +An adversary trying to interrupt two communicating parties +may have one of the four main goals: +1) Read the secret message. +2) Find the secret key, so that they can read all messages +encrypted with that key. +3) Modify the message sent by Alice and go unnoticed by +both parties. +4) Act like Alice and send a message to Bob, to make Bob +think he is communicating with Alice when in reality +he is communicating with the adversary. +In order to prevent an adversary from reaching his +goals, some security measures are Applied to cryptosystems, + +10-bit Key +4 +P10 +Encryption +Decryption +8-bit Plaintext +8-bit Plaintext +Shift +IP +IP-1 +P8 +K1 +Ki +Fk +Fk +Shift +SW +SW +P8 +K2 +K2 +Fk +Fk +IP +IP-1 +8-bit Ciphertext +8-bit Ciphertext3 +namely confidentiality, data integrity, authentication, and non- +repudiation. +1) Confidentiality means the transmitted message or infor- +mation is kept secret, and only the authorized parties +have the means to decipher the information. +2) Data integrity makes sure that the messages are not being +modified. This stops the adversary from reaching their +third goal. +3) Authentication helps Bob to correctly identify the sender +as Alice, thus stopping the adversary from posing as +Alice. +4) Non-repudiation prevents Alice from denying she sent +the message. +Cryptographic algorithms are gathered under two main +branches; symmetric algorithms and asymmetric algorithms. In +symmetric algorithms both Alice and Bob have the same key. +Since the communication channel is insecure, this key must +be previously decided on through secure ways. The encryption +and decryption keys are either the same, or very similar that +the decryption key can easily be derived from the encryption +key. But sometimes Alice and Bob cannot agree on a key +beforehand. They could be very far away from each other +and cannot get together to determine a secret key, and there +may not be a secure way for Alice to send Bob the secret +key. She cannot just send Bob a secret key through any open +channel, because an adversary can interrupt the channel and +get their hands on the key. Thus making the key useless. To +get around this problem asymmetric algorithms, usually called +public key algorithms, are used. In public key algorithms each +party has their key pairs, one public and one private key. As +can be understood from their names, private keys are kept +secret, and public keys can be known by everyone. The public +key is computed from the private key in a way that finding the +private key from the public key is infeasible. Alice encrypts +the message she wants to send using Bob’s public key. The +message can only be decrypted with the corresponding private +key, which only Bob has. Therefore Alice can send a secret +message even though they are far away and cannot decide on +a common key together [2]. +Further, the details of the DES cipher is given in the next +chpater [3]. +II. DATA ENCRYPTION STANDARD +Developed in 1974 by IBM in cooperation with the National +Securities Agency (NSA), DES has been the worldwide en- +cryption standard for more than 20 years. For these 20 years, +it has held up against cryptanalysis remarkably well and is still +secure against all but possibly the most powerful adversaries. +Because of its prevalence throughout the encryption market, +DES [4] is an excellent interoperability standard between +different encryption equipment. The predominant weakness of +DES is its 56-bit key which, more than sufficient for the time +period which it was developed [5], has become insufficient to +protect against brute-force attacks modern computers [6]. As +a result of the need for a greater encryption strength, DES +evolved into triple-DES [7]. +Fig. 3: Encryption and Decryption +III. DES ENCRYPTION +The Data Encryption Standard is a Feistel cipher. In which +round function consists of an expansion, a bitwise XOR- +operation XOR operation round key, an S-box layer and a +permutation [3]. In encryption n scheme, there are two inputs +to the encryption function [8]. the plaintext to be encrypted +and the key. In this case, the plaintext must be 64 bits in length +and the key is 56 bits in length [9]. +On the left-hand side of the figure, we can see that the +plaintext processing proceeds in three phases. First, the 64- +bit plaintext passes through an initial permutation (IP) that +rearranges the Bits to produce the permuted input [10]. This +is followed by a phase consisting of sixteen rounds of the same +function, which involves both permutation and substitution +functions. The output of the last (sixteenth) round consists of +64 bits that are a function of the input plaintext and the key. +The left and right halves of the output are swapped to produce +the pre-output. Finally, the pre-output is passed through a per- +mutation [IP −1], inverse of the initial permutation function, +to produce the make ciphertext. With the exception of initial +and final permutations. +On the right-hand portion of figure 6-5,6-a-bitkey is used. +Initially, the key is passed through a permutation function. +Then, for each of the sixteen rounds, a subkey (Ki) is produced +by the combination f.t Initially, the key is passed through a +permutation function. Then, for each of the sixteen rounds, a +subkey (Ki) is produced by the combination of a left. +A. Initial Permutation and Final Permutation +Each of these permutations takes a 64-bit input and per- +mutes them according to a predefined rule. These permutations +are keyless straight permutations that are the inverse of each +other. For example, in the initial permutation [IP], the 58th bit +in the input becomes the first bit in the output. Similarly, in the +final permutation [IP −1], the first bit in the input becomes the +58th bit in the output. In other words, if the rounds between +these two permutations do not exist, the 58th bit entering the +initial permutation is the same as the 58th bit leaving the final +permutation. The initial permutation is given in TABLE I +The final permutation is given in TABLE II. + +DES Reverse +DES Cipher +Cipher4 +Fig. 4: Structure of DES +Fig. 5: Initial and final permutation step in DES +IV. ROUNDS +DES uses 16 rounds. Each round of DES is a Feistel cipher. +Fig. 6 shows the internal structure of a single round. Again, +begin by focusing on the left-hand side of the diagram. The left +and right halves of each 64-bit intermediate value are treated +as separate 32-bit quantities, labeled L (left) and R (right). As +in the Feistel cipher, the overall processing at each round can +be summarized in the following formulas: +Li = Ri − 1 Ri = Li − 1 XOR FE(Ri − 1, Ki) +The round takes Li − 1 and Ri − 1 from the previous game +(or the initial permutation box) and creates Li and Ri, which +go to the next round (or final permutation box). +A. Initial Permutation +A single initial permutation is needed at the beginning of +the encryption process. IP is necessary on each block of 64 +bits in DES once the entire plaintext has been divided into +TABLE I: Initial Permutation +58 +50 +42 +34 +26 +18 +10 +2 +60 +52 +44 +36 +28 +20 +12 +4 +62 +54 +46 +38 +30 +22 +14 +6 +64 +56 +48 +40 +32 +24 +16 +8 +57 +49 +41 +33 +25 +17 +9 +1 +59 +51 +43 +35 +27 +19 +11 +3 +61 +53 +45 +37 +29 +21 +13 +5 +63 +55 +47 +39 +31 +23 +15 +7 +TABLE II: Final Permutation +40 +8 +48 +16 +56 +24 +64 +32 +39 +7 +47 +15 +55 +23 +63 +31 +38 +6 +46 +14 +54 +22 +62 +30 +37 +5 +45 +13 +53 +21 +61 +29 +36 +4 +44 +12 +52 +20 +60 +28 +35 +3 +53 +11 +51 +19 +59 +27 +34 +2 +42 +10 +50 +18 +58 +26 +33 +1 +41 +9 +49 +17 +57 +25 +such blocks. The transposition process goes through this initial +permutation. Only once, just before the first round, does the +first permutation appear. As seen in the Table I, it provides +decisions for how the IP transposition process has to go. It is +possible to claim for example, that the IP replaced the first +bit of the original plain-text block with the 58th bit of the +original plain-text block, the second bit with the 50th bit of +the original plain-text block, etc. This is nothing more than +bit shuffling with respect to the original plaintext block. +B. Expansion D-Box +Since Ri − 1 is a 32-bit input and KI is a 48-bit key, we +first need to expand Ri − 1 to 48 bits. Ri − 1 is divided into +8 4-bit sections. Each 4-bit section is then expanded to 6 bits. +For each section, input bits 1, 2, 3, and 4 are copied to output +bits 2, 3, 4, and 5, respectively. Output bit ‘1’ comes from bit +4 of the previous section; output bit 6 comes from bit 1 of the +next section. If sections 1 and 8 can be considered adjacent +sections, the same rule applies to bits 1 and 32. +The main part of DES is the DES function. The DES +function applies a 48-bit key to the rightmost 32 bits (Ri −1) +to produce a 32-bit output. This function is made up of four +sections: an expansion D-box, a whitener (that adds key), a +group of S-boxes, and a straight D-box, as shown in Fig 6. +C. Whitener (XOR) +After the expansion permutation, DES uses the XOR op- +eration on the expanded right section and the round key. It +XORed expansion permutation and key input and gives 48-bit +TABLE III: Expansion Permutation +32 +1 +2 +3 +4 +5 +64 +32 +4 +5 +6 +7 +8 +9 +63 +31 +8 +9 +10 +11 +12 +13 +62 +30 +12 +13 +14 +15 +16 +17 +61 +29 +16 +17 +18 +19 +20 +21 +60 +28 +20 +21 +22 +23 +24 +25 +59 +27 +24 +25 +26 +27 +28 +29 +58 +26 +28 +29 +30 +31 +32 +1 +57 +25 + +Initial Permutation +Permuted Choice 1 +Round 1 +Permuted Choice 2 +Left circular Shift +Round 2 +Permuted Choice 2 +Left Circular Shift +Round 16 +Permuted Choice 2 +Left circular shift +32 bit Swap +Inverse Initial Permutation1 +2 +25 +40 +50 +58 +60 +Initial +Permutation +1 +2 +...-8 +25 +40 +50 +58 +60 +16 Rounds +1 +2...8 +25 +40 +50 +58 +60 +Final +Permulalion +★ +25 +40 +1 +50 +58 +605 +Fig. 6: Single round of the DES Algorithm [11]. +Fig. 7: DES function +input to s-boxes. Note that both the right section and the key +are 48-bits in length [12]. +D. S-Boxes +The S-boxes do the real mixing (confusion). DES uses 8 +S-boxes, each with a 6-bit input and a 4-bit output. The 48- +bit data from the second operation is divided into eight 6-bit +chunks, and each chunk is fed into a box [13]. The result +of each box is a 4-bit chunk; when these are combined the +result is a 32-bit text. The substitution in each box follows a +pre-determined rule based on a 4-row by 16- column table. +Fig. 8: S-box +TABLE IV: Straight Permutation Table +16 +07 +20 +21 +29 +12 +28 +17 +01 +15 +23 +26 +05 +18 +31 +10 +02 +08 +24 +14 +32 +27 +03 +09 +19 +13 +30 +06 +22 +11 +04 +25 +16 +17 +18 +19 +20 +21 +60 +28 +20 +21 +22 +23 +24 +25 +59 +27 +24 +25 +26 +27 +28 +29 +58 +26 +28 +29 +30 +31 +32 +1 +57 +25 +E. Final Permutation +The last operation in the DES function is a permutation +with a 32-bit input and a 32-bit output. The input/output +relationship for this operation is shown in Table II. +V. EXAMPLES OF DES +Let M be the plain text message M = 0123456789ABCDEF +where M is in hexadecimal (base 16) format. Rewriting M in +binary format, we get the 64-bit block of text: +M = 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 +1010 1011 1100 1101 1110 1111, +R = 1000 1001 1010 1011 1100 1101 1110 1111, +L = 0000 0001 0010 0011 0100 0101 0110 0111. +The first bit of M is ‘0’. The last bit is ‘1’. We read from +left to right. DES operates on the 64-bit blocks using key sizes +of 56- bits. The keys are actually stored as being 64 bits long, +but every 8th bit in the key is not used (i.e., its numbered 8, +16, 24, 32, 40, 48, 56, and 64). However, we will nevertheless +number the bits from 1 to 64, going left to right, in the +following calculations. But, as you will see, the eight bits just +mentioned get eliminated when we create subkeys. Let K be +the hexadecimal key K = 133457799BBCDFF1. This gives +us binary key (setting 1 = 0001, 3 = 0011, etc., and grouping +together every eight bits, of which the last one in each group +will be unused) [14]. K = 00010011 00110100 01010111 +01111001 10011011 10111100 11011111 11110001 . +VI. KEY GENERATION +The 64-bit key is permuted according to the following table, +PC−1. Since the first entry in the table is “57”, this means +that the 57th bit of the original key K becomes the first bit +of the permuted key K+. The 49th bit of the original key +becomes the second bit of the permuted key. The 4th bit of +the original key is the last bit of the permuted key. Note only +56 bits of the original key appear in the permuted key. +From the original 64-bit key +K = 0001001100110100010101110111100110011011 + +4 +32 bits ++ +32 bits ++ +4 +28 bits ++ +28 bits +Li1 +Ri.1 +C(i-1 +Expansion/Perimutation (E Table) +Left Shit(s) +Right Shit(s) +48 +F +Permutation/Contraction +XOR +48 1 +(Perimuted Choice 2) +48 +Substitution/Choice (S-Box) ++ +32 +Permutation (P) +1 +32 +XOR +7 +Li +R; +Ci +D,Expansion/Permutation (E Table +48 +F +XOR +48/ +48 +Substitution/Choice (S-Box) +32 +Permutation (P) +3248-bit input +Array of S-Boxes +S-Box +S-Box +S-Box +S-Box +S-Box +S-Box +S-Box +S-Box +32-bit output6 +TABLE V: Permuted Choice-1 +57 +49 +41 +33 +25 +17 +9 +17 +1 +58 +50 +42 +34 +26 +18 +10 +10 +2 +59 +51 +43 +35 +27 +09 +19 +11 +3 +60 +52 +44 +36 +25 +63 +55 +47 +39 +31 +23 +15 +28 +7 +62 +54 +46 +38 +30 +22 +27 +14 +6 +61 +53 +45 +37 +29 +26 +21 +13 +5 +28 +20 +12 +4 +25 +101111001101111111110001 we get the 56-bit permutation +K+ = 11110000110011001010101011110101010 +101100110011110001111 +Now, From the permuted key K+, we get +C0 = 1111000011001100101010101111 +C1 = 1110000110011001010101011111 +C2 = 1100001100110010101010111111 +C3 = 0000110011001010101011111111 +C4= 0011001100101010101111111100 +C5 = 1100110010101010111111110000 +D0 = 0101010101100110011110001111 +D1 = 1010101011001100111100011110 +D2 = 0101010110011001111000111101 +D3 = 0101011001100111100011110101 +D4 = 0101100110011110001111010101 +D5 = 0110011001111000111101010101 +C6 = 0011001010101011111111000011 +D6 = 1001100111100011110101010101 +C7 = 1100101010101111111100001100 +D7 = 0110011110001111010101010110 +C8 = 0010101010111111110000110011 +D8 = 1001111000111101010101011001 +C9 = 0101010101111111100001100110 +D9 = 0011110001111010101010110011 +C10 = 0101010111111110000110011001 +D10 = 1111000111101010101011001100 +C11 = 0101011111111000011001100101 +D11 = 1100011110101010101100110011 +C12 = 0101111111100001100110010101 +D12 = 0001111010101010110011001111 +C13 = 0111111110000110011001010101 +D13 = 0111101010101011001100111100 +C14 = 1111111000011001100101010101 +D14 = 1110101010101100110011110001 +C15 = 1111100001100110010101010111 +D15 = 1010101010110011001111000111 +C16 = 1111000011001100101010101111 +D16 = 0101010101100110011110001111 +TABLE VI: Permuted choice-2 +14 +17 +11 +24 +1 +5 +9 +17 +3 +28 +15 +6 +21 +10 +18 +10 +23 +19 +12 +4 +26 +8 +27 +09 +16 +7 +27 +20 +13 +2 +36 +25 +41 +52 +31 +37 +47 +55 +15 +28 +30 +40 +51 +45 +33 +48 +22 +27 +44 +49 +39 +56 +34 +53 +29 +26 +46 +42 +50 +36 +29 +32 +4 +25 +After we apply the permutation PC-2, it becomes +K1 = 000110 110000 001011 101111 111111 000111 000001 +110010 +K2 = 011110 011010 111011 011001 110110 111100 100111 +100101 +K3 = 010101 011111 110010 001010 010000 101100 111110 +011001 +K4 = 011100 101010 110111 010110 110110 110011 010100 +011101 +K5 = 011111 001110 110000 000111 111010 110101 001110 +101000 +K6 = 011000 111010 010100 111110 010100 000111 101100 +101111 +K7 = 111011 001000 010010 110111 111101 100001 100010 +111100 +K8 = 111101 111000 101000 111010 110000 010011 101111 +111011 +K9 = 111000 001101 101111 101011 111011 011110 011110 +000001 +K10 = 101100 011111 001101 000111 101110 100100 011001 +001111 +K11 = 001000 010101 111111 010011 110111 101101 001110 +000110 +K12 = 011101 010111 000111 110101 100101 000110 011111 +101001 +K13 = 100101 111100 010111 010001 111110 101011 101001 +000001 +K14 = 010111 110100 001110 110111 111100 101110 011100 +111010 +K15 = 101111 111001 000110 001101 001111 010011 111100 +001010 +K16 = 110010 110011 110110 001011 000011 100001 011111 +110101 +Encode each 64-bit block of data Applying the initial +permutation to the block of text M, given previously, we get +M = 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 +1010 1011 1100 1101 1110 1111 +IP = 1100 1100 0000 0000 1100 1100 1111 1111 1111 0000 +1010 1010 1111 0000 1010 1010 +Here the 58th bit of M is ‘1’, which becomes the first bit +of IP. The 50th bit of M is ‘1’, which becomes the second +bit of IP. The 7th bit of M is ‘0’, which becomes the last bit +of IP. +Next, divide the permuted block IP into a left half L0 of +32 bit, and a right half R0 of 32 bits. +From IP, we get L0 and R0 +L0 = 1100 1100 0000 0000 1100 1100 1111 1111 +R0 = 1111 0000 1010 1010 1111 0000 1010 1010 +We now proceed through 16 iterations, for 1≤ n ≤ 16, using +a function f which operates on two blocks–a data block of 32 +bits and a key Kn of 48 bits–to produce a block of 32 bits. Let ++ denote XOR addition, (bit-by-bit addition modulo 2). Then +for n going from 1 to 16 we calculate (3) and (4). +Ln = Rn − 1 +(3) +Rn = Ln−1 + f(Rn−1, Kn) +(4) + +7 +This results in a final block, for n = 16, of L16R16. That +is, in each iteration, we take the right 32 bits of the previous +result and make them the left 32 bits of the current step. For +the right 32 bits in the current step, we XOR the left 32 bits +of the previous step with the calculation f. +For n = 1, we have +K1 = 000110 110000 001011 101111 111111 000111 000001 +110010 +L1 = R0 = 1111 0000 1010 1010 1111 0000 1010 1010 +R1 = L0 + f(R0, K1) +It remains to explain how the function f works. To calculate +f, first expand each block Rn −1 from 32 bits to 48 bits. This +is done by using a selection table that repeats some of the +bits in Rn − 1. We’ll call the use of this selection table the +function E. Thus E(Rn − 1) has a 32 bit input block, and a +48 bit output block. After this, We calculate E(R0) from R0 +as follows: +R0 = 1111 0000 1010 1010 1111 0000 1010 1010 +E(R0) = 011110 100001 010101 010101 011110 100001 +010101 010101 Next in the f calculation, we XOR the output +E(Rn − 1) with the key Kn : Kn + E(Rn − 1). For K1, +E(R0), we have +K1 = 000110 110000 001011 101111 111111 000111 000001 +110010 +E(R0) = 011110 100001 010101 010101 011110 100001 +010101 010101 +K1+E(R0) = 011000 010001 011110 111010 100001 100110 +010100 100111 +To this point we have expanded Rn-1 from 32 bits to 48 bits, +using the selection table, and XORed the result with the key +Kn. We now have 48 bits, or eight groups of six bits. We now +do something strange with each group of six bits: we use them +as addresses in tables called ”S boxes”. Each group of six bits +will give us an address in a different S box. Located at that +address will be a 4-bit number. This 4-bit number will replace +the original 6 bits. The net result is that the eight groups of +6 bits are transformed into eight groups of 4 bits (the 4-bit +outputs from the S boxes) for 32 bits total. +Write the previous result, which is 48 bits, in the form: +Kn + E(Rn − 1) = B1B2B3B4B5B6B7B8 +where each Bi is a group of six bits. We now calculate it as +S1(B1)S2(B2)S3(B3)S4(B4)S5(B5)S6(B6)S7(B7)S8(B8) +where Si(Bi) refers to the output of the ith S box. +To repeat, each of the functions S1, S2, ..., S8, takes a 6-bit +block as input and yields a 4-bit block as output. For the first +round, we obtain as the output of the eight S boxes: +K1 + E(R0) = 011000 010001 011110 111010 100001 +100110 010100 100111 +S1(B1)S2(B2)S3(B3)S4(B4)S5(B5)S6(B6)S7(B7)S8(B8) += 0101 1100 1000 0010 1011 0101 1001 0111 +The final stage in the calculation of f is to do a permutation +P of the S-box output to obtain the final value of f: +f = P(S1(B1)S2(B2)...S8(B8)) +(5) +The permutation P is defined in the following table. P yields +a 32-bit output from a 32-bit input by permuting the bits of +the input block +TABLE VII: Permutation +16 +7 +20 +21 +1 +5 +9 +17 +29 +12 +28 +17 +21 +10 +18 +10 +1 +15 +23 +26 +26 +8 +27 +09 +5 +18 +31 +10 +13 +2 +36 +25 +2 +8 +24 +14 +47 +55 +15 +28 +32 +27 +3 +9 +33 +48 +22 +27 +19 +23 +30 +6 +34 +53 +29 +26 +22 +11 +4 +25 +29 +32 +4 +25 +From the output of the eight S boxes: +S1(B1)S2(B2)S3(B3)S4(B4)S5(B5)S6(B6)S7(B7)S8(B8) += 0101 1100 1000 0010 1011 0101 1001 0111 +we get, +f = 0010 0011 0100 1010 1010 1001 1011 1011 +R1 = L0 + f(R0, K1) +R1 = 1100 1100 0000 0000 1100 1100 1111 1111 + 0010 +0011 0100 1010 1010 1001 1011 1011 = 1110 1111 0100 +1010 0110 0101 0100 0100 +In the next round, we will have L2 = R1, which is the +block we just calculated, and then we must calculate R2 = +L1 + f(R1, K2), and so on for 16 rounds. At the end of the +sixteenth round we have the blocks L16 and R16. We then +reverse the order of the two blocks into the 64-bit block as +shown in equation (6). +R16L16 +(6) +Now, apply a final permutation IP−1 and the output of the +algorithm has bit 40 of the preoutput block as its first bit, bit +8 as its second bit, and so on, until bit 25 of the preoutput +block is the last bit of the output. +If we process all 16 blocks using the method defined +previously, we get, on the 16th round, +L16 = 0100 0011 0100 0010 0011 0010 0011 0100 +R16 = 0000 1010 0100 1100 1101 1001 1001 0101 +We reverse the order of these two blocks and apply the +final permutation to +R16L16 += +00001010 +01001100 +11011001 +10010101 +01000011 01000010 00110010 00110100 +IP −1 = 10000101 11101000 00010011 01010100 00001111 +00001010 10110100 00000101 which in hexadecimal format +is 85E813540F0AB405. +This is the encrypted form of M = 0123456789ABCDEF: +namely, C = 85E813540F0AB405. Decryption is simply the +inverse of encryption, following the same steps as above, but +reversing the order in which the subkeys are applied. +VII. SYMMETRIC CIPHERS +If we examine the symmetric ciphers in detail, we can see +that symmetric ciphers can be divided into two categories; +stream ciphers and block ciphers [15]. +Stream ciphers use a key-stream, obtained from the original +key, and encrypts the plain-text bit by bit. Encryption is usually +done by combining the plain-text bits with the corresponding +key-stream bits with an XOR operation. In some cases stream + +8 +ciphers have some advantages over block ciphers, because +there is no error propagation. It means that an error made +in one bit of cipher-text during transmission only affects the +decryption of that bit and doesn’t affect other bits [16]. +Block ciphers, on the other hand, take the plain-text bits +in blocks. Each block is encrypted with the same encryption +function and the cipher-text blocks are produced. When the +length of the plain-text is not a multiple of the block size +some padding is applied to the plain-text [17]. This padding +is usually done by adding a ‘1’ bit followed by necessary +amount of ‘0’ bits. Because the encryption function does not +change from one block to another, same blocks of plain- +text are encrypted to same blocks of cipher-text. When an +adversary captures the cipher-text, they can accurately guess +some information about the plaintext by using this property. +In order to stop any information leakage, some modes of +operation are used. +VIII. ASYMMETRIC CIPHERS +While modern symmetric ciphers such as AES are very +secure, they have some drawbacks in practicality, namely key +distribution problem, and the number of keys [18]. +The key distribution problem occurs when Alice and Bob +want to determine a secret key. This would be easy if they +can come together and decide, but if they have no means +to decide on a key in person, they have to decide on the +key through a secure channel [19]. Since the communication +channel is always assumed to be insecure, because it can be +easily hacked, this poses a problem. Even if they can somehow +solve this problem, they would be facing another problem, the +number of keys [20]. If there are n users in a network, and +all of the users want to communicate with each other secretly, +the number of encryption keys needed would be n∗(n−1) +2 +,and +each user would have n − 1 key pairs they need to know +and keep secret. This becomes exponentially infeasible as the +number of people increase. The usage of asymmetric ciphers +eliminate these problems. Since every user has a pair of keys, +and anything encrypted with a specific public key can only be +decrypted with the corresponding private key, Alice and Bob +doesn’t need to agree on a secret key together beforehand. In +addition, nobody would need to store n − 1 key pairs, they +only need to store their own private and public keys, and the +number of key pairs needed in the network would be reduced +to n [11]. Cryptographic protocols can be considered as a third +main branch of cryptography, and one of the most important +primitives they use is called a hash function. Therefore it +would be useful to go over the definition of hash functions. +In order to understand how public key algorithms work +we can imagine a box [21]. For Alice to send Bob a secret +message, first Bob sends Alice a box with an open padlock, +for which he has the key. Alice then can put her message in the +box, and lock it with the padlock. When Bob receives the box +he simply unlocks the padlock and reads the message [11]. +Of course there are still some security concerns, for example +an adversary can intercept the box and replace the padlock +with their own lock, or put their own message in the box and +act like Alice. To achieve authentication and to prevent these +problems, cryptographers have developed some procedures. +A. Modes of Operation +There are several modes of operation that can be used when +encrypting a plaintext with a block cipher. NIST recommends +the usage of 5 modes of operation. [11]: +• Electronic Codebook (ECB) +• Cipher Block Chaining (CBC) +• Cipher Feedback (CFB) +• Output Feedback (OFB) +• Counter (CTR) +In ECB mode, each block is encrypted and decrypted in- +dependently from each other. Because the encryption function +does not change, identical blocks of plaintext are encypted to +identical blocks of ciphertext [22]. +In CBC mode, the ciphertext of one block is XORed with +the plaintext of the next block before the encryption [23]. +For the first plaintext block an Initialization Vector IV is +used. In CFB mode, ciphertext blocks are encrypted with the +encryption function instead of the plaintext blocks. Plaintext +blocks are XORed with the results of encryption function to +obtain the ciphertext blocks. For the first block an IV is used +[24]. +In OFB mode [25], IV is repeatedly encrypted with the +encryption function and the results are XORed with the +plaintext blocks to obtain ciphertext blocks [26]. +In CTR mode, a nonce and counter is encrypted and the +result is XORed with the plaintext block [27]. The counter is +increased each time. +All of these modes while having different advantages also +have some disadvantages. For example, some of them have +parallelizable encryption and decryption but others don’t. The +decision of which modes of operation is to be used should +be made based on the desired security and performance levels +[28]. +IX. RESULTS +Implementation of DES has been performed using VHDL +and the results is shown in TABLE IX and TABLE VIII. +TABLE VIII: Performance Matrix of DES on Virtex-7 FPGA +Device. +Operating +Frequency (MHz) +Datapath Dalay +(nS) +Maximum +Frequency (MHz) +Dynamic +Power (mW) +100 +1.829 +246 +8 +TABLE IX: Resource Utilization of DES on Virtex-7 FPGA +Device. +Slices +LUTs +Flip-Flops +69 +244 +139 +X. CONCLUSION +Architecture Exploration of Simplified Data Encryption +Standard (SDES) and Data Encryption Standard (DES) has +been done. Simplified DES (SDES) was designed for educa- +tional purposes only, to help learn about modern cryptanalytic +techniques. SDES has similar properties and structure as DES +but has been simplified to make it much easier to perform + +9 +encryption and decryption by hand with pencil and paper. +Some people feel that learning SDES gives insight into DES +and other block ciphers, and insight into various cryptanalytic +attacks against them [29]. In DES, 64-bit input is encrypted +and decrypted using 56-bit key. At the encryption site, DES +takes a 64-bit plaintext and creates a 64-bit ciphertext; at the +decryption site, DES takes a 64- bit ciphertext and creates a +64-bit block of plaintext. The same 56-bit cipher key is used +for both encryption and decryption. Implementation of SDES +and DES has been performed using Python 3.7 version and +VHDL. During this project I have learned thoroughly about +various cryptography techniques and ciphers. +REFERENCES +[1] Z. Lu, “Encryption management of accounting data based on des +algorithm of wireless sensor network,” Wireless Communications and +Mobile Computing, vol. 2022, 2022. +[2] F. Pub, “Data encryption standard (des),” FIPS PUB, pp. 46–3, 1999. +[3] S.-J. Han, H.-S. Oh, and J. Park, “The improved data encryption standard +(des) algorithm,” in Proceedings of ISSSTA’95 International Symposium +on Spread Spectrum Techniques and Applications, vol. 3. +IEEE, 1996, +pp. 1310–1314. +[4] K. Rabah, “Theory and implementation of data encryption standard: A +review,” Information Technology Journal, vol. 4, 04 2005. +[5] T. Nie and T. Zhang, “A study of des and blowfish encryption algorithm,” +in Tencon 2009-2009 IEEE Region 10 Conference. +IEEE, 2009, pp. +1–4. +[6] A. A. Yazdeen, S. R. Zeebaree, M. M. Sadeeq, S. F. Kak, O. M. +Ahmed, and R. R. Zebari, “Fpga implementations for data encryption +and decryption via concurrent and parallel computation: A review,” +Qubahan Academic Journal, vol. 1, no. 2, pp. 8–16, 2021. +[7] D. Coppersmith, D. B. Johnson, and S. M. Matyas, “A proposed mode +for triple-des encryption,” IBM Journal of Research and Development, +vol. 40, no. 2, pp. 253–262, 1996. +[8] J. Thakur and N. Kumar, “Des, aes and blowfish: Symmetric key +cryptography algorithms simulation based performance analysis,” In- +ternational journal of emerging technology and advanced engineering, +vol. 1, no. 2, pp. 6–12, 2011. +[9] D. Coppersmith, “The data encryption standard (des) and its strength +against attacks,” IBM journal of research and development, vol. 38, +no. 3, pp. 243–250, 1994. +[10] P. Mahajan and A. Sachdeva, “A study of encryption algorithms aes, +des and rsa for security,” Global Journal of Computer Science and +Technology, 2013. +[11] W. Stallings, Cryptography and network security, 4/E. +Pearson Educa- +tion India, 2006. +[12] K. Bhatia and S. Som, “Study on white-box cryptography: key whitening +and entropy attacks,” in 2016 5th International Conference on Re- +liability, Infocom Technologies and Optimization (Trends and Future +Directions)(ICRITO). +IEEE, 2016, pp. 323–327. +[13] K. Mohamed, M. N. M. Pauzi, F. H. H. M. Ali, S. Ariffin, and +N. H. N. Zulkipli, “Study of s-box properties in block cipher,” in 2014 +International Conference on Computer, Communications, and Control +Technology (I4CT). +IEEE, 2014, pp. 362–366. +[14] “Des +examples.” +[Online]. +Available: +https://page.math. +tu-berlin.de/∼kant/teaching/hess/krypto-ws2006/des.htm#:∼: +text=For%20example%2C%20if%20we%20take,the%20original% +20plaintext%20%228787878787878787%22. +[15] A. Biryukov, “Block ciphers and stream ciphers: The state of the art,” +Cryptology EPrint Archive, 2004. +[16] J. Burke, J. McDonald, and T. Austin, “Architectural support for fast +symmetric-key cryptography,” in Proceedings of the ninth international +conference on Architectural support for programming languages and +operating systems, 2000, pp. 178–189. +[17] A. Schubert and W. Anheier, “Efficient vlsi implementation of modern +symmetric block ciphers,” in ICECS’99. Proceedings of ICECS’99. 6th +IEEE International Conference on Electronics, Circuits and Systems +(Cat. No. 99EX357), vol. 2. +IEEE, 1999, pp. 757–760. +[18] B. Lee, “The key distribution problem: Prior advances and future +challenges,” 2020. +[19] Y. Yusfrizal, A. Meizar, H. Kurniawan, and F. Agustin, “Key man- +agement using combination of diffie–hellman key exchange with aes +encryption,” in 2018 6th International Conference on Cyber and IT +Service Management (CITSM), 2018, pp. 1–6. +[20] U. SenthilKumar and U. Senthilkumaran, “Review of asymmetric key +cryptography in wireless sensor networks,” International Journal of +Engineering and Technology, vol. 8, no. 2, pp. 859–862, 2016. +[21] S. Chandra, S. Paira, S. S. Alam, and G. Sanyal, “A comparative survey +of symmetric and asymmetric key cryptography,” in 2014 international +conference on electronics, communication and computational engineer- +ing (ICECCE). +IEEE, 2014, pp. 83–93. +[22] E. Celikel, J. Davidson, and C. Kern, “Parallel performance of des in +ecb mode,” in 2006 International Symposium on Computer Networks. +IEEE, 2006, pp. 134–139. +[23] C. Tan, X. Deng, and L. Zhang, “Identification of block ciphers under +cbc mode,” Procedia Computer Science, vol. 131, pp. 65–71, 2018. +[24] S. Mister and R. Zuccherato, “An attack on cfb mode encryption as +used by openpgp,” in International Workshop on Selected Areas in +Cryptography. +Springer, 2006, pp. 82–94. +[25] Y.-L. Huang, F.-Y. Leu, J.-C. Liu, J.-H. Yang, C.-W. Yu, C.-C. Chu, and +C.-T. Yang, “Building a block cipher mode of operation with feedback +keys,” in 2013 IEEE International Symposium on Industrial Electronics. +IEEE, 2013, pp. 1–4. +[26] T. Iwata, K. Minematsu, J. Guo, S. Morioka, and E. Kobayashi, “Silc: +simple lightweight cfb,” CAESAR submission, 2014. +[27] H. Lipmaa, P. Rogaway, and D. Wagner, “Ctr-mode encryption,” in First +NIST Workshop on Modes of Operation, vol. 39. +Citeseer. MD, 2000. +[28] S. Almuhammadi and I. Al-Hejri, “A comparative analysis of aes +common modes of operation,” in 2017 IEEE 30th Canadian Conference +on Electrical and Computer Engineering (CCECE), 2017, pp. 1–4. +[29] R. C. Merkle and M. E. Hellman, “On the security of multiple encryp- +tion,” Communications of the ACM, vol. 24, no. 7, pp. 465–467, 1981. +R uby Kumari, Integrated Dual Degree Ph.D (IDDP) +scholar at Academy of Scientific Innovative Re- +search (AcSIR), working area Integrated Circuits +and System Group, CSIR-CEERI. Completed B.tech +in Electronics and Communication Engineering from +Maulana Abul Kalam Azad University of Technol- +ogy. Her research interest includes Cryptography, +Lightweight Ciphers, Digital Logic Design, VLSI +Architecture and RTL Design. +Jai Gopal Pandey is a Principal Scientist and work- +ing in the CSIR-CEERI, Pilani, India since 2005. He +is an M.Tech. (Electronics Design and Technology) +from U. P. Technical University, Lucknow, in 2003 +and a Ph.D. in Electronics Engineering from Birla +Institute of Technology and Science (BITS), Pilani, +India in 2015. +His research interests include High-performance +Architecture, System-on-chips (SoCs), Embedded +Systems, Cryptography, FPGAs, and ASIC designs. +Dr. Pandey is a Senior Member of IEEE and an IETE +Fellow. + +10 +Abhijit Karmakar received the B.E. degree in +Electronics and Telecommunication Engineering in +1993 from Jadavpur University, India, and M.Tech. +degree in Electrical Engineering from Indian Insti- +tute of Technology (IIT), Madras, India, in 1995. He +recieved the Ph.D. degree in Electrical Engineering +from IIT, Delhi, India, in 2007. Since 1995, he has +been working with the CSIR - Central Electronics +Engineering Research Institute (CEERI), Pilani, In- +dia. His research interest span the area of VLSI +Design, Signal Processing and related areas. + diff --git a/AdE5T4oBgHgl3EQfSg_z/content/tmp_files/load_file.txt b/AdE5T4oBgHgl3EQfSg_z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7b67856dc9853976b23e1fb49c63f3a852a723b4 --- /dev/null +++ b/AdE5T4oBgHgl3EQfSg_z/content/tmp_files/load_file.txt @@ -0,0 +1,1009 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf,len=1008 +page_content='1 An RTL Implementation of the Data Encryption Standard (DES) Ruby Kumari, Student Member, IEEE, Jai Gopal Pandey, Senior Member,IEEE, and Abhijit Karmakar, Abstract—Data Encryption Standard (DES) is based on the Feistel block cipher, developed in 1971 by IBM cryptography researcher Horst Feistel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' DES uses 16 rounds of the Feistel structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' But with the changes in recent years, the internet is starting to be used more to connect devices to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' These devices can range from powerful computing devices, such as desktop computers and tablets, to resource constrained devices, When it comes to these constrained devices, using a different key for each round cryptography algorithms fail to provide necessary security and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Index Terms—Keywords: Cryptography, DES , SDES, Feistel block Cipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' INTRODUCTION T HIS Security is a prevalent concern in information and data systems of all types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Historically, military and na- tional security issues drove the need for secure communica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Recently, security issues have pervaded the business and private sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' E-commerce has driven the need for secure internet communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Many businesses have fire- walls to protect internal corporate information from competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' In the private sector, personal privacy is a growing concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Products are available to scramble both e-mail and telephone communi- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' One means of providing security in communications is through encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' By encryption, data is transformed in a way that it is rendered unrecognizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Only by decryption can this data be recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Ostensibly, the process of decryption can only be performed correctly by the intended recipients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The validity of this assertion determines the “strength” or “security” of the encryption scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Many communications products incorporate encryption as a feature to provide se- curity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' This application report studies the implementation of one of the most historically famous and widely implemented encryption algorithms, the Data Encryption Standard (DES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The Data Encryption Standard is a symmetric-key block cipher published by the National Institute of Standards and Technology (NIST) for the encryption of digital data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' DES is probably one of the best-known cryptographic algorithms and has been widely used since its introduction in 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Although its short key length of 56 bits makes it too inse- 1 cure for applications, it has been highly influential in the advancement of cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The DES must be stronger than the other cryptosystems in security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The goal of this project is to develop a python code for SDES and DES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Before building our design, we need an overview of cryptography, followed by a description of the DES algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Manuscript received December 19, 2022 1) Overview of Cryptography: Cryptography is a type of rule or technique by which private or sensitive information is secured from the public or other members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' It plays a vital role in preserving data integrity, confidentiality and user privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' An encryption algorithm can convert imported essential data to encrypted data (plaintext into cipher- text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' This data would be of no use to a person that does not possess the encryption key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The use of Cryptography in passwords is a very famous example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Cryptography is based on mathematical theory and some Computer Science principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' There are many terminologies related to cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Some terms are defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Ciphertext: Conversion of plain text into intelligible text is called ciphertext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Cipher: It is a technique of encryption and decryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Critical and algorithms play vital role in this technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Symmetric: It is a kind of cryptosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' It uses same key for encryption and decryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' It is faster than asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Asymmetric: It is also a kind of cryptosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' It uses a public key for the encryption and a private key for the decryption of any message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Cryptanalysis: It studies cracking the encryption of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Symmetric Ciphers Model Symmetric-key (or private-key) encryption can be simply illustrated with the schematic shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 1: Symmetric Cryptosystem model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' A symmetric encryption scheme has five main parts, that is, Encryption algorithm: The encryption algorithm per- forms various substitutions and transformations on plain- text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Secret key: The secret key is also input to the encryption algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The key is a value independent of the plaintext and the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The algorithm will produce a different arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='05530v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='CR] 13 Jan 2023 Cryptanalyst Message Encryption Decryption Sources Destination Algorithm Algorithm Secure Channel Key Sources2 output depending on the specific key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The exact substi- tutions and transformations performed by the algorithm depend on the key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Ciphertext: This is the scrambled message produced as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' It depends on the plaintext and the secret key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' For a given message, two different keys will produce two different ciphertexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The ciphertext is an random stream of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Decryption algorithm: This is essentially the encryption algorithm run in reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' It takes the ciphertext and the secret key and produces the original plaintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Alice and Bob want to communicate over an un-secure channel, but Oscar is trying to read the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' So Alice and Bob must use a cryptosystem to prevent Oscar from reading the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Let us take a closer look at the essential elements of a symmetric encryption scheme using Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' A source produces a message in plaintext, X = [X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' , XM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The M elements of X are letters in some finite alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Traditionally, the alphabet usually consisted of t6 capital letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Nowadays, the binary alphabet 0, 1 is typically used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' For encryption, a key of the form K = [K1, K2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='., KJ] is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' If the key is generated at the message source, then it must also be provided to the destination using some secure channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Alternatively, a third party could generate key and securely deliver it to both source and destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The encryption algorithm forms the ciphertext as given in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Y = [Y 1, Y 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='., Y N] (1) with the message X and the encryption key K as it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' We can write this as given in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Y = E(K, X) (2) This notation indicates that Y is produced by using en- cryption algorithm E as a function of the plaintext X, with the specific process determined by the value of the key K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The intended receiver, in possession of the key, can invert the transformation: X = D(K, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' An opponent, observing Y but not having access to K or X, may attempt to recover X or K or both X and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' It is assumed that the opponent knows the encryption (E) and decryption (D) algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' If the opponent is interested in only this particular message, then the focus of the effort is to recover X by generating a plaintext estimate X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Often, however, the opponent is interested in being able to read future messages as well, in which case an attempt is made to recover K by generating an estimate K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Simplified Data Encryption Standard The S-DES encryption algorithm takes an 8-bit block of plaintext and a 10-bit key as input and produces an 8-bit block of ciphertext as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The S-DES decryption algorithm takes an 8- bit block of ciphertext and the same 10-bit key used to produce that ciphertext as input and produces the original 8-bit block of plaintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Simplified DES (SDES) was designed for educational purposes only, to help students learn about modern cryptanalytic techniques [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' SDES has similar properties and structure as DES but has been simplified to make it much easier to perform encryption and decryption by hand with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 2: Simplified DES (SDES) Pencil and paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Some people feel that learning SDES gives insight into DES and other block ciphers, and insight into various cryptanalytic attacks against them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' An adversary trying to interrupt two communicating parties may have one of the four main goals: 1) Read the secret message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 2) Find the secret key, so that they can read all messages encrypted with that key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 3) Modify the message sent by Alice and go unnoticed by both parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 4) Act like Alice and send a message to Bob, to make Bob think he is communicating with Alice when in reality he is communicating with the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' In order to prevent an adversary from reaching his goals, some security measures are Applied to cryptosystems, 10-bit Key 4 P10 Encryption Decryption 8-bit Plaintext 8-bit Plaintext Shift IP IP-1 P8 K1 Ki Fk Fk Shift SW SW P8 K2 K2 Fk Fk IP IP-1 8-bit Ciphertext 8-bit Ciphertext3 namely confidentiality, data integrity, authentication, and non- repudiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 1) Confidentiality means the transmitted message or infor- mation is kept secret, and only the authorized parties have the means to decipher the information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 2) Data integrity makes sure that the messages are not being modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' This stops the adversary from reaching their third goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 3) Authentication helps Bob to correctly identify the sender as Alice, thus stopping the adversary from posing as Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 4) Non-repudiation prevents Alice from denying she sent the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Cryptographic algorithms are gathered under two main branches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' symmetric algorithms and asymmetric algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' In symmetric algorithms both Alice and Bob have the same key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Since the communication channel is insecure, this key must be previously decided on through secure ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The encryption and decryption keys are either the same, or very similar that the decryption key can easily be derived from the encryption key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' But sometimes Alice and Bob cannot agree on a key beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' They could be very far away from each other and cannot get together to determine a secret key, and there may not be a secure way for Alice to send Bob the secret key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' She cannot just send Bob a secret key through any open channel, because an adversary can interrupt the channel and get their hands on the key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Thus making the key useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' To get around this problem asymmetric algorithms, usually called public key algorithms, are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' In public key algorithms each party has their key pairs, one public and one private key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' As can be understood from their names, private keys are kept secret, and public keys can be known by everyone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The public key is computed from the private key in a way that finding the private key from the public key is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Alice encrypts the message she wants to send using Bob’s public key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The message can only be decrypted with the corresponding private key, which only Bob has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Therefore Alice can send a secret message even though they are far away and cannot decide on a common key together [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Further, the details of the DES cipher is given in the next chpater [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' DATA ENCRYPTION STANDARD Developed in 1974 by IBM in cooperation with the National Securities Agency (NSA), DES has been the worldwide en- cryption standard for more than 20 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' For these 20 years, it has held up against cryptanalysis remarkably well and is still secure against all but possibly the most powerful adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Because of its prevalence throughout the encryption market, DES [4] is an excellent interoperability standard between different encryption equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The predominant weakness of DES is its 56-bit key which, more than sufficient for the time period which it was developed [5], has become insufficient to protect against brute-force attacks modern computers [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' As a result of the need for a greater encryption strength, DES evolved into triple-DES [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 3: Encryption and Decryption III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' DES ENCRYPTION The Data Encryption Standard is a Feistel cipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' In which round function consists of an expansion, a bitwise XOR- operation XOR operation round key, an S-box layer and a permutation [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' In encryption n scheme, there are two inputs to the encryption function [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' the plaintext to be encrypted and the key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' In this case, the plaintext must be 64 bits in length and the key is 56 bits in length [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' On the left-hand side of the figure, we can see that the plaintext processing proceeds in three phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' First, the 64- bit plaintext passes through an initial permutation (IP) that rearranges the Bits to produce the permuted input [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' This is followed by a phase consisting of sixteen rounds of the same function, which involves both permutation and substitution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The output of the last (sixteenth) round consists of 64 bits that are a function of the input plaintext and the key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The left and right halves of the output are swapped to produce the pre-output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Finally, the pre-output is passed through a per- mutation [IP −1], inverse of the initial permutation function, to produce the make ciphertext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' With the exception of initial and final permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' On the right-hand portion of figure 6-5,6-a-bitkey is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Initially, the key is passed through a permutation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Then, for each of the sixteen rounds, a subkey (Ki) is produced by the combination f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='t Initially, the key is passed through a permutation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Then, for each of the sixteen rounds, a subkey (Ki) is produced by the combination of a left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Initial Permutation and Final Permutation Each of these permutations takes a 64-bit input and per- mutes them according to a predefined rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' These permutations are keyless straight permutations that are the inverse of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' For example, in the initial permutation [IP], the 58th bit in the input becomes the first bit in the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Similarly, in the final permutation [IP −1], the first bit in the input becomes the 58th bit in the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' In other words, if the rounds between these two permutations do not exist, the 58th bit entering the initial permutation is the same as the 58th bit leaving the final permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The initial permutation is given in TABLE I The final permutation is given in TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' DES Reverse DES Cipher Cipher4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 4: Structure of DES Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 5: Initial and final permutation step in DES IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' ROUNDS DES uses 16 rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Each round of DES is a Feistel cipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 6 shows the internal structure of a single round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Again, begin by focusing on the left-hand side of the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The left and right halves of each 64-bit intermediate value are treated as separate 32-bit quantities, labeled L (left) and R (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' As in the Feistel cipher, the overall processing at each round can be summarized in the following formulas: Li = Ri − 1 Ri = Li − 1 XOR FE(Ri − 1, Ki) The round takes Li − 1 and Ri − 1 from the previous game (or the initial permutation box) and creates Li and Ri, which go to the next round (or final permutation box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Initial Permutation A single initial permutation is needed at the beginning of the encryption process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' IP is necessary on each block of 64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='bits in DES once the entire plaintext has been divided into ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='TABLE I: Initial Permutation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='2 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='59 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='43 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='61 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='TABLE II: Final Permutation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='24 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='49 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='57 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='such blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The transposition process goes through this initial permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Only once, just before the first round, does the first permutation appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' As seen in the Table I, it provides decisions for how the IP transposition process has to go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' It is possible to claim for example, that the IP replaced the first bit of the original plain-text block with the 58th bit of the original plain-text block, the second bit with the 50th bit of the original plain-text block, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' This is nothing more than bit shuffling with respect to the original plaintext block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Expansion D-Box Since Ri − 1 is a 32-bit input and KI is a 48-bit key, we first need to expand Ri − 1 to 48 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Ri − 1 is divided into 8 4-bit sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Each 4-bit section is then expanded to 6 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' For each section, input bits 1, 2, 3, and 4 are copied to output bits 2, 3, 4, and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Output bit ‘1’ comes from bit 4 of the previous section;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' output bit 6 comes from bit 1 of the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' If sections 1 and 8 can be considered adjacent sections, the same rule applies to bits 1 and 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The main part of DES is the DES function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The DES function applies a 48-bit key to the rightmost 32 bits (Ri −1) to produce a 32-bit output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' This function is made up of four sections: an expansion D-box, a whitener (that adds key), a group of S-boxes, and a straight D-box, as shown in Fig 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Whitener (XOR) After the expansion permutation, DES uses the XOR op- eration on the expanded right section and the round key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' It ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='XORed expansion permutation and key input and gives 48-bit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='TABLE III: Expansion Permutation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='57 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Initial Permutation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Permuted Choice 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Round 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Permuted Choice 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Left circular Shift ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Round 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Permuted Choice 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Left Circular Shift ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Round 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Permuted Choice 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Left circular shift ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='32 bit Swap ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Inverse Initial Permutation1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Initial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Permutation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='-8 25 40 50 58 60 16 Rounds 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='8 25 40 50 58 60 Final Permulalion ★ 25 40 1 50 58 605 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 6: Single round of the DES Algorithm [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 7: DES function input to s-boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Note that both the right section and the key are 48-bits in length [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' S-Boxes The S-boxes do the real mixing (confusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' DES uses 8 S-boxes, each with a 6-bit input and a 4-bit output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The 48- bit data from the second operation is divided into eight 6-bit chunks, and each chunk is fed into a box [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The result of each box is a 4-bit chunk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' when these are combined the result is a 32-bit text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The substitution in each box follows a pre-determined rule based on a 4-row by 16- column table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 8: S-box TABLE IV: Straight Permutation Table 16 07 20 21 29 12 28 17 01 15 23 26 05 18 31 10 02 08 24 14 32 27 03 09 19 13 30 06 22 11 04 25 16 17 18 19 20 21 60 28 20 21 22 23 24 25 59 27 24 25 26 27 28 29 58 26 28 29 30 31 32 1 57 25 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Final Permutation The last operation in the DES function is a permutation with a 32-bit input and a 32-bit output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The input/output relationship for this operation is shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' EXAMPLES OF DES Let M be the plain text message M = 0123456789ABCDEF where M is in hexadecimal (base 16) format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Rewriting M in binary format, we get the 64-bit block of text: M = 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111, R = 1000 1001 1010 1011 1100 1101 1110 1111, L = 0000 0001 0010 0011 0100 0101 0110 0111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The first bit of M is ‘0’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The last bit is ‘1’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' We read from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' DES operates on the 64-bit blocks using key sizes of 56- bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The keys are actually stored as being 64 bits long, but every 8th bit in the key is not used (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=', its numbered 8, 16, 24, 32, 40, 48, 56, and 64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' However, we will nevertheless number the bits from 1 to 64, going left to right, in the following calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' But, as you will see, the eight bits just mentioned get eliminated when we create subkeys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Let K be the hexadecimal key K = 133457799BBCDFF1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' This gives us binary key (setting 1 = 0001, 3 = 0011, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=', and grouping together every eight bits, of which the last one in each group will be unused) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' K = 00010011 00110100 01010111 01111001 10011011 10111100 11011111 11110001 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' KEY GENERATION The 64-bit key is permuted according to the following table, PC−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Since the first entry in the table is “57”, this means that the 57th bit of the original key K becomes the first bit of the permuted key K+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The 49th bit of the original key becomes the second bit of the permuted key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The 4th bit of the original key is the last bit of the permuted key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Note only 56 bits of the original key appear in the permuted key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' From the original 64-bit key K = 0001001100110100010101110111100110011011 4 32 bits + 32 bits + 4 28 bits + 28 bits Li1 Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='1 C(i-1 Expansion/Perimutation (E Table) Left Shit(s) Right Shit(s) 48 F Permutation/Contraction XOR 48 1 (Perimuted Choice 2) 48 Substitution/Choice (S-Box) + 32 Permutation (P) 1 32 XOR 7 Li R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Ci D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Expansion/Permutation (E Table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='XOR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='48/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Substitution/Choice (S-Box) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Permutation (P) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='3248-bit input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Array of S-Boxes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='S-Box ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='S-Box ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='S-Box ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='S-Box ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='S-Box ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='S-Box ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='S-Box ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='S-Box ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='32-bit output6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='TABLE V: Permuted Choice-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='57 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='49 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='59 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='43 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='09 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='62 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='54 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='46 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='61 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='101111001101111111110001 we get the 56-bit permutation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='K+ = 11110000110011001010101011110101010 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='101100110011110001111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' From the permuted key K+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' we get ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='C0 = 1111000011001100101010101111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='C1 = 1110000110011001010101011111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='C2 = 1100001100110010101010111111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='C3 = 0000110011001010101011111111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='C4= 0011001100101010101111111100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='C5 = 1100110010101010111111110000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D0 = 0101010101100110011110001111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D1 = 1010101011001100111100011110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D2 = 0101010110011001111000111101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D3 = 0101011001100111100011110101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D4 = 0101100110011110001111010101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D5 = 0110011001111000111101010101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='C6 = 0011001010101011111111000011 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D6 = 1001100111100011110101010101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='C7 = 1100101010101111111100001100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D7 = 0110011110001111010101010110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='C8 = 0010101010111111110000110011 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D8 = 1001111000111101010101011001 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='C9 = 0101010101111111100001100110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D9 = 0011110001111010101010110011 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='C10 = 0101010111111110000110011001 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D10 = 1111000111101010101011001100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='C11 = 0101011111111000011001100101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D11 = 1100011110101010101100110011 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='C12 = 0101111111100001100110010101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D12 = 0001111010101010110011001111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='C13 = 0111111110000110011001010101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D13 = 0111101010101011001100111100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='C14 = 1111111000011001100101010101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D14 = 1110101010101100110011110001 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='C15 = 1111100001100110010101010111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D15 = 1010101010110011001111000111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='C16 = 1111000011001100101010101111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D16 = 0101010101100110011110001111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='TABLE VI: Permuted choice-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='09 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='49 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='46 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='After we apply the permutation PC-2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' it becomes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='K1 = 000110 110000 001011 101111 111111 000111 000001 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='110010 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='K2 = 011110 011010 111011 011001 110110 111100 100111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='100101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='K3 = 010101 011111 110010 001010 010000 101100 111110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='011001 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='K4 = 011100 101010 110111 010110 110110 110011 010100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='011101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='K5 = 011111 001110 110000 000111 111010 110101 001110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='101000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='K6 = 011000 111010 010100 111110 010100 000111 101100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='101111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='K7 = 111011 001000 010010 110111 111101 100001 100010 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='111100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='K8 = 111101 111000 101000 111010 110000 010011 101111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='111011 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='K9 = 111000 001101 101111 101011 111011 011110 011110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='000001 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='K10 = 101100 011111 001101 000111 101110 100100 011001 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='001111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='K11 = 001000 010101 111111 010011 110111 101101 001110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='000110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='K12 = 011101 010111 000111 110101 100101 000110 011111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='101001 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='K13 = 100101 111100 010111 010001 111110 101011 101001 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='000001 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='K14 = 010111 110100 001110 110111 111100 101110 011100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='111010 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='K15 = 101111 111001 000110 001101 001111 010011 111100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='001010 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='K16 = 110010 110011 110110 001011 000011 100001 011111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='110101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Encode each 64-bit block of data Applying the initial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='permutation to the block of text M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' given previously,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' we get M = 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 IP = 1100 1100 0000 0000 1100 1100 1111 1111 1111 0000 1010 1010 1111 0000 1010 1010 Here the 58th bit of M is ‘1’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' which becomes the first bit of IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The 50th bit of M is ‘1’, which becomes the second bit of IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The 7th bit of M is ‘0’, which becomes the last bit of IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Next, divide the permuted block IP into a left half L0 of 32 bit, and a right half R0 of 32 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' From IP, we get L0 and R0 L0 = 1100 1100 0000 0000 1100 1100 1111 1111 R0 = 1111 0000 1010 1010 1111 0000 1010 1010 We now proceed through 16 iterations, for 1≤ n ≤ 16, using a function f which operates on two blocks–a data block of 32 bits and a key Kn of 48 bits–to produce a block of 32 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Let + denote XOR addition, (bit-by-bit addition modulo 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Then for n going from 1 to 16 we calculate (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Ln = Rn − 1 (3) Rn = Ln−1 + f(Rn−1, Kn) (4) 7 This results in a final block, for n = 16, of L16R16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' That is, in each iteration, we take the right 32 bits of the previous result and make them the left 32 bits of the current step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' For the right 32 bits in the current step, we XOR the left 32 bits of the previous step with the calculation f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' For n = 1, we have K1 = 000110 110000 001011 101111 111111 000111 000001 110010 L1 = R0 = 1111 0000 1010 1010 1111 0000 1010 1010 R1 = L0 + f(R0, K1) It remains to explain how the function f works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' To calculate f, first expand each block Rn −1 from 32 bits to 48 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' This is done by using a selection table that repeats some of the bits in Rn − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' We’ll call the use of this selection table the function E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Thus E(Rn − 1) has a 32 bit input block, and a 48 bit output block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' After this, We calculate E(R0) from R0 as follows: R0 = 1111 0000 1010 1010 1111 0000 1010 1010 E(R0) = 011110 100001 010101 010101 011110 100001 010101 010101 Next in the f calculation, we XOR the output E(Rn − 1) with the key Kn : Kn + E(Rn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' For K1, E(R0), we have K1 = 000110 110000 001011 101111 111111 000111 000001 110010 E(R0) = 011110 100001 010101 010101 011110 100001 010101 010101 K1+E(R0) = 011000 010001 011110 111010 100001 100110 010100 100111 To this point we have expanded Rn-1 from 32 bits to 48 bits, using the selection table, and XORed the result with the key Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' We now have 48 bits, or eight groups of six bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' We now do something strange with each group of six bits: we use them as addresses in tables called ”S boxes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Each group of six bits will give us an address in a different S box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Located at that address will be a 4-bit number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' This 4-bit number will replace the original 6 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The net result is that the eight groups of 6 bits are transformed into eight groups of 4 bits (the 4-bit outputs from the S boxes) for 32 bits total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Write the previous result, which is 48 bits, in the form: Kn + E(Rn − 1) = B1B2B3B4B5B6B7B8 where each Bi is a group of six bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' We now calculate it as S1(B1)S2(B2)S3(B3)S4(B4)S5(B5)S6(B6)S7(B7)S8(B8) where Si(Bi) refers to the output of the ith S box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' To repeat, each of the functions S1, S2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=', S8, takes a 6-bit block as input and yields a 4-bit block as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' For the first round, we obtain as the output of the eight S boxes: K1 + E(R0) = 011000 010001 011110 111010 100001 100110 010100 100111 S1(B1)S2(B2)S3(B3)S4(B4)S5(B5)S6(B6)S7(B7)S8(B8) = 0101 1100 1000 0010 1011 0101 1001 0111 The final stage in the calculation of f is to do a permutation P of the S-box output to obtain the final value of f: f = P(S1(B1)S2(B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='S8(B8)) (5) The permutation P is defined in the following table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' P yields a 32-bit output from a 32-bit input by permuting the bits of the input block TABLE VII: Permutation 16 7 20 21 1 5 9 17 29 12 28 17 21 10 18 10 1 15 23 26 26 8 27 09 5 18 31 10 13 2 36 25 2 8 24 14 47 55 15 28 32 27 3 9 33 48 22 27 19 23 30 6 34 53 29 26 22 11 4 25 29 32 4 25 From the output of the eight S boxes: S1(B1)S2(B2)S3(B3)S4(B4)S5(B5)S6(B6)S7(B7)S8(B8) = 0101 1100 1000 0010 1011 0101 1001 0111 we get,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' f = 0010 0011 0100 1010 1010 1001 1011 1011 R1 = L0 + f(R0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' K1) R1 = 1100 1100 0000 0000 1100 1100 1111 1111 + 0010 0011 0100 1010 1010 1001 1011 1011 = 1110 1111 0100 1010 0110 0101 0100 0100 In the next round,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' we will have L2 = R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' which is the block we just calculated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' and then we must calculate R2 = L1 + f(R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' K2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' and so on for 16 rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' At the end of the sixteenth round we have the blocks L16 and R16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' We then reverse the order of the two blocks into the 64-bit block as shown in equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' R16L16 (6) Now, apply a final permutation IP−1 and the output of the algorithm has bit 40 of the preoutput block as its first bit, bit 8 as its second bit, and so on, until bit 25 of the preoutput block is the last bit of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' If we process all 16 blocks using the method defined previously, we get, on the 16th round, L16 = 0100 0011 0100 0010 0011 0010 0011 0100 R16 = 0000 1010 0100 1100 1101 1001 1001 0101 We reverse the order of these two blocks and apply the final permutation to R16L16 = 00001010 01001100 11011001 10010101 01000011 01000010 00110010 00110100 IP −1 = 10000101 11101000 00010011 01010100 00001111 00001010 10110100 00000101 which in hexadecimal format is 85E813540F0AB405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' This is the encrypted form of M = 0123456789ABCDEF: namely, C = 85E813540F0AB405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Decryption is simply the inverse of encryption, following the same steps as above, but reversing the order in which the subkeys are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' SYMMETRIC CIPHERS If we examine the symmetric ciphers in detail, we can see that symmetric ciphers can be divided into two categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' stream ciphers and block ciphers [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Stream ciphers use a key-stream, obtained from the original key, and encrypts the plain-text bit by bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Encryption is usually done by combining the plain-text bits with the corresponding key-stream bits with an XOR operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' In some cases stream 8 ciphers have some advantages over block ciphers, because there is no error propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' It means that an error made in one bit of cipher-text during transmission only affects the decryption of that bit and doesn’t affect other bits [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Block ciphers, on the other hand, take the plain-text bits in blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Each block is encrypted with the same encryption function and the cipher-text blocks are produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' When the length of the plain-text is not a multiple of the block size some padding is applied to the plain-text [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' This padding is usually done by adding a ‘1’ bit followed by necessary amount of ‘0’ bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Because the encryption function does not change from one block to another, same blocks of plain- text are encrypted to same blocks of cipher-text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' When an adversary captures the cipher-text, they can accurately guess some information about the plaintext by using this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' In order to stop any information leakage, some modes of operation are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' ASYMMETRIC CIPHERS While modern symmetric ciphers such as AES are very secure, they have some drawbacks in practicality, namely key distribution problem, and the number of keys [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The key distribution problem occurs when Alice and Bob want to determine a secret key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' This would be easy if they can come together and decide, but if they have no means to decide on a key in person, they have to decide on the key through a secure channel [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Since the communication channel is always assumed to be insecure, because it can be easily hacked, this poses a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Even if they can somehow solve this problem, they would be facing another problem, the number of keys [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' If there are n users in a network, and all of the users want to communicate with each other secretly, the number of encryption keys needed would be n∗(n−1) 2 ,and each user would have n − 1 key pairs they need to know and keep secret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' This becomes exponentially infeasible as the number of people increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The usage of asymmetric ciphers eliminate these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Since every user has a pair of keys, and anything encrypted with a specific public key can only be decrypted with the corresponding private key, Alice and Bob doesn’t need to agree on a secret key together beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' In addition, nobody would need to store n − 1 key pairs, they only need to store their own private and public keys, and the number of key pairs needed in the network would be reduced to n [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Cryptographic protocols can be considered as a third main branch of cryptography, and one of the most important primitives they use is called a hash function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Therefore it would be useful to go over the definition of hash functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' In order to understand how public key algorithms work we can imagine a box [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' For Alice to send Bob a secret message, first Bob sends Alice a box with an open padlock, for which he has the key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Alice then can put her message in the box, and lock it with the padlock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' When Bob receives the box he simply unlocks the padlock and reads the message [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Of course there are still some security concerns, for example an adversary can intercept the box and replace the padlock with their own lock, or put their own message in the box and act like Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' To achieve authentication and to prevent these problems, cryptographers have developed some procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Modes of Operation There are several modes of operation that can be used when encrypting a plaintext with a block cipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' NIST recommends the usage of 5 modes of operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' [11]: Electronic Codebook (ECB) Cipher Block Chaining (CBC) Cipher Feedback (CFB) Output Feedback (OFB) Counter (CTR) In ECB mode, each block is encrypted and decrypted in- dependently from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Because the encryption function does not change, identical blocks of plaintext are encypted to identical blocks of ciphertext [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' In CBC mode, the ciphertext of one block is XORed with the plaintext of the next block before the encryption [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' For the first plaintext block an Initialization Vector IV is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' In CFB mode, ciphertext blocks are encrypted with the encryption function instead of the plaintext blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Plaintext blocks are XORed with the results of encryption function to obtain the ciphertext blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' For the first block an IV is used [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' In OFB mode [25], IV is repeatedly encrypted with the encryption function and the results are XORed with the plaintext blocks to obtain ciphertext blocks [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' In CTR mode, a nonce and counter is encrypted and the result is XORed with the plaintext block [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The counter is increased each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' All of these modes while having different advantages also have some disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' For example, some of them have parallelizable encryption and decryption but others don’t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The decision of which modes of operation is to be used should be made based on the desired security and performance levels [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' RESULTS Implementation of DES has been performed using VHDL and the results is shown in TABLE IX and TABLE VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' TABLE VIII: Performance Matrix of DES on Virtex-7 FPGA Device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Operating Frequency (MHz) Datapath Dalay (nS) Maximum Frequency (MHz) Dynamic Power (mW) 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='829 246 8 TABLE IX: Resource Utilization of DES on Virtex-7 FPGA Device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Slices LUTs Flip-Flops 69 244 139 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' CONCLUSION Architecture Exploration of Simplified Data Encryption Standard (SDES) and Data Encryption Standard (DES) has been done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Simplified DES (SDES) was designed for educa- tional purposes only, to help learn about modern cryptanalytic techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' SDES has similar properties and structure as DES but has been simplified to make it much easier to perform 9 encryption and decryption by hand with pencil and paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Some people feel that learning SDES gives insight into DES and other block ciphers, and insight into various cryptanalytic attacks against them [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' In DES, 64-bit input is encrypted and decrypted using 56-bit key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' At the encryption site, DES takes a 64-bit plaintext and creates a 64-bit ciphertext;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' at the decryption site, DES takes a 64- bit ciphertext and creates a 64-bit block of plaintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' The same 56-bit cipher key is used for both encryption and decryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Implementation of SDES and DES has been performed using Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='7 version and VHDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' During this project I have learned thoroughly about various cryptography techniques and ciphers.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 465–467, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' R uby Kumari, Integrated Dual Degree Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D (IDDP) scholar at Academy of Scientific Innovative Re- search (AcSIR), working area Integrated Circuits and System Group, CSIR-CEERI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Completed B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='tech in Electronics and Communication Engineering from Maulana Abul Kalam Azad University of Technol- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Her research interest includes Cryptography, Lightweight Ciphers, Digital Logic Design, VLSI Architecture and RTL Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Jai Gopal Pandey is a Principal Scientist and work- ing in the CSIR-CEERI, Pilani, India since 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' He is an M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' (Electronics Design and Technology) from U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Technical University, Lucknow, in 2003 and a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' in Electronics Engineering from Birla Institute of Technology and Science (BITS), Pilani, India in 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' His research interests include High-performance Architecture, System-on-chips (SoCs), Embedded Systems, Cryptography, FPGAs, and ASIC designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Pandey is a Senior Member of IEEE and an IETE Fellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' 10 Abhijit Karmakar received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' degree in Electronics and Telecommunication Engineering in 1993 from Jadavpur University, India, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' degree in Electrical Engineering from Indian Insti- tute of Technology (IIT), Madras, India, in 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' He recieved the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' degree in Electrical Engineering from IIT, Delhi, India, in 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' Since 1995, he has been working with the CSIR - Central Electronics Engineering Research Institute (CEERI), Pilani, In- dia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} +page_content=' His research interest span the area of VLSI Design, Signal Processing and related areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE5T4oBgHgl3EQfSg_z/content/2301.05530v1.pdf'} diff --git a/AtE5T4oBgHgl3EQfSw9y/content/tmp_files/2301.05531v1.pdf.txt b/AtE5T4oBgHgl3EQfSw9y/content/tmp_files/2301.05531v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9e73e6e73a1691ab64cf45a54c0da0e85bad21fb --- /dev/null +++ b/AtE5T4oBgHgl3EQfSw9y/content/tmp_files/2301.05531v1.pdf.txt @@ -0,0 +1,781 @@ +Strain-mediated ion-ion interaction in +rare-earth-doped solids +A. Louchet-Chauvet +E-mail: anne.louchet-chauvet@espci.fr +ESPCI Paris, Universit´e PSL, CNRS, Institut Langevin, 75005 Paris, France +T. Chaneli`ere +Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut N´eel, 38000 Grenoble, France +Abstract. +It was recently shown that the optical excitation of rare-earth ions produces a +local change of the host matrix shape, attributed to a change of the rare-earth ion’s +electronic orbital geometry. In this work we investigate the consequences of this piezo- +orbital backaction and show from a macroscopic model how it yields a disregarded +ion-ion interaction mediated by mechanical strain. +This interaction scales as 1/r3, +similarly to the other archetypal ion-ion interactions, namely electric and magnetic +dipole-dipole interactions. We quantitatively assess and compare the magnitude of these +three interactions from the angle of the instantaneous spectral diffusion mechanism, and +reexamine the scientific literature in a range of rare-earth doped systems in the light of +this generally underestimated contribution. +arXiv:2301.05531v1 [cond-mat.mtrl-sci] 13 Jan 2023 + +Strain-mediated ion-ion interaction in rare-earth-doped solids +2 +1. Introduction +Atomic ensembles have attracted a lot of attention over more than 20 years due to their +inherent capacity to efficiently interact with light [1]. They are at the center of a number of +quantum storage protocols, in the form of gases [2, 3] or solid state [4, 5]. Among the most +interesting solid state candidates, rare-earth ion-doped crystals (REIC) are particularly +attractive due to their long optical coherence lifetimes at cryogenic temperatures [6] and +are at the center of a number of actively developed quantum memory protocols [7, 8, 9]. +In these light-matter interfaces, the optical depth of the medium is an important figure of +merit since it enables high storage and retrieval efficiency [10, 11, 5]. However, reaching +large optical depths usually comes with working with large atomic concentrations, leading +to reinforced ion-ion interactions and thereby enhanced decoherence [12, 13, 14, 15, 16, 17]. +Interestingly, although ion-ion interactions are potentially detrimental to the +performance of quantum devices, they have also emerged as the foundational mechanism +for quantum computing architectures because they allow multiqubit gate operations [18]. +Quantum computing was initially developed in physical systems with strong readout +capacity ranging from dilute systems (trapped atoms) to condensed matter (quantum +dots or superconducting qubits). +Rare-earth ion-doped crystals were only recently +considered as relevant for such applications [19] thanks to the elaboration of efficient +single ion readout schemes that compensate for the optical transition’s weak oscillator +strength [20, 21]. +Either way, proper understanding and quantifying of ion-ion interactions in REIC +are crucial. +In most systems, the order of magnitude of the measured interaction +strength is compatible with magnetic and/or electric dipole-dipole interaction. In both +mechanisms, the excitation of some ions induces a change in their electric or magnetic +dipole moment, modifying the local field accordingly in their vicinity. +This modified +field affects the surrounding ions in proportion to the Stark or Zeeman sensitivity of +their energy levels. +In a limited number of REIC, however, the actual magnitude of +this interaction is larger than expected by several orders of magnitude [22, 23]. In this +paper we consider a strain-mediated ion-ion interaction that has not been investigated +so far and that may explain this discrepancy. The interaction we consider stems from +the apparition of an excitation-induced stress field, affecting the surrounding ions via +their piezospectroscopic sensivity. This fundamental effect has been pointed out early +in the context of paramagnetic resonance under RF excitation, named virtual phonon +exchange interaction initially considering transition metals ions [24, 25, 26]. At the time, +the description exploited the equivalent operators formalism for the spin-lattice coupled +system, but the different modeling parameters are difficult to infer from experimental +measurements. Despite a series of studies including rare-earth salts [27, and references], +phonon-mediated interactions seem to have been overlooked for years, before reappearing +in the context of quantum technologies with original proposals of phononic engineering +[28, 29]. + +Strain-mediated ion-ion interaction in rare-earth-doped solids +3 +This paper is organized as follows. In Sec. 2, the physical origin of this interaction +is presented. A scaling law is given and an estimation of its strength is provided. In +Sec. 3, we quantitatively compare the magnitude of this strain-mediated interaction with +respect to the electromagnetic dipole-dipole interactions in a number of host-dopant +combinations and confront our predictions with the experimental data found in the +scientific literature. The strength of the instantaneous spectral diffusion mechanism is +used as a comparison criterion. Finally, in Sec. 4, we discuss the implications of this work +in quantum technology-related applications. +2. Strain-mediated ion-ion interaction +2.1. General description +When an atomic particle is promoted to a different electronic level, its outer shape and +size are bound to change due to the modification of its electronic wavefunction. If the +particle is embedded in a solid matrix, this piezo-orbital backaction effect will additionally +give rise to a stress field around the excited particle, making it detectable via a change +of shape of the solid itself. This was recently evidenced in a bulk rare-earth ion-doped +crystal in which only a finite volume of the crystal was illuminated, leading to a distortion +of the nearby crystal surface [30]. +Conversely, it has been known for several decades that the optical lines of rare-earth +ions in solids are sensitive to stress via the piezospectroscopic effect [31, 32]. Indeed, +in the elastic regime, a compressive or tensile stress modifies the interatomic distances. +This affects the crystal field and in turn shifts the rare-earth ion’s energy levels. This +sensitivity to stress was recently proposed as a tool to sense mechanical vibrations in a +cryogenic environment [33, 34]. +The piezo-orbital backaction and the piezospectroscopic effect are in fact two facets +of the same coupling mechanism. Their combination leads to a shift of the transition +frequency in the ions surrounding a given excited rare-earth ion. In the following we coin +this as the strain-mediated ion-ion interaction. +2.2. Magnitude of the strain-mediated interaction +For simplicity we assume that the ion is spherical with a radius r1 and that the piezo- +orbital backaction acts as a simple ionic radius change (see Fig. 1). This impacts the +surrounding matrix by creating a radial stress field σ(r) around the ion. With a spherically +symmetric continuum mechanics model detailed in appendix Appendix A, we calculate +the elastic strain energy that is necessary to establish this stress field. Due to energy +conservation, this elastic energy corresponds to the energy shift of the electronic levels +due to the internal stress within the ionic volume (see appendix Appendix B). This allows + +Strain-mediated ion-ion interaction in rare-earth-doped solids +4 +Figure 1. Simplified view of the piezo-orbital backaction around a spherical rare-earth +ion. The stress field is symbolized by a radial color gradient around the ion. +us to relate the ionic radius variation ∆r to the piezospectroscopic sensitivity κ: +∆r = hκ +4πr2 +1 +(1) +where h is the Planck constant. With this we derive the radial stress field σ(r) and write +the strain-mediated ion-ion interaction energy between two ions separated by a distance +r: +Estr(r) = hκσ(r) = +E +1 + ν +(hκ)2 +2πr3 +(2) +where E is the Young modulus and ν is the Poisson’s ratio of the crystal. We emphasize +the relatively strong hypotheses underlying this result: the piezo-orbital backaction is +assumed to occur in the form of a mere ionic radius change of the spherical excited ion, +and the piezospectroscopic sensitivity is assumed to be scalar. The anisotropic nature of +the crystalline matrix may naturally translate into a non-spherical strain field. Because +of the large distance between dopants (larger than the crystal cell parameter at low +concentration levels), we may expect the strain anisotropy to be weak. On the contrary, +the piezospectroscopic sensitivity is usually described by a tensor [35], so its scalar nature +appears as a crude assumption in order to derive an order of magnitude. +Using Eq. 1 we can estimate the corresponding relative ionic radius change due +to the piezo-orbital backaction in rare-earth-doped crystals. +Taking typical values +κ = 100 Hz/Pa [33] and r1 = 1 ˚A [36], we obtain ∆r/r1 ≃ 5·10−3. This value remarkably +agrees with the relative ionic radius change of a free ion among its 4f states that can be +derived from a Hartree-Fock calculation (see [37] for Ce3+ and [38] for Eu3+) even if the +exact rearrangement of the outer shell defining the ionic radius surrounding the 4f-shell +deserves further analysis. +3. Comparing the three ion-ion interactions +In this section we propose to study how this interaction compares with the other usual +ion-ion interactions generally considered in rare-earth doped systems, i.e. electric and +magnetic dipole-dipole interactions. +We also confront these estimations to published +measurements of ion-ion interactions in rare-earth doped crystals. + +(a) lon at rest +(b) Excited ion +ri ＋△r +r1Strain-mediated ion-ion interaction in rare-earth-doped solids +5 +Interestingly, the strain-mediated interaction scales as 1/r3, exactly like the electric +and magnetic dipole-dipole interactions, although originating from a radically different +physical mechanism. All three interactions can be characterized by a coefficient Ai such +that their energies read as: +Ei(r) = hAi +2πr3 +(3) +where the {Ai} are defined by the following: +Ael += π¯h +ϵ0ϵr +∆µ2 +el +(4) +Amag = µ0¯h +4π ∆µ2 +mag +(5) +Astr = +E +1 + ν hκ2 +(6) +∆µel and ∆µmag are the electric and magnetic dipole moment variation when the ion +is promoted to the excited state. +They are expressed in Hz m V−1 and in Hz T−1, +respectively. We note that in all three expressions, the sensitivity of the optical transition +to electric field, magnetic field or strain (respectively) appears as a square law, in +agreement with what is expected in an ion-ion interaction. +This scaling has been early derived for paramagnetic impurities under RF +excitation [24, 25], in the so-called zero-retardation limit of the virtual phonon exchange +(VPE) interaction [26]. At the time, McMahon et al. also showed that VPE and magnetic +interactions have the same order of magnitude for transition metal paramagnetic dopants +[25]. +With these expressions one can anticipate the variability of the three considered +interactions amongst REIC. For example, the strain-mediated interaction should be +quite constant across different ions or crystals, since the mechanical properties of typical +crystalline rare-earth doped oxides and the piezospectroscopic sensitivity of their optical +lines are rather similar [33]. On the other hand, the magnetic dipole-dipole interactions +vary by several orders of magnitude between Kramers and non-Kramers ions because some +exhibit an electronic spin while some only have a nuclear spin behaviour. A variability +of 6 orders of magnitude is expected, since (µB/µN)2 ≃ 3 · 106, where µB and µN are +the Bohr magneton and the nuclear magneton, respectively. A large variability is also +expected for the electric dipole-dipole interaction because the appearance of an electric +dipole is only permitted by the crystal field that weakly perturbs the free ion electronic +structure. Indeed different hosts, depending on the crystal site symmetry, allow or inhibit +permanent electric dipole moment for the rare earth ion dopant. +The overall ion-ion interactions taking place in an ensemble of rare-earth ions can be +probed experimentally by the observation of instantaneous spectral diffusion (ISD) [39], +ie the effect of a substantial population transfer on the spectral width of a narrow subset +of ions within a given volume. When it builds upon interactions scaling as 1/r3, ISD + +Strain-mediated ion-ion interaction in rare-earth-doped solids +6 +manifests as an additional term in the homogeneous linewidth Γh that is proportional to +the volumic density of excited particles ne: +Γeff = Γh + β +2 ne +(7) +where β += +� +i βi is the sum of individual contributions due to each considered +interaction [40, 14]: +βi = 8π +9 +√ +3106Ai, +(8) +the 106 factor stemming from the choice of units (m3 s−1 for Ai and Hz cm3 for βi). +ISD is generally observed via high-resolution spectroscopy experiments such as +photon echoes [41, 42, 43]. It has been evidenced in a large number of rare-earth ion- +doped materials, but a quantitative value is only given in a handful of them. The published +experimental values for β in ten different rare-earth ion-doped crystals are given in Table 1. +These materials represent a rather complete sampling of the REIC diversity, including +non-Kramers and Kramers ions, different crystal and site symmetries, and crystals with +and without charge compensation. The measured values of β are contained within a 2 +order-of-magnitude range (between 10−13 and 10−11 Hz cm3). +Cristal +βexp (Hz cm3) +Tm:YAG +2.3 · 10−12 [43] +Tm:YGG +2.6 · 10−13 [43] +Tm:LiNbO3 +1.0 · 10−11 [43] +Er:YSO +1.3 · 10−12 [44] +Er:LiNbO3 +1.0 · 10−13 [45] +Eu:YSO +9.0 · 10−13 [46] +Eu:Y2O3 +1.3 · 10−13 [47] +EuCl3·6D2O +4.6 · 10−13 [22] +Pr:YSO +1.2 · 10−11 [48] +Pr:La2(WO4)3 +7.0 · 10−12 [49] +Table 1. Measured values for the ISD coefficient βexp in a selection of rare-earth ion- +doped crystals. +The three ion-ion interactions considered in this work (see equations 4, 5 and 6) +should contribute to ISD. Focusing our attention to the selection of REICs for which +quantitative ISD measurements are available, we calculate the βi parameters for each ion- +ion interactions using Eq. 8 and display the results graphically in Figure 2. The material +parameters relevant to this calculation and resulting numerical estimations are given in +appendix Appendix C. Again, we point out that the values are mostly indicative since +they rely on a very simplified modelization of the interactions. +We verify that both magnetic and electric dipole-dipole interactions vary among +the materials within a 6 order-of-magnitude span depending on the existence of an + +Strain-mediated ion-ion interaction in rare-earth-doped solids +7 +Figure 2. +Calculated (bars) and measured (triangles) values for βi and Ai (where +i = {mag, el, str, exp}) for the 10 rare-earth ion-doped crystals for which quantitative +values for βexp are available (see Table 1). Note the logarithmic vertical scale spanning +10 orders of magnitude. +electronic spin and the presence of a centrosymmetry in the doping site (βmag is comprised +between 7 · 10−20 and 2 · 10−13 Hz cm3and βel between 8 · 10−19 and 8 · 10−12 Hz cm3, +respectively). Conversely, the strain-mediated interaction is comprised within a much +smaller interval (βstr between 6 · 10−13 and 2 · 10−12 Hz cm3) since we use a typical value +for the piezospectroscopic sensitivity κ for all and because of the very similar values of +the Young moduli between crystals. +It is interesting to note that the strain-mediated interaction is of the order of +the largest of the electric or magnetic dipole-dipole contributions found among all +materials. +This means that when one or both of the dipole-dipole interactions are +strong, they should coexist with the strain-mediated interaction. On the other hand, +when both electromagnetic contributions are in the low range (weak Stark effect and +no electronic spin for the non-Kramers ions, e.g. Tm-doped garnets or in a lesser part +stoechiometric europium chloride), the strain-mediated interaction dominates by several +orders of magnitude. +In all considered REICs, the sum of the three predicted effects +satisfactorily accounts for the measured values of βexp, finally providing an explanation +to the previously large discrepancy between theoretical estimations and measurements of +the ISD mechanism in some media. + +10-10 +(Hz cm²) +） 10-12 +S +10-20 +A +10-14 +ISD coeficient β +Magnetic +Electric +Strain +10-22 +Exp +10 +16 +Interaction +10- +24 +10-18 +10-26 +10-20 +YSO +Tm:YAG +Tm:YGG +Eu:YSO +EuCl'Strain-mediated ion-ion interaction in rare-earth-doped solids +8 +4. Discussion +Besides providing a better understanding of the physical origin and strength of ISD in +REIC, the strain-mediated interaction in REIC could have interesting applications in +the field of quantum computing. +Indeed, quantum computing schemes rest upon the +existence of a long-range atom-atom interaction enabling multiqubit operations [18]. This +interaction is often spontaneously assimilated to dipole-dipole interaction, and particularly +in REIC [50, 19]. +We argue that the strain-mediated interaction could play this role +in quantum-computing to replace the dipole-dipole coupling. Such a scheme could be +performed in almost any rare-earth ion-doped crystal since the strength of the interaction +is rather similar over a broad variety of ion and host combinations [33]. In particular, +the possession of a permanent electric dipole moment would not be an exclusive criteria +for quantum computing compatibility. On the contrary, in some crystals (namely Tm- +doped garnets, although there could be other candidates) this strain-mediated interaction +dominates its electromagnetic counterparts by at least 4 orders of magnitude. This means +that not only is the ion-ion interaction particularly pure in such media, but these strain- +coupled qubits would be more robust against other types of decoherence process, such +as magnetic field fluctuations due to spin flips in the host matrix [51, 13], or electric +field noise due to charge fluctuations that may occur at the surface of rare-earth-doped +nanoparticles [52]. +One may argue that since the strain-mediated interaction is intrinsically slow since it +relies on the propagation of stress at the speed of sound (between 3000 and 8000 m/s in +most considered crystals). We estimate its propagation delay by considering the average +ion-ion distance dion−ion, given by +3√nRE (where nRE is the volumic density of rare-earth +ions). dion−ion ranges from a few nm in highly concentrated materials (e.g. ∼ 2 nm in +1% doped YAG) up to hundreds of nm in low concentration materials (e.g. ∼ 650 nm in +200ppm doped YSO). This leads to a strain propagation time shorter than the nanosecond, +to be compared with typical µs-scale light-matter interactions occuring in such media. +Therefore, the strain-mediated interaction can still be considered instantaneous within +rare-earth-doped crystals with typical concentrations. +5. Conclusion +Based on recent evidence of a conservative optomechanical backaction mechanism in rare- +earth ion-doped crystals, we have unveiled a disregarded ion-ion interaction based on a +physical mechanism fundamentally different from generally considered electromagnetic +dipole-dipole interactions: the sensitivity of rare-earth ions to piezo-orbitally induced +stress. With a simple mechanical model, we have estimated the strength of this strain- +mediated interaction and shown that it is largely dominant in some rare-earth ion-doped +crystals, opening interesting perspectives for strain-based quantum computing in solids. + +Strain-mediated ion-ion interaction in rare-earth-doped solids +9 +6. Acknowledgments +The authors are grateful to Lars Rippe and Xiaoping Jia for helpful discussions. +The authors acknowledge support from the French National Research Agency (ANR) +through the projects ATRAP (ANR-19-CE24-0008), MIRESPIN (ANR-19-CE47-0011) +and MARS (ANR-20-CE92-0041). This work has received support under the program +“Investissements d’Avenir” launched by the French Government. +Appendix A. Continuum mechanics in a REIC +Appendix A.1. Spherical defect in an isotropic elastic medium +Let us consider a sphere with radius r1, embedded in an infinite, isotropic, continuous +elastic medium with a Young modulus E and a Poisson’s ratio ν. When a homogeneous +radial stress σ0 is applied in the sphere, the displacement field at a distance r outside the +sphere is radial and obeys [53]: +u(r) = σ0 +1 + ν +2E +r3 +1 +r2, +(A.1) +while the stress field around the sphere is also radial and reads as: +σ(r) = σ0 +r3 +1 +r3 for r > r1 +(A.2) +Defining ∆r = u(r1) as the radius change, we obtain a simple relationship between +σ0 et ∆r: +σ0 = ∆r +r1 +2E +1 + ν +(A.3) +Figure A1. +Radial stress σ(r) corresponding to the dilation of a sphere within an +infinite isotropic medium. The stress is homogeneous within the sphere and decays with +a 1/r3 law outside. +The radial stress can then be written: +σ(r) = −∆r +r1 +2E +1 + ν +r3 +1 +r3 +(A.4) + +/0 +() +1 +1 stress ( +0.8 +Normalized radial +0.6 +0.4 +0.2 +0 +0 +1 +2 +3 +4 +5 +Normalized distance to ion r/riStrain-mediated ion-ion interaction in rare-earth-doped solids +10 +Appendix A.2. Strain energy +We now want to assess the elastic strain energy contained in the medium when the stress +σ0 is applied. +We successively calculate the energy contained inside and outside the +sphere [54]. +Inside the sphere: +The strain energy inside the sphere reads as: +Uint = 1 +2σ0ε0V +(A.5) +where ε0 = ∆V/V = 3∆r/r1 is the volumic change of the sphere. We finally obtain: +Uint = 2π∆rσ0r2 +1 +(A.6) +Outside the sphere: +The sphere being included in an infinite medium, we must also +consider the strain energy that was necessary to establish the whole stress field around +the sphere. The volumic energy outside the sphere at a distance r reads as: +uV (r) = 1 +2σ(r)ε(r) +(A.7) +The volumic strain ε(r) is related to the local stress via the volumic Hooke’s law +σ(r) = Kε(r) (where K = +E +1−2ν is the bulk modulus). Using Eq. A.2 we get ε(r) = ε0r3 +1/r3, +which finally leads to: +uV (r) = 3∆r +2 σ0 +r5 +1 +r6 +(A.8) +We integrate this volumic energy over the infinite volume of the medium: +Uext = 4π +� ∞ +r=r1 +uV (r)r2dr = 6π∆rσ0r5 +1 +� ∞ +r=r1 +1 +r4dr +(A.9) +and obtain: +Uext = 2π∆rσ0r2 +1 +(A.10) +Total strain energy: +Based on Eqs. A.6 and A.10, we derive the total strain energy that +is necessary to distort the sphere and generate the associated stress field around it: +Ustrain = Uint + Uext = 4π∆rσ0r2 +1 +(A.11) +Appendix B. Relating the strain energy with the atomic energy +Due to piezo-orbital backaction [30], the size of a rare-earth ion is expected to change +under optical excitation. Assuming this change of shape is merely a change of radius on +a spherical ion, we can apply the calculation presented in appendix Appendix A. Due +to energy conservation, this elastic energy is taken from the ion’s energy levels. We can + +Strain-mediated ion-ion interaction in rare-earth-doped solids +11 +therefore write the following, considering that the shift in the ion’s energy levels ∆E is +related to the stress within the ion σ0. +Ustrain = ∆E = hκσ0 +(B.1) +where κ is the piezospectroscopic sensitivity, assumed scalar. Injecting the expression of +Ustrain given by Eq. A.11, we get: +∆r = hκ +4πr2 +1 +(B.2) +This allows us to obtain a simple relationship between the radial stress σ(r) and the ionic +radius change ∆r, using Eq. A.4: +σ(r) = +2E +1 + ν +hκ +4πr3 +(B.3) +Appendix C. Instantaneous Spectral Diffusion +In Table C1 we list the physical parameters that are needed for the estimation of ISD +strengths for an ensemble of 10 REIC. Note that some values had to be estimated using +data measured in similar materials. The value for the piezospectroscopic sensitivity κ +being basically unknown for most REICs, we choose to take κ = 100 Hz/Pa for all. This +choice is supported by the observed similarity of the value of κ in a broad variety of hosts, +dopants and transitions [33]. +Crystal +E +ν +∆µmag +ϵr +∆µel +Tm:YAG +270 GPa [55] +0.256 [55] +320 MHz/T [56] +10.6 [57] +65 Hz cm/V [58] +Tm:YGG +224 GPa [59] +0.28 [59] +44 MHz/T [60] +12 [57] +⋆65 Hz cm/V +Tm:LiNbO3 +170 GPa [61] +0.25 [61] +1.3 GHz/T [62] +†65 [63] +18 kHz cm/V [62] +Er:YSO +150 GPa [64] +0.26 [64] +26 GHz/T [65] +10 [66] +50 kHz cm/V [67] +Er:LiNbO3 +170 GPa [61] +0.25 [61] +16 GHz/T [68] +†65 [63] +25 kHz cm/V [69] +Eu:YSO +150 GPa [64] +0.26 [64] +10 MHz/T [70] +10 [66] +35 kHz cm/V [71, 72] +Eu:Y2O3 +⋆120 GPa +⋆0.25 +⋆10 MHz/T +15 [73] +⋆35 kHz cm/V +EuCl3·6D2O +⋆120 GPa +⋆0.25 +8 MHz/T [74] +3.6 [22] +1.57 kHz cm/V [22] +Pr:YSO +150 GPa [64] +0.26 [64] +150 MHz/T [75] +10 [66] +111 kHz cm/V [76] +Pr:La2(WO4)3 +90 GPa [77] +0.3 [77] +⋆150 MHz/T +20 [78] +⋆100 kHz cm/V +Table C1. Material parameters used to compute the interaction strengths. E is the +mechanical Young modulus, ν the Poisson’s ratio, and ϵr is the dielectric constant of +the host matrix. +∆µmag and ∆µel are the difference in the rare earth magnetic or +electric moments between ground and excited states. When no values could be found +in the literature, we chose a value among similar materials and indicated this with the +symbol ”⋆”. †: The lithium niobate (LNO) crystal is anisotropic and exhibits different +values of ϵr depending on the crystallographic direction [63]. For simplicity we choose +an intermediate value. + +Strain-mediated ion-ion interaction in rare-earth-doped solids +12 +In Table C2 we present the different ISD coefficients βi calculated for the three +possible ion-ion interactions (magnetic dipole-dipole interaction, electric dipole-dipole +interaction, and strain-mediated interaction), using the parameters listed in Table C1 +and Eqs. 4, 5 and 6. +Crystal +βmag +βel +βstr +Tm:YAG +6.9 · 10−17 +2.0 · 10−18 +2.3 · 10−12 +Tm:YGG +1.3 · 10−18 +8.3 · 10−19 +1.9 · 10−12 +Tm:LiNbO3 +1.1 · 10−15 +3.0 · 10−14 +1.5 · 10−12 +Er:YSO +4.5 · 10−13 +3.8 · 10−13 +1.3 · 10−12 +Er:LiNbO3 +1.7 · 10−13 +5.8 · 10−14 +1.5 · 10−12 +Eu:YSO +6.7 · 10−20 +7.4 · 10−13 +1.3 · 10−12 +Eu:Y2O3 +6.7 · 10−20 +4.9 · 10−13 +1.2 · 10−12 +EuCl3·6D2O +4.3 · 10−20 +4.1 · 10−15 +8.2 · 10−13 +Pr:YSO +1.5 · 10−17 +7.5 · 10−12 +1.3 · 10−12 +Pr:La2(WO4)3 +1.5 · 10−17 +3.0 · 10−12 +6.3 · 10−13 +Table C2. 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Data 2 313–410 + diff --git a/AtE5T4oBgHgl3EQfSw9y/content/tmp_files/load_file.txt b/AtE5T4oBgHgl3EQfSw9y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..132951ba5b4dc67ceb8dc40b1c7bb1ea5c97c580 --- /dev/null +++ b/AtE5T4oBgHgl3EQfSw9y/content/tmp_files/load_file.txt @@ -0,0 +1,452 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf,len=451 +page_content='Strain-mediated ion-ion interaction in rare-earth-doped solids A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Louchet-Chauvet E-mail: anne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='louchet-chauvet@espci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='fr ESPCI Paris, Universit´e PSL, CNRS, Institut Langevin, 75005 Paris, France T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Chaneli`ere Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Grenoble Alpes, CNRS, Grenoble INP, Institut N´eel, 38000 Grenoble, France Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' It was recently shown that the optical excitation of rare-earth ions produces a local change of the host matrix shape, attributed to a change of the rare-earth ion’s electronic orbital geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' In this work we investigate the consequences of this piezo- orbital backaction and show from a macroscopic model how it yields a disregarded ion-ion interaction mediated by mechanical strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' This interaction scales as 1/r3, similarly to the other archetypal ion-ion interactions, namely electric and magnetic dipole-dipole interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' We quantitatively assess and compare the magnitude of these three interactions from the angle of the instantaneous spectral diffusion mechanism, and reexamine the scientific literature in a range of rare-earth doped systems in the light of this generally underestimated contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='05531v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='mtrl-sci] 13 Jan 2023 Strain-mediated ion-ion interaction in rare-earth-doped solids 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Introduction Atomic ensembles have attracted a lot of attention over more than 20 years due to their inherent capacity to efficiently interact with light [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' They are at the center of a number of quantum storage protocols, in the form of gases [2, 3] or solid state [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Among the most interesting solid state candidates, rare-earth ion-doped crystals (REIC) are particularly attractive due to their long optical coherence lifetimes at cryogenic temperatures [6] and are at the center of a number of actively developed quantum memory protocols [7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' In these light-matter interfaces, the optical depth of the medium is an important figure of merit since it enables high storage and retrieval efficiency [10, 11, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' However, reaching large optical depths usually comes with working with large atomic concentrations, leading to reinforced ion-ion interactions and thereby enhanced decoherence [12, 13, 14, 15, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Interestingly, although ion-ion interactions are potentially detrimental to the performance of quantum devices, they have also emerged as the foundational mechanism for quantum computing architectures because they allow multiqubit gate operations [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Quantum computing was initially developed in physical systems with strong readout capacity ranging from dilute systems (trapped atoms) to condensed matter (quantum dots or superconducting qubits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Rare-earth ion-doped crystals were only recently considered as relevant for such applications [19] thanks to the elaboration of efficient single ion readout schemes that compensate for the optical transition’s weak oscillator strength [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Either way, proper understanding and quantifying of ion-ion interactions in REIC are crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' In most systems, the order of magnitude of the measured interaction strength is compatible with magnetic and/or electric dipole-dipole interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' In both mechanisms, the excitation of some ions induces a change in their electric or magnetic dipole moment, modifying the local field accordingly in their vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' This modified field affects the surrounding ions in proportion to the Stark or Zeeman sensitivity of their energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' In a limited number of REIC, however, the actual magnitude of this interaction is larger than expected by several orders of magnitude [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' In this paper we consider a strain-mediated ion-ion interaction that has not been investigated so far and that may explain this discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' The interaction we consider stems from the apparition of an excitation-induced stress field, affecting the surrounding ions via their piezospectroscopic sensivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' This fundamental effect has been pointed out early in the context of paramagnetic resonance under RF excitation, named virtual phonon exchange interaction initially considering transition metals ions [24, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' At the time, the description exploited the equivalent operators formalism for the spin-lattice coupled system, but the different modeling parameters are difficult to infer from experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Despite a series of studies including rare-earth salts [27, and references], phonon-mediated interactions seem to have been overlooked for years, before reappearing in the context of quantum technologies with original proposals of phononic engineering [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Strain-mediated ion-ion interaction in rare-earth-doped solids 3 This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' 2, the physical origin of this interaction is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' A scaling law is given and an estimation of its strength is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' 3, we quantitatively compare the magnitude of this strain-mediated interaction with respect to the electromagnetic dipole-dipole interactions in a number of host-dopant combinations and confront our predictions with the experimental data found in the scientific literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' The strength of the instantaneous spectral diffusion mechanism is used as a comparison criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' 4, we discuss the implications of this work in quantum technology-related applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Strain-mediated ion-ion interaction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' General description When an atomic particle is promoted to a different electronic level, its outer shape and size are bound to change due to the modification of its electronic wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' If the particle is embedded in a solid matrix, this piezo-orbital backaction effect will additionally give rise to a stress field around the excited particle, making it detectable via a change of shape of the solid itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' This was recently evidenced in a bulk rare-earth ion-doped crystal in which only a finite volume of the crystal was illuminated, leading to a distortion of the nearby crystal surface [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Conversely, it has been known for several decades that the optical lines of rare-earth ions in solids are sensitive to stress via the piezospectroscopic effect [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Indeed, in the elastic regime, a compressive or tensile stress modifies the interatomic distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' This affects the crystal field and in turn shifts the rare-earth ion’s energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' This sensitivity to stress was recently proposed as a tool to sense mechanical vibrations in a cryogenic environment [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' The piezo-orbital backaction and the piezospectroscopic effect are in fact two facets of the same coupling mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Their combination leads to a shift of the transition frequency in the ions surrounding a given excited rare-earth ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' In the following we coin this as the strain-mediated ion-ion interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Magnitude of the strain-mediated interaction For simplicity we assume that the ion is spherical with a radius r1 and that the piezo- orbital backaction acts as a simple ionic radius change (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' This impacts the surrounding matrix by creating a radial stress field σ(r) around the ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' With a spherically symmetric continuum mechanics model detailed in appendix Appendix A, we calculate the elastic strain energy that is necessary to establish this stress field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Due to energy conservation, this elastic energy corresponds to the energy shift of the electronic levels due to the internal stress within the ionic volume (see appendix Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' This allows Strain-mediated ion-ion interaction in rare-earth-doped solids 4 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Simplified view of the piezo-orbital backaction around a spherical rare-earth ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' The stress field is symbolized by a radial color gradient around the ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' us to relate the ionic radius variation ∆r to the piezospectroscopic sensitivity κ: ∆r = hκ 4πr2 1 (1) where h is the Planck constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' With this we derive the radial stress field σ(r) and write the strain-mediated ion-ion interaction energy between two ions separated by a distance r: Estr(r) = hκσ(r) = E 1 + ν (hκ)2 2πr3 (2) where E is the Young modulus and ν is the Poisson’s ratio of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' We emphasize the relatively strong hypotheses underlying this result: the piezo-orbital backaction is assumed to occur in the form of a mere ionic radius change of the spherical excited ion, and the piezospectroscopic sensitivity is assumed to be scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' The anisotropic nature of the crystalline matrix may naturally translate into a non-spherical strain field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Because of the large distance between dopants (larger than the crystal cell parameter at low concentration levels), we may expect the strain anisotropy to be weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' On the contrary, the piezospectroscopic sensitivity is usually described by a tensor [35], so its scalar nature appears as a crude assumption in order to derive an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' 1 we can estimate the corresponding relative ionic radius change due to the piezo-orbital backaction in rare-earth-doped crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Taking typical values κ = 100 Hz/Pa [33] and r1 = 1 ˚A [36], we obtain ∆r/r1 ≃ 5·10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' This value remarkably agrees with the relative ionic radius change of a free ion among its 4f states that can be derived from a Hartree-Fock calculation (see [37] for Ce3+ and [38] for Eu3+) even if the exact rearrangement of the outer shell defining the ionic radius surrounding the 4f-shell deserves further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Comparing the three ion-ion interactions In this section we propose to study how this interaction compares with the other usual ion-ion interactions generally considered in rare-earth doped systems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' electric and magnetic dipole-dipole interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' We also confront these estimations to published measurements of ion-ion interactions in rare-earth doped crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' (a) lon at rest (b) Excited ion ri ＋△r r1Strain-mediated ion-ion interaction in rare-earth-doped solids 5 Interestingly, the strain-mediated interaction scales as 1/r3, exactly like the electric and magnetic dipole-dipole interactions, although originating from a radically different physical mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' All three interactions can be characterized by a coefficient Ai such that their energies read as: Ei(r) = hAi 2πr3 (3) where the {Ai} are defined by the following: Ael = π¯h ϵ0ϵr ∆µ2 el (4) Amag = µ0¯h 4π ∆µ2 mag (5) Astr = E 1 + ν hκ2 (6) ∆µel and ∆µmag are the electric and magnetic dipole moment variation when the ion is promoted to the excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' They are expressed in Hz m V−1 and in Hz T−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' We note that in all three expressions, the sensitivity of the optical transition to electric field, magnetic field or strain (respectively) appears as a square law, in agreement with what is expected in an ion-ion interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' This scaling has been early derived for paramagnetic impurities under RF excitation [24, 25], in the so-called zero-retardation limit of the virtual phonon exchange (VPE) interaction [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' At the time, McMahon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' also showed that VPE and magnetic interactions have the same order of magnitude for transition metal paramagnetic dopants [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' With these expressions one can anticipate the variability of the three considered interactions amongst REIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' For example, the strain-mediated interaction should be quite constant across different ions or crystals, since the mechanical properties of typical crystalline rare-earth doped oxides and the piezospectroscopic sensitivity of their optical lines are rather similar [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' On the other hand, the magnetic dipole-dipole interactions vary by several orders of magnitude between Kramers and non-Kramers ions because some exhibit an electronic spin while some only have a nuclear spin behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' A variability of 6 orders of magnitude is expected, since (µB/µN)2 ≃ 3 · 106, where µB and µN are the Bohr magneton and the nuclear magneton, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' A large variability is also expected for the electric dipole-dipole interaction because the appearance of an electric dipole is only permitted by the crystal field that weakly perturbs the free ion electronic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Indeed different hosts, depending on the crystal site symmetry, allow or inhibit permanent electric dipole moment for the rare earth ion dopant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' The overall ion-ion interactions taking place in an ensemble of rare-earth ions can be probed experimentally by the observation of instantaneous spectral diffusion (ISD) [39], ie the effect of a substantial population transfer on the spectral width of a narrow subset of ions within a given volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' When it builds upon interactions scaling as 1/r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' ISD Strain-mediated ion-ion interaction in rare-earth-doped solids 6 manifests as an additional term in the homogeneous linewidth Γh that is proportional to the volumic density of excited particles ne: Γeff = Γh + β 2 ne (7) where β = � i βi is the sum of individual contributions due to each considered interaction [40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' 14]: βi = 8π 9 √ 3106Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' (8) the 106 factor stemming from the choice of units (m3 s−1 for Ai and Hz cm3 for βi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' ISD is generally observed via high-resolution spectroscopy experiments such as photon echoes [41, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' It has been evidenced in a large number of rare-earth ion- doped materials, but a quantitative value is only given in a handful of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' The published experimental values for β in ten different rare-earth ion-doped crystals are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' These materials represent a rather complete sampling of the REIC diversity, including non-Kramers and Kramers ions, different crystal and site symmetries, and crystals with and without charge compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' The measured values of β are contained within a 2 order-of-magnitude range (between 10−13 and 10−11 Hz cm3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Cristal βexp (Hz cm3) Tm:YAG 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='3 · 10−12 [43] Tm:YGG 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='6 · 10−13 [43] Tm:LiNbO3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='0 · 10−11 [43] Er:YSO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='3 · 10−12 [44] Er:LiNbO3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='0 · 10−13 [45] Eu:YSO 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='0 · 10−13 [46] Eu:Y2O3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='3 · 10−13 [47] EuCl3·6D2O 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='6 · 10−13 [22] Pr:YSO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='2 · 10−11 [48] Pr:La2(WO4)3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='0 · 10−12 [49] Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Measured values for the ISD coefficient βexp in a selection of rare-earth ion- doped crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' The three ion-ion interactions considered in this work (see equations 4, 5 and 6) should contribute to ISD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Focusing our attention to the selection of REICs for which quantitative ISD measurements are available, we calculate the βi parameters for each ion- ion interactions using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' 8 and display the results graphically in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' The material parameters relevant to this calculation and resulting numerical estimations are given in appendix Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Again, we point out that the values are mostly indicative since they rely on a very simplified modelization of the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' We verify that both magnetic and electric dipole-dipole interactions vary among the materials within a 6 order-of-magnitude span depending on the existence of an Strain-mediated ion-ion interaction in rare-earth-doped solids 7 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Calculated (bars) and measured (triangles) values for βi and Ai (where i = {mag, el, str, exp}) for the 10 rare-earth ion-doped crystals for which quantitative values for βexp are available (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Note the logarithmic vertical scale spanning 10 orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' electronic spin and the presence of a centrosymmetry in the doping site (βmag is comprised between 7 · 10−20 and 2 · 10−13 Hz cm3and βel between 8 · 10−19 and 8 · 10−12 Hz cm3, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Conversely, the strain-mediated interaction is comprised within a much smaller interval (βstr between 6 · 10−13 and 2 · 10−12 Hz cm3) since we use a typical value for the piezospectroscopic sensitivity κ for all and because of the very similar values of the Young moduli between crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' It is interesting to note that the strain-mediated interaction is of the order of the largest of the electric or magnetic dipole-dipole contributions found among all materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' This means that when one or both of the dipole-dipole interactions are strong, they should coexist with the strain-mediated interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' On the other hand, when both electromagnetic contributions are in the low range (weak Stark effect and no electronic spin for the non-Kramers ions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Tm-doped garnets or in a lesser part stoechiometric europium chloride), the strain-mediated interaction dominates by several orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' In all considered REICs, the sum of the three predicted effects satisfactorily accounts for the measured values of βexp, finally providing an explanation to the previously large discrepancy between theoretical estimations and measurements of the ISD mechanism in some media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=" 10-10 (Hz cm²) ） 10-12 S 10-20 A 10-14 ISD coeficient β Magnetic Electric Strain 10-22 Exp 10 16 Interaction 10- 24 10-18 10-26 10-20 YSO Tm:YAG Tm:YGG Eu:YSO EuCl'Strain-mediated ion-ion interaction in rare-earth-doped solids 8 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Discussion Besides providing a better understanding of the physical origin and strength of ISD in REIC, the strain-mediated interaction in REIC could have interesting applications in the field of quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Indeed, quantum computing schemes rest upon the existence of a long-range atom-atom interaction enabling multiqubit operations [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' This interaction is often spontaneously assimilated to dipole-dipole interaction, and particularly in REIC [50, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' We argue that the strain-mediated interaction could play this role in quantum-computing to replace the dipole-dipole coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Such a scheme could be performed in almost any rare-earth ion-doped crystal since the strength of the interaction is rather similar over a broad variety of ion and host combinations [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' In particular, the possession of a permanent electric dipole moment would not be an exclusive criteria for quantum computing compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' On the contrary, in some crystals (namely Tm- doped garnets, although there could be other candidates) this strain-mediated interaction dominates its electromagnetic counterparts by at least 4 orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' This means that not only is the ion-ion interaction particularly pure in such media, but these strain- coupled qubits would be more robust against other types of decoherence process, such as magnetic field fluctuations due to spin flips in the host matrix [51, 13], or electric field noise due to charge fluctuations that may occur at the surface of rare-earth-doped nanoparticles [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' One may argue that since the strain-mediated interaction is intrinsically slow since it relies on the propagation of stress at the speed of sound (between 3000 and 8000 m/s in most considered crystals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' We estimate its propagation delay by considering the average ion-ion distance dion−ion, given by 3√nRE (where nRE is the volumic density of rare-earth ions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' dion−ion ranges from a few nm in highly concentrated materials (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' ∼ 2 nm in 1% doped YAG) up to hundreds of nm in low concentration materials (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' ∼ 650 nm in 200ppm doped YSO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' This leads to a strain propagation time shorter than the nanosecond, to be compared with typical µs-scale light-matter interactions occuring in such media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Therefore, the strain-mediated interaction can still be considered instantaneous within rare-earth-doped crystals with typical concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Conclusion Based on recent evidence of a conservative optomechanical backaction mechanism in rare- earth ion-doped crystals, we have unveiled a disregarded ion-ion interaction based on a physical mechanism fundamentally different from generally considered electromagnetic dipole-dipole interactions: the sensitivity of rare-earth ions to piezo-orbitally induced stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' With a simple mechanical model, we have estimated the strength of this strain- mediated interaction and shown that it is largely dominant in some rare-earth ion-doped crystals, opening interesting perspectives for strain-based quantum computing in solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Strain-mediated ion-ion interaction in rare-earth-doped solids 9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Acknowledgments The authors are grateful to Lars Rippe and Xiaoping Jia for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' The authors acknowledge support from the French National Research Agency (ANR) through the projects ATRAP (ANR-19-CE24-0008), MIRESPIN (ANR-19-CE47-0011) and MARS (ANR-20-CE92-0041).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' This work has received support under the program “Investissements d’Avenir” launched by the French Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Continuum mechanics in a REIC Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Spherical defect in an isotropic elastic medium Let us consider a sphere with radius r1, embedded in an infinite, isotropic, continuous elastic medium with a Young modulus E and a Poisson’s ratio ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' When a homogeneous radial stress σ0 is applied in the sphere, the displacement field at a distance r outside the sphere is radial and obeys [53]: u(r) = σ0 1 + ν 2E r3 1 r2, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='1) while the stress field around the sphere is also radial and reads as: σ(r) = σ0 r3 1 r3 for r > r1 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='2) Defining ∆r = u(r1) as the radius change, we obtain a simple relationship between σ0 et ∆r: σ0 = ∆r r1 2E 1 + ν (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='3) Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Radial stress σ(r) corresponding to the dilation of a sphere within an infinite isotropic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' The stress is homogeneous within the sphere and decays with a 1/r3 law outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' The radial stress can then be written: σ(r) = −∆r r1 2E 1 + ν r3 1 r3 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='4) /0 () 1 1 stress ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='8 Normalized radial 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='2 0 0 1 2 3 4 5 Normalized distance to ion r/riStrain-mediated ion-ion interaction in rare-earth-doped solids 10 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Strain energy We now want to assess the elastic strain energy contained in the medium when the stress σ0 is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' We successively calculate the energy contained inside and outside the sphere [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Inside the sphere: The strain energy inside the sphere reads as: Uint = 1 2σ0ε0V (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='5) where ε0 = ∆V/V = 3∆r/r1 is the volumic change of the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' We finally obtain: Uint = 2π∆rσ0r2 1 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='6) Outside the sphere: The sphere being included in an infinite medium, we must also consider the strain energy that was necessary to establish the whole stress field around the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' The volumic energy outside the sphere at a distance r reads as: uV (r) = 1 2σ(r)ε(r) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='7) The volumic strain ε(r) is related to the local stress via the volumic Hooke’s law σ(r) = Kε(r) (where K = E 1−2ν is the bulk modulus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='2 we get ε(r) = ε0r3 1/r3, which finally leads to: uV (r) = 3∆r 2 σ0 r5 1 r6 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='8) We integrate this volumic energy over the infinite volume of the medium: Uext = 4π � ∞ r=r1 uV (r)r2dr = 6π∆rσ0r5 1 � ∞ r=r1 1 r4dr (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='9) and obtain: Uext = 2π∆rσ0r2 1 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='10) Total strain energy: Based on Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='6 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='10, we derive the total strain energy that is necessary to distort the sphere and generate the associated stress field around it: Ustrain = Uint + Uext = 4π∆rσ0r2 1 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='11) Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Relating the strain energy with the atomic energy Due to piezo-orbital backaction [30], the size of a rare-earth ion is expected to change under optical excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Assuming this change of shape is merely a change of radius on a spherical ion, we can apply the calculation presented in appendix Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Due to energy conservation, this elastic energy is taken from the ion’s energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' We can Strain-mediated ion-ion interaction in rare-earth-doped solids 11 therefore write the following, considering that the shift in the ion’s energy levels ∆E is related to the stress within the ion σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Ustrain = ∆E = hκσ0 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='1) where κ is the piezospectroscopic sensitivity, assumed scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Injecting the expression of Ustrain given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='11, we get: ∆r = hκ 4πr2 1 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='2) This allows us to obtain a simple relationship between the radial stress σ(r) and the ionic radius change ∆r, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='4: σ(r) = 2E 1 + ν hκ 4πr3 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='3) Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Instantaneous Spectral Diffusion In Table C1 we list the physical parameters that are needed for the estimation of ISD strengths for an ensemble of 10 REIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Note that some values had to be estimated using data measured in similar materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' The value for the piezospectroscopic sensitivity κ being basically unknown for most REICs, we choose to take κ = 100 Hz/Pa for all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' This choice is supported by the observed similarity of the value of κ in a broad variety of hosts, dopants and transitions [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Crystal E ν ∆µmag ϵr ∆µel Tm:YAG 270 GPa [55] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='256 [55] 320 MHz/T [56] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='6 [57] 65 Hz cm/V [58] Tm:YGG 224 GPa [59] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='28 [59] 44 MHz/T [60] 12 [57] ⋆65 Hz cm/V Tm:LiNbO3 170 GPa [61] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='25 [61] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='3 GHz/T [62] †65 [63] 18 kHz cm/V [62] Er:YSO 150 GPa [64] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='26 [64] 26 GHz/T [65] 10 [66] 50 kHz cm/V [67] Er:LiNbO3 170 GPa [61] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='25 [61] 16 GHz/T [68] †65 [63] 25 kHz cm/V [69] Eu:YSO 150 GPa [64] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='26 [64] 10 MHz/T [70] 10 [66] 35 kHz cm/V [71, 72] Eu:Y2O3 ⋆120 GPa ⋆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='25 ⋆10 MHz/T 15 [73] ⋆35 kHz cm/V EuCl3·6D2O ⋆120 GPa ⋆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='25 8 MHz/T [74] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='6 [22] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='57 kHz cm/V [22] Pr:YSO 150 GPa [64] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='26 [64] 150 MHz/T [75] 10 [66] 111 kHz cm/V [76] Pr:La2(WO4)3 90 GPa [77] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='3 [77] ⋆150 MHz/T 20 [78] ⋆100 kHz cm/V Table C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Material parameters used to compute the interaction strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' E is the mechanical Young modulus, ν the Poisson’s ratio, and ϵr is the dielectric constant of the host matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' ∆µmag and ∆µel are the difference in the rare earth magnetic or electric moments between ground and excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' When no values could be found in the literature, we chose a value among similar materials and indicated this with the symbol ”⋆”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' †: The lithium niobate (LNO) crystal is anisotropic and exhibits different values of ϵr depending on the crystallographic direction [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' For simplicity we choose an intermediate value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Strain-mediated ion-ion interaction in rare-earth-doped solids 12 In Table C2 we present the different ISD coefficients βi calculated for the three possible ion-ion interactions (magnetic dipole-dipole interaction, electric dipole-dipole interaction, and strain-mediated interaction), using the parameters listed in Table C1 and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' 4, 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Crystal βmag βel βstr Tm:YAG 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='9 · 10−17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='0 · 10−18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='3 · 10−12 Tm:YGG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='3 · 10−18 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='3 · 10−19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='9 · 10−12 Tm:LiNbO3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='1 · 10−15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='0 · 10−14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='5 · 10−12 Er:YSO 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='5 · 10−13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='8 · 10−13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='3 · 10−12 Er:LiNbO3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='7 · 10−13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='8 · 10−14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='5 · 10−12 Eu:YSO 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='7 · 10−20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='4 · 10−13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='3 · 10−12 Eu:Y2O3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='7 · 10−20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='9 · 10−13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='2 · 10−12 EuCl3·6D2O 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='3 · 10−20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='1 · 10−15 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='2 · 10−13 Pr:YSO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='5 · 10−17 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='5 · 10−12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='3 · 10−12 Pr:La2(WO4)3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='5 · 10−17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='0 · 10−12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content='3 · 10−13 Table C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Estimated values for the ISD coefficients βi in Hz cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' [1] Hammerer K, Sørensen A S and Polzik E S 2010 Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE5T4oBgHgl3EQfSw9y/content/2301.05531v1.pdf'} +page_content=' Mod.' metadata={'source': 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The cold case +Alejandro Ayala1,2,3, Aritra Bandyopadhyay3,4,5, Ricardo L. S. Farias3, L. A. Hern´andez6,2, Jos´e Luis Hern´andez7,8 +1Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, Apartado Postal 70-543, CdMx 04510, Mexico. +2Centre for Theoretical and Mathematical Physics, and Department of Physics, +University of Cape Town, Rondebosch 7700, South Africa. +3Departamento de F´ısica, Universidade Federal de Santa Maria, Santa Maria, RS 97105-900, Brazil. +4Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, +South China Normal University, Guangzhou 510006, China. +5Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany. +6Departamento de F´ısica, Universidad Aut´onoma Metropolitana-Iztapalapa, +Av. +San Rafael Atlixco 186, CdMx 09340, Mexico. +7Instituto de Ciencias del Espacio (ICE, CSIC), +c.Can Magrans s.n., 08193 Cerdanyola del Vall`es, Catalonia, Spain. +8Facultat de F´ısica, Universitat de Barcelona, Mart´ı i Franqu`es 1, 08028 Barcelona, Spain. +We use the two-flavor linear sigma model with quarks to study the phase structure of isospin +asymmetric matter at zero temperature. +The meson degrees of freedom provide the mean field +chiral- and isospin-condensates on top of which we compute the effective potential accounting for +quark fluctuations at one-loop order. +Using the renormalizability of the model, we absorb the +ultraviolet divergences into suitable counter-terms that are added respecting the original structure +of the theory. These counter-terms are determined from the stability conditions which require the +effective potential to have minima in the condensates directions at the classical values, as well as +the transition from the non-condensed to the condensed phase to be smooth as a function of the +isospin chemical potential. We use the model to study the evolution of the condensates as well as +the pressure, energy and isospin densities and the sound velocity as functions of the isospin chemical +potential. The approach does a good average description up to isospin chemical potentials values +not too large as compared to the vacuum pion mass. +Keywords: Quantum Chromodynamics, Linear Sigma Model with Quarks, Isospin Asymmetry +I. +INTRODUCTION +Multiple implications of the remarkably rich phase +structure of Quantum Chromodynamics (QCD) have +been extensively explored over the last years. QCD at +finite density is usually characterized by the baryon µB +and the isospin µI chemical potentials. Nature provides +us with physical systems at finite baryon densities with +non zero µI in the form of isospin asymmetric matter, for +example, compact astrophysical objects such as neutron +stars. Because of this, along with the imminent arrival +of new generation relativistic heavy-ion collision experi- +ments at the FAIR [1] and NICA [2] facilities, the study of +the phase structure in the temperature T and the chem- +ical potentials µB and µI has become an ideal subject of +scrutiny within the heavy-ion and astroparticle physics +communities [3, 4]. +A typical T − µB − µI phase diagram is anticipated to +be full of rich phase structures [5]. However, from the +theoretical perspective, systems with finite µB are not +straightforwardly accessible to the first-principle meth- +ods of Lattice QCD (LQCD), due to the well-known +fermion determinant sign problem [6, 7]. Hence, studies +on the µB − µI plane have been performed mainly using +low energy effective models. These models have revealed +the existence of an exciting phase structure that includes +Gapless Pion Condensates (GPC), a Bose-Einstein Con- +densed (BEC) phase with gaped single particle excita- +tions, a BEC-BCS crossover, etc [8, 9]. +On the other hand, LQCD calculations for vanishing +and even small µB do not suffer from the sign problem. +These calculations have predicted the existence of a su- +perfluid pion condensate phase for high enough µI [10– +15]. At zero temperature, they show that a second order +phase transition at a critical isospin chemical potential +(corresponding to the vacuum pion mass), separates the +hadron from the pion condensate phase [14]. +In addi- +tion to LQCD, these phases are also found using chiral +perturbation theory (χPT) [16–28], Hard Thermal Loop +perturbation theory (HTLPt) [29], the Nambu-Jona- +Lasinio (NJL) model [9, 30–45] and its Polyakov loop +(PNJL) extended version [46, 47], the quark meson model +(QMM) [48–51] and other low energy effective models ex- +ploiting functional RG studies [52]. Calculations using a +LQCD equation of state for finite µI have investigated +the viability of the existence of pion stars, with a pion +condensate as the dominant core constituent [24, 53]. +Since LQCD calculations with µI ̸= 0, µB = µs = T = 0 +can be carried out without being hindered by the sign +problem, they can be used as a benchmark to test effec- +tive model predictions. For example, recently, the NJL +model has been used in this domain and it has been +found that results agree exceptionally well with LQCD +results [54, 55]. +In this work we study another effective QCD model, +the Linear Sigma Model with quarks (LSMq), extended + +2 +to consider a finite µI to describe the properties of +strongly interacting systems with an isospin imbalance. +The LSMq is a renormalizable theory that explicitly im- +plements the QCD chiral symmetry. +It has been suc- +cessfully employed to study the chiral phase transition +at finite T and µB [56–59], as well as in the presence +of a magnetic field [60–67]. +The Linear Sigma Model +has been used at finite µI, albeit considering the meson +degrees of freedom as an effective classical background, +in the Hartree or Hartree Fock approximations within +the Cornwall-Jackiw-Tomboulis (CJT) formalism [68]. In +contrast, in the LSMq mesons are treated as dynamical +fields able to contribute to quantum fluctuations. Part of +the reason for other models to avoid considering mesons +as dynamical fields, for example the QMM, is that when +mesons become true quantum fields and chiral symmetry +is only spontaneously broken, their masses are subject to +change as a result of medium effects. During this change, +the meson square masses can become zero or even neg- +ative. +At zero temperature, this drawback is avoided +by considering an explicit symmetry breaking term that +provides pions with a vacuum finite mass. At finite tem- +perature, the plasma screening effects need to also be +included. +In this work we use the LSMq to describe the evolu- +tion of the chiral and isospin (pion) condensates, as well +as thermodynamical quantities such as pressure, isospin +and energy densities and the sound velocity at zero tem- +perature and finite µI. We restrict ourselves to consider- +ing only the effects of fermion quantum fluctuations, re- +serving for a future work the inclusion of meson quantum +fluctuations effects. We make use of the renormalizability +of the LSMq and describe in detail the renormalization +procedure which is achieved by implementing the stabil- +ity conditions. The results thus obtained are valid for +the case where µ2 +I) is small compared to the sum of the +squares of the chiral and isospin condensates multiplied +by the square of the boson-fermion coupling constant g. +The work is organized as follows: In Sec. II we write the +LSMq Lagrangian using degrees of freedom appropriate +to describe an isospin imbalanced system. We work with +an explicit breaking of the chiral symmetry introducing a +vacuum pion mass and expanding the charged pion fields +around the values of their condensates. The effective po- +tential is constructed by adding to the tree-level poten- +tial the one-loop contribution from the fermion degrees +of freedom. Renormalization is carried out by introduc- +ing counter-terms to enforce that the tree-level structure +of the effective potential is preserved by loop corrections. +We first work out explicitly the treatment in the con- +densed phase to then work out the non-condensed phase. +In Sec. III we study the condensates evolution with µI as +well as that of the pressure, isospin and energy density +and the sound velocity, and compare to recent LQCD +results. We finally summarize and conclude in Sec. IV. +We reserve for a follow up work the computation of the +meson quantum fluctuations as well as finite temperature +effects. The appendix is devoted to the explicit computa- +tion of the one-loop fermion contribution to the effective +potential. +II. +LSMQ AT FINITE ISOSPIN CHEMICAL +POTENTIAL +The LSMq is an effective theory that captures the ap- +proximate chiral symmetry of QCD. It describes the in- +teractions among small-mass mesons and quarks. +We +start with a Lagrangian invariant under SU(2)L × +SU(2)R chiral transformations +L = 1 +2(∂µσ)2 + 1 +2(∂µ⃗π)2 + a2 +2 (σ2 + ⃗π2) − λ +4 (σ2 + ⃗π2)2 ++ i ¯ψγµ∂µψ − ig ¯ψγ5⃗τ · ⃗πψ − g ¯ψψσ, +(1) +where ⃗τ = (τ1, τ2, τ3) are the Pauli matrices, +ψL,R = +� +u +d +� +L,R +, +(2) +is a SU(2)L,R doublet, σ is a real scalar field and ⃗π = +(π1, π2, π3) is a triplet of real scalar fields. π3 corresponds +to the neutral pion whereas the charged ones are repre- +sented by the combinations +π− = +1 +√ +2 +(π1 + iπ2), +π+ = +1 +√ +2 +(π1 − iπ2). +(3) +The parameters a2, λ and g are real and positive definite. +Equation (1) can be written in terms of the charged and +neutral-pion degrees of freedom as +L = 1 +2[(∂µσ)2 + (∂µπ0)2] + ∂µπ−∂µπ+ + a2 +2 (σ2 + π2 +0) ++ a2π−π+ − λ +4 (σ4 + 4σ2π−π+ + 2σ2π2 +0 + 4π2 +−π2 ++ ++ 4π−π+π2 +0 + π4 +0) + i ¯ψ/∂ψ − g ¯ψψσ − ig ¯ψγ5(τ+π+ ++ τ−π− + τ3π0)ψ, +(4) +where we introduced the combination of Pauli matrices +τ+ = +1 +√ +2(τ1 + iτ2), +τ− = +1 +√ +2(τ1 − iτ2). +(5) +The Lagrangian in Eq. (4) possesses the following sym- +metries: A SU(Nc) global color symmetry, a SU(2)L × +SU(2)R chiral symmetry and a U(1)B symmetry. The +sub-index of the latter emphasizes that the conserved +charge is the baryon number B. +A conserved isospin +charge can be added to the LSMq Hamiltonian, multi- +plied by the isospin chemical potential µI. The result is +that the Lagrangian gets modified such that the ordinary +derivative becomes a covariant derivative [69] +∂µ → Dµ = ∂µ + iµIδ0 +µ, +∂µ → Dµ = ∂µ − iµIδµ +0 , (6) + +3 +As a result, Eq. (4) is modified to read as +L = 1 +2[(∂µσ)2 + (∂µπ0)2] + Dµπ−Dµπ+ + a2 +2 (σ2 + π2 +0) ++ a2π−π+ − λ +4 +� +σ4 + 4σ2π−π+ + 2σ2π2 +0 + 4π2 +−π2 ++ ++ 4π−π+π2 +0 + π4 +0 +� ++ i ¯ψ/∂ψ − g ¯ψψσ + ¯ψµIτ3γ0ψ +− ig ¯ψγ5(τ+π+ + τ−π− + τ3π0)ψ. +(7) +Because of the spontaneous breaking of the chiral sym- +metry in the Lagrangian given in Eq. (7), the σ field ac- +quires a non-vanishing vacuum expectation value +σ → σ + v. +To make better contact with the meson vacuum proper- +ties and to include a finite vacuum pion mass, m0, we +can add an explicit symmetry breaking term in the La- +grangian such that +L → L′ = L + h(σ + v). +(8) +The constant h is fixed by requiring that the model ex- +pression for the neutral vacuum pion mass squared in +the non-condensed phase, Eq. (11a), corresponds to m2 +0. +This yields +h = m2 +0 +� +a2 + m2 +0 +λ +, +≡ m2 +0fπ, +(9) +where fπ is the pion decay constant and have used its ex- +plicit model expression. Equation (9) provides a relation +for the model parameters a and λ in terms of fπ. +Before diving into the formalism details, here we first +pause to discuss the symmetry properties of the theory. +Notice that the introduction of µI and h modifies the +structure of the effective Lagrangian given in Eq. (8). In +the presence of a finite µI, the U(1)B ×SU(2)L×SU(2)R +symmetry is reduced to U(1)B × U(1)I3L × U(1)I3R for +h = 0, and to U(1)B × U(1)I3 for h ̸= 0, thereby repre- +senting the explicit breaking of the chiral symmetry [70]. +The notation also emphasizes that the third component +of the isospin charge, I3, corresponds to the generator +of the remaining symmetry U(1)I3. Since in the present +work, we are interested in the dynamics of the pion fields, +further simplifications in the pseudoscalar channels can +be obtained using the ansatz ⟨ ¯ψiγ5τ3ψ⟩ = 0 combined +with ⟨¯uiγ5d⟩ = ⟨ ¯diγ5u⟩∗ ̸= 0 [9]. This further breaks the +residual U(1)I3 symmetry and corresponds to a Bose- +Einstein condensation of the charged pions. Then, the +charged pion fields can be referred from their conden- +sates as +π+ → π+ + ∆ +√ +2eiθ, +π− → π− + ∆ +√ +2e−iθ, +(10) +where the phase factor θ indicates the direction of the +U(1)I3 symmetry breaking. We take θ = π for defini- +tiveness. The shift in the sigma field produces that the +fermions and neutral bosons acquire masses given by +mf = gv +(11a) +m2 +π0 = λv2 − a2 + λ∆2 +(11b) +m2 +σ = 3λv2 − a2 + λ∆2. +(11c) +The charged pions also acquire masses. +However, in +the condensed phase (∆ ̸= 0) they need to be described +in terms of the π1,2 fields [71]. Since for our purposes, +pions are not treated as quantum fluctuations, hereby +we just notice that, as a consequence of the breaking +of the U(1)I3 symmetry, one of these fields becomes a +Goldstone boson. In the absence of the explicit symmetry +breaking term in the Lagrangian of Eq. (8), this mode’s +mass would vanish. +However, a finite h prevents this +mode from being massless. +A. +Condensed phase +In the condensed phase the tree-level potential, ex- +tracted from Eqs. (7) and (8), can be written as +Vtree = −a2 +2 +� +v2 + ∆2� ++ λ +4 +� +v2 + ∆2�2 − 1 +2µ2 +I∆2 − hv. +(12) +The fermion contribution to the one-loop effective po- +tential becomes +� +f=u,d +V 1 +f = −2Nc +� +d3k +(2π)3 +� +Eu +∆ + Ed +∆ +� +, +(13) +with (see Appendix A) +Eu +∆ = +��� +k2 + m2 +f + µI +�2 ++ g2∆2 +�1/2 +, +(14a) +Ed +∆ = +��� +k2 + m2 +f − µI +�2 ++ g2∆2 +�1/2 +, +(14b) +where we chose that +µd = µI +µu = −µI. +(15) +Equation (13) is ultraviolet divergent. Ultraviolet diver- +gences are a common feature of loop vacuum contribu- +tions. However, since Eq. (13) depends on µI, this di- +vergence needs to be carefully treated given that mat- +ter contributions cannot contain ultraviolet divergences. +To identify the divergent terms, we work in the approx- +imation whereby the fermion energies, Eqs. (14), are ex- +panded in powers of µ2 +I/[g2(v2 +∆2)]. Considering terms +up to O(µ4 +I), we obtain +� +f=u,d +Ef +∆ ≃ 2 +� +k2 + m2 +f + g2∆2 + +µ2 +Ig2∆2 +(k2 + m2 +f + g2∆2)3/2 ++ +µ4 +I +� +4(k2 + m2 +f)g2∆2 − g4∆4� +4 +� +k2 + m2 +f + g2∆2 +�7/2 ++ O(µ6 +I). +(16) + +4 +Notice that the ultraviolet divergent part corresponds +only to the first and second terms on the right-hand side +of Eq. (16). In this approximation, and up to terms of +order µ2 +I, the expression for the leading fermion contri- +bution to the one-loop effective potential is given by +� +f=u,d +V 1 +f = −2Nc +� +d3k +(2π)3 +� +2 +� +k2 + m2 +f + g2∆2 ++ +µ2 +Ig2∆2 +(k2 + m2 +f + g2∆2)3/2 +� +(17) +This expression can be readily computed using dimen- +sional regularization in the MS scheme, with the result +(see Appendix A) +� +f=u,d +V 1 +f = 2Nc +g4 � +v2 + ∆2�2 +(4π)2 +�1 +ǫ + 3 +2 + ln +� Λ2/g2 +v2 + ∆2 +�� +− 2Nc +g2µ2 +I∆2 +(4π)2 +�1 +ǫ + ln +� Λ2/g2 +v2 + ∆2 +�� +, +(18) +where Nc = 3 is the number of colors, Λ is the dimen- +sional regularization ultraviolet scale and the limit ǫ → 0 +is to be understood. The explicit computation of Eq. (18) +is described also in Appendix A. Notice that Eq. (18) +contains an ultraviolet divergence proportional to µ2 +I∆2. +Since a term with this same structure is already present +in the tree-level potential, Eq. (12), it is not surpris- +ing that this ultraviolet divergence can be handled by +the renormalization procedure with the introduction of a +counter-term with the same structure, as we proceed to +show. +To carry out the renormalization of the effective po- +tential up to one-loop order, we introduce counter-terms +that respect the structure of the tree-level potential and +determine them by accounting for the stability condi- +tions. The latter are a set of conditions satisfied by the +tree-level potential and that must be preserved when con- +sidering loop corrections. These conditions require that +the position of the minimum in the v- and ∆-directions +remain the same as the tree-level potential ones. +The tree-level minimum in the v, ∆ plane is found from +∂Vtree +∂v += +� +λv3 − (a2 − λ∆2)v − h +����� +v0, ∆0 += 0 +(19a) +∂Vtree +∂∆ += +� +λ∆2 − (µ2 +I − λv2 + a2) +����� +v0, ∆0 += 0. (19b) +Notice that the second of Eqs. (19) admits a real, non- +vanishing solution, only when +µ2 +I > λv2 − a2 = m2 +0, +(20) +which means that a non-zero isospin condensate is devel- +oped only when, for positive values of the isospin chem- +ical potential, the latter is larger than the vacuum pion +mass. This is what we identify as the condensed phase. +The simultaneous solutions of Eqs. (19) are +v0 = h +µ2 +I +, +(21a) +∆0 = +� +µ2 +I +λ − h2 +µ4 +I ++ a2 +λ . +(21b) +Hereafter, we refer to the expressions in Eq. (21) as the +classical solution. +The effective potential, up to one-loop order in the +fermion fluctuations, including the counter-terms, can be +written as +Veff = Vtree + +� +f=u,d +V 1 +f − δλ +4 (v2 + ∆2)2 ++ δa +2 (v2 + ∆2) + δ +2∆2µ2 +I. +(22) +The counter-terms δλ and δ are determined from the gap +equations +∂Veff +∂v +���� +v0, ∆0 += 0, +(23a) +∂Veff +∂∆ +���� +v0, ∆0 += 0. +(23b) +These conditions suffice to absorb the infinities of +Eq. (18). The counter-term δa is determined by requiring +that the slope of Veff vanishes at µI = m0, +∂Veff +∂µI +���� +µI=m0 += 0, +(24) +or in other words, that the transition from the non- +condensed to the condensed phase be smooth. The ef- +fective potential thus obtained is ultraviolet finite as well +as Λ-independent. +B. +Non-condensed phase +In the non-condensed phase, 0 ≤ µI ≤ m0, the only +allowed solution for the second of Eqs. (19) is ∆ = 0. For +this case, the first of Eqs. (19) becomes a cubic equation +in v. The only real solution is +˜v0 = ( +√ +3 +√ +27h2λ4 − 4a6λ3 + 9hλ2)1/3 +(18)2/3λ ++ +(2/3)1/3a2 +( +√ +3 +√ +27h2λ4 − 4a6λ3 + 9hλ2)1/3 . +(25) +In the limit when h is taken as small one gets +˜v0 ≃ +a +√ +λ ++ h +2a2 , +(26) +an approximation that some times is considered. How- +ever, hereafter we work instead with the full expression +given by Eq. (25). + +5 +Δ [GeV] +v [GeV] +0.0 +0.5 +1.0 +1.5 +2.0 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +μI/m0 +Figure 1: v- and ∆-condensates as functions of the +scaled variable µI/m0. For µI ≥ m0, the v-condensate +decreases while the ∆-condensate increases. +The effective potential V noncond +eff +up to one-loop order +can be obtained from the corresponding one in the con- +densed phase, by setting ∆ = 0. Therefore, we can write +V noncond +eff += λ +4 v4 − a2 +2 v2 − hv − +˜δ1 +4 v4 + +˜δ2 +2 v2 ++ 2Nc +g4v4 +(4π)2 +�1 +ǫ + 3 +2 + ln +� Λ2 +g2v2 +�� +. (27) +In this case, only two conditions are needed to stabilize +the vacuum. We take these as the requirement that the +position and curvature of V noncond +eff +remain at its classical +value when evaluated at ˜v0, namely, +∂V noncond +eff +∂v +���� +˜v0 += 0 +(28a) +∂2V noncond +eff +∂v2 +���� +˜v0 += 3λ˜v2 +0 − a2, +(28b) +from where the counter-terms ˜δ1, ˜δ2 can be determined. +Therefore, in the non-condensed phase, in addition to +∆ = 0, the v-condensate is simply given by the constant +˜v0 given in Eq. (25). As for the case of the condensed +phase, in the non-condensed phase the effective potential +is ultraviolet finite as well as Λ-independent. +III. +THERMODYNAMICS OF THE +CONDENSED PHASE +Armed with the expressions for the effective potential, +we can now proceed to study the dependence of the con- +densates as well as of the thermodynamical quantities as +functions of µI. +Since the µI-dependence in the non- +condensed phase is trivial, we concentrate in the descrip- +tion of the behavior of these quantities in the condensed +phase. +LQCD 24×48 +LQCD 32×48 +Tree Level +One loop +SU(2) NJL +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +μI/m0 +PN/m0 +4 +Figure 2: Normalized pressure as a function of the +scaled variable µI/m0. Shown are the tree-level and +one-loop fermion improved pressures compared to the +results from Ref. [54] together with the LQCD results +from Ref. [72]. +The model requires fixing three independent parame- +ters: the boson self-coupling λ, the boson-fermion cou- +pling g and the mass parameter a. For a vacuum pion +mass m0 = 135 MeV, these parameters are fixed by re- +quiring that the pion vacuum decay constant is fπ = 93 +MeV, the light quark mass is mq = 235 MeV and the +sigma mass is mσ = 400 MeV. The phase space for these +parameters is limited since for certain combinations, the +gap equation conditions in the v-∆ plane become saddle +points rather than global minima. +Figure 1 shows the v- and ∆-condensates as functions +of the scaled variable µI/m0. The behavior is qualita- +tively as expected: for µI ≥ m0, the v-condensate de- +creases while the ∆-condensate increases. +Figure 2 shows the normalized pressure, defined as the +negative of the effective potential referred from its value +at µI = m0, as a function of the scaled variable µI/m0 +and divided by m4 +0. Shown are the results obtained by +using the tree-level and the fermion one-loop corrected +effective potentials, compared to the results from Ref. [54] +and the LQCD results from [72]. Notice that the one-loop +improved calculation does a better description than the +tree-level one and that deviations from the LQCD result +appear for µI ≳ 1.5 m0. +Figure 3 shows the normalized isospin density, nI = +dP/dµI, divided by m3 +0 as a function of the scaled vari- +able µI/m0 compared to results obtained using the tree- +level potential as well as to the results from Ref. [54] +together with the LQCD results from Ref. [72]. Notice +that the one-loop improved calculation is close to the NJL +one up to µI ∼ 1.5 m0 but the latter does a better job +describing the LQCD results for µI ≳ 1.5 m0. However, +it is fair to say that neither the current calculation nor +the NJL result reproduce the change of curvature that + +6 +Tree Level +One loop +SU(2) NJL +1.0 +1.2 +1.4 +1.6 +1.8 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +μI/m0 +nI/m0 +3 +Figure 3: Normalized isospin density as a function of +the scaled variable µI/m0. Shown are the tree-level and +one-loop fermion improved effective potentials +compared to a recent SU(2) NJL calculation [54] and +the LQCD results from Ref. [72]. +seems to be present in the LQCD result. +Figure 4 shows the normalized energy density, ǫ/m4 +0, +as a function of the scaled variable µI/m0, compared to +the results from Ref. [54] together with the LQCD results +from Ref. [72]. Although the change in curvature shown +by the LQCD results is not described by the present cal- +culation, it is fair to say that neither the NJL calculation +captures such trend. The one-loop improved calculation +does a better average description of the LQCD result al- +though deviations appear for µI ≳ 1.5 m0. +Figure 5 shows the equation of state, pressure vs. en- +ergy density, compared to the results from Ref. [54] to- +gether with the LQCD results from Ref. [72]. +Notice +that for the latter, the vacuum pion mass is taken as +m0 = 135 MeV. As can be seen, the initial increasing +trend of LQCD results is properly described by the low- +energy models considered. Given that the accuracy of +our results is limited to the low µI domain the NJL cal- +culation does a better description of the LQCD results. +Figure 6 shows the square of the speed of sound, c2 +s, as +a function of the scaled variable µI/m0. Shown are the +one-loop results compared to the results from Ref. [54] to- +gether with the LQCD results from Ref. [72]. The appar- +ent peak in the LQCD results is not reproduced by any +model. However, notice that for the range of shown µI +values, the one-loop improved result is above, although +closer to the conformal bound, shown as a horizontal line +at c2 +s = 1/3. +IV. +SUMMARY AND CONCLUSIONS +In this work we have used the LSMq, with two quark +flavors, to study the phase structure of isospin asymmet- +LQCD 24×48 +LQCD 32×48 +Tree Level +One loop +SU(2) NJL +1.0 +1.2 +1.4 +1.6 +1.8 +0.0 +0.5 +1.0 +1.5 +μI/m0 +ϵ/m0 +4 +Figure 4: Normalized energy density as a function of +the scaled variable µI/m0. Shown are the tree-level and +one-loop fermion improved effective potentials +compared to the results from Ref. [54] together with the +LQCD results from Ref. [72]. +ric matter at zero temperature. The meson degrees of +freedom are taken as providing the mean field on top of +which we include quantum quark fluctuations at one-loop +order. We have used the renormalization of the LSMq to +absorb the ultraviolet divergences with the addition of +counter-terms that respect the original structure of the +theory. An interesting aspect of the method is that it al- +lows the proper handling of the disturbing µI-dependent +ultraviolet divergence. The one-loop quark contributions +are treated in the approximation whereby µ2 +I is taken as +small compared to g2(v2 +∆2) and working up to O(µ2 +I). +After determining the model parameters, we have stud- +ied the evolution of the chiral and isospin condensates +as well as the pressure, energy and isospin densities and +the sound velocity. We have compared the model results +with a recent NJL calculation of the same quantities and +with LQCD data. The model does a good description +for µI ≲ 1.5 m0, except perhaps for the sound veloc- +ity for which it does not reproduce the peak seemingly +appearing in the LQCD calculations. +The results are encouraging and set the stage to ex- +plore whether the method can be used to incorporate the +effect of meson fluctuations. The method also lends itself +to include in the description higher powers of µ2 +I as well +as finite temperature effects. We are currently exploring +these avenues and will report on the findings elsewhere +in the near future. +ACKNOWLEDGMENTS +The authors are grateful to G. Endr¨odi and B. B. +Brandt for kindly sharing their LQCD data in tabular +form. +Support for this work was received in part by + +LQCD 24×48 +LQCD32×487 +LQCD 24×48 +LQCD 32×48 +Tree level +One loop +SU(2) NJL +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.5 +1.0 +1.5 +PN/m0 +4 +ϵ/m0 +4 +Figure 5: Equation of state, pressure vs. energy +density. Shown are the tree-level and one-loop fermion +improved effective potentials compared to the results +from Ref. [54] together with the LQCD results from +Ref. [72]. For the latter, the vacuum pion mass is taken +as m0 = 135 MeV. +UNAM-PAPIIT IG100322 and by Consejo Nacional de +Ciencia y Tecnolog´ıa grant number A1-S-7655. L. A. H. +acknowledges support from a PAPIIT-DGAPA-UNAM +fellowship. This work was partially supported by Con- +selho Nacional de Desenvolvimento Cient´ıfico e Tecno- +l´ogico (CNPq), Grant No. +309598/2020-6 (R.L.S.F.); +Funda¸c˜ao de Amparo `a Pesquisa do Estado do Rio +Grande do Sul (FAPERGS), Grants Nos. +19/2551- +0000690-0 and 19/2551-0001948-3 (R.L.S.F.). A.B. ac- +knowledges the support from the Alexander von Hum- +boldt Foundation postdoctoral research fellowship in +Germany. +Appendix A: One-loop quark contribution to the +effective potential +The thermodynamic potential accounting for the quark +contribution at one-loop order is given by +V 1 +f = iV −1 ln +� +Z1 +f +� +, +(A1) +where +ln (Z1 +f ) = ln +� +det +�� +S−1 +mf +��� +, +(A2) +and V is the space-time volume. Also here, S−1 +mf is the +inverse propagator of the two light-quark species. There- +fore, we are bound to compute the determinant of a ma- +trix M of the form +M = +� +A B +C D +� +, +(A3) +where A, B, C, D can be thought of as p×p, p×q, q ×p, +and q × q complex matrices, respectively. When A, and +NJL SU(2) +Conformal Bound +LQCD 24×48 +LQCD +1.0 +1.2 +1.4 +1.6 +1.8 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +μI/m0 +cs +2 +Figure 6: Square of the speed of sound as a function of +the scaled variable µI/m0. Shown are the tree-level and +one-loop fermion improved effective potentials +compared to a recent SU(2) NJL calculation [54] and +the LQCD results from Ref. [72]. +D, are invertible, the determinant of M is given by +det{(M)} = det{(A)} det +� +(D − CA−1B) +� +, +(A4) +det{(M)} = det{(D)} det +� +(A − BD−1C) +� +. +(A5) +Equation (A4) can be written as +det{(M)} = det{(A)} det +� +(D − CA−1B) +� += det{(A)} det +� +(C−1C) +� +det +� +(D − CA−1B) +� += det +� +(−C2A−1BC−1A + CDC−1A) +� +, +(A6) +whereas Eq. (A5) as +det{(M)} = det{(D)} det +� +(A − BD−1C) +� += det{(D)} det +� +(C−1C) +� +det +� +(A − BD−1C) +� += det +� +(−CB + CAC−1D) +� +. +(A7) +For our purposes, B = C = ig∆γ5. +Thus, from +Eqs. (A6) and (A7), we obtain +det{(M)} = det +� +(−C2 + CDC−1A) +� +, +(A8) +det{(M)} = det +� +(−C2 + CAC−1D) +� +. +(A9) +We explicitly compute both expressions. +Fist, we use +that the standard spin projectors Λ± satisfy +γ0Λ±γ0 = ˜Λ∓, +(A10) +and +γ5Λ±γ5 = ˜Λ±, +(A11) + +LQCD32X48 +Tree Level +One Loop8 +with the projectors ˜Λ± defined as +˜Λ± = 1 +2 +� +1 ± γ0(⃗γ · ⃗k − gv) +Ek +� +. +(A12) +Next, we notice that A = S−1 +u +and D = S−1 +d . There- +fore, working first in the absence of an isospin chemical +potential, for which +S−1 +u += S−1 +d += k0γ0 − ⃗γ · ⃗k − gv, +(A13) +D1 ≡ −C2 + CDC−1A += g2∆2 + (ig∆γ5)S−1 +d +� 1 +ig∆γ5 +� +S−1 +u += g2∆2 − +� +k2 +0 − (Eu +k )2� +Λ− − +� +k0 − +� +Ed +k +�2� +Λ+, +(A14) +and +D2 ≡ −C2 + CAC−1D += g2∆2 + γ5S−1 +u γ5S−1 +d += g2∆2 − +� +k2 +0 − +� +Ed +k +�2� +Λ− − +� +k2 +0 − (Eu +k )2� +Λ+. +(A15) +Thus, using that Λ+ + Λ− = 11 and defining Eq +∆ = +� +(Eq +k)2 + g2∆2, we have +D1 = − +� +k2 +0 − (Eu +∆)2� +Λ− − +� +k0 − +� +Ed +∆ +�2� +Λ+, (A16) +D2 = − +� +k2 +0 − +� +Ed +∆ +�2� +Λ− − +� +k2 +0 − (Eu +∆)2� +Λ+, (A17) +and +det +� +(S−1 +mf ) +� += det{(D1)} = det{(D2)}. +(A18) +Note that +ln (Z1 +f ) = ln +� +det +�� +S−1 +mf +��� += 1 +2 ln +� +det +�� +S−1 +mf +�2�� += 1 +2 ln (det{(D1D2)}) += 1 +2Tr [ln (D1D2)] , +(A19) +and since the product D1D2 is given by +D1D2 = +� +k2 +0 − (Eu +∆)2� � +k2 +0 − +� +Ed +∆ +�2� +, +(A20) +we get +ln (Z1 +f ) = 1 +2 +� +q=u,d +Tr +� +ln +� +k2 +0 − (Eq +∆)2 �� +, +(A21) +where the trace is taken in Dirac, color (factors of 4 and +Nc, respectively), and in coordinate spaces, namely, +ln (Z1 +f ) = 2Nc +� +q=u,d +� +d4x +� +x +��� ln +� +k2 +0 − (Eq +∆)2 ����x +� += 2Nc +� +q=u,d +� +d4x +� +d4k +(2π)4 ln +� +k2 +0 − (Eq +∆)2 � +. +(A22) +Therefore +ln (Z1 +f) = 2V Nc +� +q=u,d +� +d4k +(2π)4 ln +� +k2 +0 − (Eq +∆)2 � +. +(A23) +In order to obtain a more compact expression, we inte- +grate and differentiate with respect to Eq +∆ as follows +ln (Z1 +f) = 2V Nc +� +q=u,d +� +d4k +(2π)4 +� +dEq +∆ +Eq +∆ +k2 +0 − (Eq +∆)2 . +(A24) +Performing a Wick rotation k0 → ik4, we obtain +ln (Z1 +f) = 4iV Nc +� +q=u,d +� +d4kE +(2π)4 +� +dEq +∆ +Eq +∆ +k2 +0 − (Eq +∆)2 , +(A25) +and integrating over k4 and Eq +∆, in this order, we get +ln (Z1 +f) = 2iV Nc +� +q=u,d +� +d3k +(2π)3 Eq +∆, +(A26) +with Re[(Eq +∆)2] ≥ 0. Therefore, the quark contribution +to the effective potential at one-loop order is given by +V 1 +f = iV −1 ln (Z1 +f ). +(A27) +Thus, +V 1 +f = −2Nc +� +q=u,d +� +d3k +(2π)3 Eq +∆. +(A28) +In the presence of an isospin chemical potential for which +S−1 +u += (k0 + µI)γ0 − ⃗γ · ⃗k − gv, +S−1 +d += (k0 − µI)γ0 − ⃗γ · ⃗k − gv, +(A29) +and repeating the steps starting from Eq. (A14), we ob- +tain Eq. (A28), with the energies Eu +∆ and Ed +∆ given by +Eqs. (14). +We now proceed to the explicit computation of +Eq. (13). In the limit where µ2 +I/[g2(v2 + ∆2)] is small, +Eq. (A28) can be written as in Eq. (17). We use dimen- +sional regularization. The first of the integrals on the +right hand side of Eq. (17) is expressed as +� +d3k +(2π)3 +� +k2 + g2v2 + g2∆2 → Λ3−d Γ +� +− 1 +2 − d +2 +� +(4π) +d +2 Γ +� +− 1 +2 +� +× +� +1 +g2v2 + g2∆2 +�− 1 +2 − d +2 +. +(A30) + +9 +Taking d → 3 − 2ǫ and working in the MS scheme +Λ2 → Λ2eγE +4π +, +(A31) +where γE is the Euler-Mascheroni constant, we get +� +d3k +(2π)3 +� +k2 + g2v2 + g2∆2 → −(g2v2 + g2∆2)2 +2(4π)2 +�1 +ǫ + 3 +2 + ln +� +Λ2 +g2v2 + g2∆2 +�� +. +(A32) +The second of the integrals on the right hand side of Eq. (17) is expressed as +� +d3k +(2π)3 +1 +(k2 + g2v2 + g2∆2)3/2 → Λ3−d Γ +� 3 +2 − d +2 +� +(4π) +d +2 Γ +� 3 +2 +� +� +1 +g2v2 + g2∆2 +� 3 +2 − d +2 +. +(A33) +Taking d → 3 − 2ǫ and working in the MS scheme we get +� +d3k +(2π)3 +1 +(k2 + g2v2 + g2∆2)3/2 → +2 +(4π)2 +�1 +ǫ + ln +� +Λ2 +g2v2 + g2∆2 +�� +, +(A34) +from where the result of Eq. (18) follows. +[1] K. Agarwal (CBM), (2022), arXiv:2207.14585 [hep-ex]. +[2] V. +Abgaryan +et +al. +(MPD), +Eur. Phys. J. 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Schmitt, Introduction to Superfluidity: Field-theoretical approach and applications, +Vol. 888 (2015) arXiv:1404.1284 [hep-ph]. +[72] B. +B. +Brandt, +F. +Cuteri, +and +G. +Endr¨odi, +PoS LATTICE2022, 144 (2023), +arXiv:2212.01431 [hep-lat]. + diff --git a/BdFRT4oBgHgl3EQfvDhO/content/tmp_files/load_file.txt b/BdFRT4oBgHgl3EQfvDhO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2df8c21c187f25681964ff62df4981725e53f95a --- /dev/null +++ b/BdFRT4oBgHgl3EQfvDhO/content/tmp_files/load_file.txt @@ -0,0 +1,971 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf,len=970 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='13633v1 [hep-ph] 31 Jan 2023 QCD equation of state at finite isospin density from the linear sigma model with quarks: The cold case Alejandro Ayala1,2,3, Aritra Bandyopadhyay3,4,5, Ricardo L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Farias3, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Hern´andez6,2, Jos´e Luis Hern´andez7,8 1Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, Apartado Postal 70-543, CdMx 04510, Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' 2Centre for Theoretical and Mathematical Physics, and Department of Physics, University of Cape Town, Rondebosch 7700, South Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' 3Departamento de F´ısica, Universidade Federal de Santa Maria, Santa Maria, RS 97105-900, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' 4Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' 5Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' 6Departamento de F´ısica, Universidad Aut´onoma Metropolitana-Iztapalapa, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' San Rafael Atlixco 186, CdMx 09340, Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' 7Instituto de Ciencias del Espacio (ICE, CSIC), c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='Can Magrans s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=', 08193 Cerdanyola del Vall`es, Catalonia, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' 8Facultat de F´ısica, Universitat de Barcelona, Mart´ı i Franqu`es 1, 08028 Barcelona, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' We use the two-flavor linear sigma model with quarks to study the phase structure of isospin asymmetric matter at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The meson degrees of freedom provide the mean field chiral- and isospin-condensates on top of which we compute the effective potential accounting for quark fluctuations at one-loop order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Using the renormalizability of the model, we absorb the ultraviolet divergences into suitable counter-terms that are added respecting the original structure of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' These counter-terms are determined from the stability conditions which require the effective potential to have minima in the condensates directions at the classical values, as well as the transition from the non-condensed to the condensed phase to be smooth as a function of the isospin chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' We use the model to study the evolution of the condensates as well as the pressure, energy and isospin densities and the sound velocity as functions of the isospin chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The approach does a good average description up to isospin chemical potentials values not too large as compared to the vacuum pion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Keywords: Quantum Chromodynamics, Linear Sigma Model with Quarks, Isospin Asymmetry I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' INTRODUCTION Multiple implications of the remarkably rich phase structure of Quantum Chromodynamics (QCD) have been extensively explored over the last years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' QCD at finite density is usually characterized by the baryon µB and the isospin µI chemical potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Nature provides us with physical systems at finite baryon densities with non zero µI in the form of isospin asymmetric matter, for example, compact astrophysical objects such as neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Because of this, along with the imminent arrival of new generation relativistic heavy-ion collision experi- ments at the FAIR [1] and NICA [2] facilities, the study of the phase structure in the temperature T and the chem- ical potentials µB and µI has become an ideal subject of scrutiny within the heavy-ion and astroparticle physics communities [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' A typical T − µB − µI phase diagram is anticipated to be full of rich phase structures [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' However, from the theoretical perspective, systems with finite µB are not straightforwardly accessible to the first-principle meth- ods of Lattice QCD (LQCD), due to the well-known fermion determinant sign problem [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Hence, studies on the µB − µI plane have been performed mainly using low energy effective models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' These models have revealed the existence of an exciting phase structure that includes Gapless Pion Condensates (GPC), a Bose-Einstein Con- densed (BEC) phase with gaped single particle excita- tions, a BEC-BCS crossover, etc [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' On the other hand, LQCD calculations for vanishing and even small µB do not suffer from the sign problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' These calculations have predicted the existence of a su- perfluid pion condensate phase for high enough µI [10– 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' At zero temperature, they show that a second order phase transition at a critical isospin chemical potential (corresponding to the vacuum pion mass), separates the hadron from the pion condensate phase [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' In addi- tion to LQCD, these phases are also found using chiral perturbation theory (χPT) [16–28], Hard Thermal Loop perturbation theory (HTLPt) [29], the Nambu-Jona- Lasinio (NJL) model [9, 30–45] and its Polyakov loop (PNJL) extended version [46, 47], the quark meson model (QMM) [48–51] and other low energy effective models ex- ploiting functional RG studies [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Calculations using a LQCD equation of state for finite µI have investigated the viability of the existence of pion stars, with a pion condensate as the dominant core constituent [24, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Since LQCD calculations with µI ̸= 0, µB = µs = T = 0 can be carried out without being hindered by the sign problem, they can be used as a benchmark to test effec- tive model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' For example, recently, the NJL model has been used in this domain and it has been found that results agree exceptionally well with LQCD results [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' In this work we study another effective QCD model, the Linear Sigma Model with quarks (LSMq), extended 2 to consider a finite µI to describe the properties of strongly interacting systems with an isospin imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The LSMq is a renormalizable theory that explicitly im- plements the QCD chiral symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' It has been suc- cessfully employed to study the chiral phase transition at finite T and µB [56–59], as well as in the presence of a magnetic field [60–67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The Linear Sigma Model has been used at finite µI, albeit considering the meson degrees of freedom as an effective classical background, in the Hartree or Hartree Fock approximations within the Cornwall-Jackiw-Tomboulis (CJT) formalism [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' In contrast, in the LSMq mesons are treated as dynamical fields able to contribute to quantum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Part of the reason for other models to avoid considering mesons as dynamical fields, for example the QMM, is that when mesons become true quantum fields and chiral symmetry is only spontaneously broken, their masses are subject to change as a result of medium effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' During this change, the meson square masses can become zero or even neg- ative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' At zero temperature, this drawback is avoided by considering an explicit symmetry breaking term that provides pions with a vacuum finite mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' At finite tem- perature, the plasma screening effects need to also be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' In this work we use the LSMq to describe the evolu- tion of the chiral and isospin (pion) condensates, as well as thermodynamical quantities such as pressure, isospin and energy densities and the sound velocity at zero tem- perature and finite µI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' We restrict ourselves to consider- ing only the effects of fermion quantum fluctuations, re- serving for a future work the inclusion of meson quantum fluctuations effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' We make use of the renormalizability of the LSMq and describe in detail the renormalization procedure which is achieved by implementing the stabil- ity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The results thus obtained are valid for the case where µ2 I) is small compared to the sum of the squares of the chiral and isospin condensates multiplied by the square of the boson-fermion coupling constant g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The work is organized as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' II we write the LSMq Lagrangian using degrees of freedom appropriate to describe an isospin imbalanced system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' We work with an explicit breaking of the chiral symmetry introducing a vacuum pion mass and expanding the charged pion fields around the values of their condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The effective po- tential is constructed by adding to the tree-level poten- tial the one-loop contribution from the fermion degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Renormalization is carried out by introduc- ing counter-terms to enforce that the tree-level structure of the effective potential is preserved by loop corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' We first work out explicitly the treatment in the con- densed phase to then work out the non-condensed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' III we study the condensates evolution with µI as well as that of the pressure, isospin and energy density and the sound velocity, and compare to recent LQCD results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' We finally summarize and conclude in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' We reserve for a follow up work the computation of the meson quantum fluctuations as well as finite temperature effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The appendix is devoted to the explicit computa- tion of the one-loop fermion contribution to the effective potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' LSMQ AT FINITE ISOSPIN CHEMICAL POTENTIAL The LSMq is an effective theory that captures the ap- proximate chiral symmetry of QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' It describes the in- teractions among small-mass mesons and quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' We start with a Lagrangian invariant under SU(2)L × SU(2)R chiral transformations L = 1 2(∂µσ)2 + 1 2(∂µ⃗π)2 + a2 2 (σ2 + ⃗π2) − λ 4 (σ2 + ⃗π2)2 + i ¯ψγµ∂µψ − ig ¯ψγ5⃗τ · ⃗πψ − g ¯ψψσ, (1) where ⃗τ = (τ1, τ2, τ3) are the Pauli matrices, ψL,R = � u d � L,R , (2) is a SU(2)L,R doublet, σ is a real scalar field and ⃗π = (π1, π2, π3) is a triplet of real scalar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' π3 corresponds to the neutral pion whereas the charged ones are repre- sented by the combinations π− = 1 √ 2 (π1 + iπ2), π+ = 1 √ 2 (π1 − iπ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (3) The parameters a2, λ and g are real and positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Equation (1) can be written in terms of the charged and neutral-pion degrees of freedom as L = 1 2[(∂µσ)2 + (∂µπ0)2] + ∂µπ−∂µπ+ + a2 2 (σ2 + π2 0) + a2π−π+ − λ 4 (σ4 + 4σ2π−π+ + 2σ2π2 0 + 4π2 −π2 + + 4π−π+π2 0 + π4 0) + i ¯ψ/∂ψ − g ¯ψψσ − ig ¯ψγ5(τ+π+ + τ−π− + τ3π0)ψ, (4) where we introduced the combination of Pauli matrices τ+ = 1 √ 2(τ1 + iτ2), τ− = 1 √ 2(τ1 − iτ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (5) The Lagrangian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (4) possesses the following sym- metries: A SU(Nc) global color symmetry, a SU(2)L × SU(2)R chiral symmetry and a U(1)B symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The sub-index of the latter emphasizes that the conserved charge is the baryon number B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' A conserved isospin charge can be added to the LSMq Hamiltonian, multi- plied by the isospin chemical potential µI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The result is that the Lagrangian gets modified such that the ordinary derivative becomes a covariant derivative [69] ∂µ → Dµ = ∂µ + iµIδ0 µ, ∂µ → Dµ = ∂µ − iµIδµ 0 , (6) 3 As a result, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (4) is modified to read as L = 1 2[(∂µσ)2 + (∂µπ0)2] + Dµπ−Dµπ+ + a2 2 (σ2 + π2 0) + a2π−π+ − λ 4 � σ4 + 4σ2π−π+ + 2σ2π2 0 + 4π2 −π2 + + 4π−π+π2 0 + π4 0 � + i ¯ψ/∂ψ − g ¯ψψσ + ¯ψµIτ3γ0ψ − ig ¯ψγ5(τ+π+ + τ−π− + τ3π0)ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (7) Because of the spontaneous breaking of the chiral sym- metry in the Lagrangian given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (7), the σ field ac- quires a non-vanishing vacuum expectation value σ → σ + v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' To make better contact with the meson vacuum proper- ties and to include a finite vacuum pion mass, m0, we can add an explicit symmetry breaking term in the La- grangian such that L → L′ = L + h(σ + v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (8) The constant h is fixed by requiring that the model ex- pression for the neutral vacuum pion mass squared in the non-condensed phase, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (11a), corresponds to m2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' This yields h = m2 0 � a2 + m2 0 λ , ≡ m2 0fπ, (9) where fπ is the pion decay constant and have used its ex- plicit model expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Equation (9) provides a relation for the model parameters a and λ in terms of fπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Before diving into the formalism details, here we first pause to discuss the symmetry properties of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Notice that the introduction of µI and h modifies the structure of the effective Lagrangian given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' In the presence of a finite µI, the U(1)B ×SU(2)L×SU(2)R symmetry is reduced to U(1)B × U(1)I3L × U(1)I3R for h = 0, and to U(1)B × U(1)I3 for h ̸= 0, thereby repre- senting the explicit breaking of the chiral symmetry [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The notation also emphasizes that the third component of the isospin charge, I3, corresponds to the generator of the remaining symmetry U(1)I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Since in the present work, we are interested in the dynamics of the pion fields, further simplifications in the pseudoscalar channels can be obtained using the ansatz ⟨ ¯ψiγ5τ3ψ⟩ = 0 combined with ⟨¯uiγ5d⟩ = ⟨ ¯diγ5u⟩∗ ̸= 0 [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' This further breaks the residual U(1)I3 symmetry and corresponds to a Bose- Einstein condensation of the charged pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Then, the charged pion fields can be referred from their conden- sates as π+ → π+ + ∆ √ 2eiθ, π− → π− + ∆ √ 2e−iθ, (10) where the phase factor θ indicates the direction of the U(1)I3 symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' We take θ = π for defini- tiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The shift in the sigma field produces that the fermions and neutral bosons acquire masses given by mf = gv (11a) m2 π0 = λv2 − a2 + λ∆2 (11b) m2 σ = 3λv2 − a2 + λ∆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (11c) The charged pions also acquire masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' However, in the condensed phase (∆ ̸= 0) they need to be described in terms of the π1,2 fields [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Since for our purposes, pions are not treated as quantum fluctuations, hereby we just notice that, as a consequence of the breaking of the U(1)I3 symmetry, one of these fields becomes a Goldstone boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' In the absence of the explicit symmetry breaking term in the Lagrangian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (8), this mode’s mass would vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' However, a finite h prevents this mode from being massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Condensed phase In the condensed phase the tree-level potential, ex- tracted from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (7) and (8), can be written as Vtree = −a2 2 � v2 + ∆2� + λ 4 � v2 + ∆2�2 − 1 2µ2 I∆2 − hv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (12) The fermion contribution to the one-loop effective po- tential becomes � f=u,d V 1 f = −2Nc � d3k (2π)3 � Eu ∆ + Ed ∆ � , (13) with (see Appendix A) Eu ∆ = ��� k2 + m2 f + µI �2 + g2∆2 �1/2 , (14a) Ed ∆ = ��� k2 + m2 f − µI �2 + g2∆2 �1/2 , (14b) where we chose that µd = µI µu = −µI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (15) Equation (13) is ultraviolet divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Ultraviolet diver- gences are a common feature of loop vacuum contribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' However, since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (13) depends on µI, this di- vergence needs to be carefully treated given that mat- ter contributions cannot contain ultraviolet divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' To identify the divergent terms, we work in the approx- imation whereby the fermion energies, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (14), are ex- panded in powers of µ2 I/[g2(v2 +∆2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Considering terms up to O(µ4 I), we obtain � f=u,d Ef ∆ ≃ 2 � k2 + m2 f + g2∆2 + µ2 Ig2∆2 (k2 + m2 f + g2∆2)3/2 + µ4 I � 4(k2 + m2 f)g2∆2 − g4∆4� 4 � k2 + m2 f + g2∆2 �7/2 + O(µ6 I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (16) 4 Notice that the ultraviolet divergent part corresponds only to the first and second terms on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' In this approximation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' and up to terms of order µ2 I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' the expression for the leading fermion contri- bution to the one-loop effective potential is given by � f=u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='d V 1 f = −2Nc � d3k (2π)3 � 2 � k2 + m2 f + g2∆2 + µ2 Ig2∆2 (k2 + m2 f + g2∆2)3/2 � (17) This expression can be readily computed using dimen- sional regularization in the MS scheme,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' with the result (see Appendix A) � f=u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='d V 1 f = 2Nc g4 � v2 + ∆2�2 (4π)2 �1 ǫ + 3 2 + ln � Λ2/g2 v2 + ∆2 �� − 2Nc g2µ2 I∆2 (4π)2 �1 ǫ + ln � Λ2/g2 v2 + ∆2 �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (18) where Nc = 3 is the number of colors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Λ is the dimen- sional regularization ultraviolet scale and the limit ǫ → 0 is to be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The explicit computation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (18) is described also in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Notice that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (18) contains an ultraviolet divergence proportional to µ2 I∆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Since a term with this same structure is already present in the tree-level potential, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (12), it is not surpris- ing that this ultraviolet divergence can be handled by the renormalization procedure with the introduction of a counter-term with the same structure, as we proceed to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' To carry out the renormalization of the effective po- tential up to one-loop order, we introduce counter-terms that respect the structure of the tree-level potential and determine them by accounting for the stability condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The latter are a set of conditions satisfied by the tree-level potential and that must be preserved when con- sidering loop corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' These conditions require that the position of the minimum in the v- and ∆-directions remain the same as the tree-level potential ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The tree-level minimum in the v, ∆ plane is found from ∂Vtree ∂v = � λv3 − (a2 − λ∆2)v − h ����� v0, ∆0 = 0 (19a) ∂Vtree ∂∆ = � λ∆2 − (µ2 I − λv2 + a2) ����� v0, ∆0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (19b) Notice that the second of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (19) admits a real, non- vanishing solution, only when µ2 I > λv2 − a2 = m2 0, (20) which means that a non-zero isospin condensate is devel- oped only when, for positive values of the isospin chem- ical potential, the latter is larger than the vacuum pion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' This is what we identify as the condensed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The simultaneous solutions of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (19) are v0 = h µ2 I , (21a) ∆0 = � µ2 I λ − h2 µ4 I + a2 λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (21b) Hereafter, we refer to the expressions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (21) as the classical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The effective potential, up to one-loop order in the fermion fluctuations, including the counter-terms, can be written as Veff = Vtree + � f=u,d V 1 f − δλ 4 (v2 + ∆2)2 + δa 2 (v2 + ∆2) + δ 2∆2µ2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (22) The counter-terms δλ and δ are determined from the gap equations ∂Veff ∂v ���� v0, ∆0 = 0, (23a) ∂Veff ∂∆ ���� v0, ∆0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (23b) These conditions suffice to absorb the infinities of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The counter-term δa is determined by requiring that the slope of Veff vanishes at µI = m0, ∂Veff ∂µI ���� µI=m0 = 0, (24) or in other words, that the transition from the non- condensed to the condensed phase be smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The ef- fective potential thus obtained is ultraviolet finite as well as Λ-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Non-condensed phase In the non-condensed phase, 0 ≤ µI ≤ m0, the only allowed solution for the second of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (19) is ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' For this case, the first of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (19) becomes a cubic equation in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The only real solution is ˜v0 = ( √ 3 √ 27h2λ4 − 4a6λ3 + 9hλ2)1/3 (18)2/3λ + (2/3)1/3a2 ( √ 3 √ 27h2λ4 − 4a6λ3 + 9hλ2)1/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (25) In the limit when h is taken as small one gets ˜v0 ≃ a √ λ + h 2a2 , (26) an approximation that some times is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' How- ever, hereafter we work instead with the full expression given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' 5 Δ [GeV] v [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='12 μI/m0 Figure 1: v- and ∆-condensates as functions of the scaled variable µI/m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' For µI ≥ m0, the v-condensate decreases while the ∆-condensate increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The effective potential V noncond eff up to one-loop order can be obtained from the corresponding one in the con- densed phase, by setting ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Therefore, we can write V noncond eff = λ 4 v4 − a2 2 v2 − hv − ˜δ1 4 v4 + ˜δ2 2 v2 + 2Nc g4v4 (4π)2 �1 ǫ + 3 2 + ln � Λ2 g2v2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (27) In this case, only two conditions are needed to stabilize the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' We take these as the requirement that the position and curvature of V noncond eff remain at its classical value when evaluated at ˜v0, namely, ∂V noncond eff ∂v ���� ˜v0 = 0 (28a) ∂2V noncond eff ∂v2 ���� ˜v0 = 3λ˜v2 0 − a2, (28b) from where the counter-terms ˜δ1, ˜δ2 can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Therefore, in the non-condensed phase, in addition to ∆ = 0, the v-condensate is simply given by the constant ˜v0 given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' As for the case of the condensed phase, in the non-condensed phase the effective potential is ultraviolet finite as well as Λ-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' THERMODYNAMICS OF THE CONDENSED PHASE Armed with the expressions for the effective potential, we can now proceed to study the dependence of the con- densates as well as of the thermodynamical quantities as functions of µI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Since the µI-dependence in the non- condensed phase is trivial, we concentrate in the descrip- tion of the behavior of these quantities in the condensed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' LQCD 24×48 LQCD 32×48 Tree Level One loop SU(2) NJL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='6 μI/m0 PN/m0 4 Figure 2: Normalized pressure as a function of the scaled variable µI/m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Shown are the tree-level and one-loop fermion improved pressures compared to the results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [54] together with the LQCD results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The model requires fixing three independent parame- ters: the boson self-coupling λ, the boson-fermion cou- pling g and the mass parameter a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' For a vacuum pion mass m0 = 135 MeV, these parameters are fixed by re- quiring that the pion vacuum decay constant is fπ = 93 MeV, the light quark mass is mq = 235 MeV and the sigma mass is mσ = 400 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The phase space for these parameters is limited since for certain combinations, the gap equation conditions in the v-∆ plane become saddle points rather than global minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Figure 1 shows the v- and ∆-condensates as functions of the scaled variable µI/m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The behavior is qualita- tively as expected: for µI ≥ m0, the v-condensate de- creases while the ∆-condensate increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Figure 2 shows the normalized pressure, defined as the negative of the effective potential referred from its value at µI = m0, as a function of the scaled variable µI/m0 and divided by m4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Shown are the results obtained by using the tree-level and the fermion one-loop corrected effective potentials, compared to the results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [54] and the LQCD results from [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Notice that the one-loop improved calculation does a better description than the tree-level one and that deviations from the LQCD result appear for µI ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='5 m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Figure 3 shows the normalized isospin density, nI = dP/dµI, divided by m3 0 as a function of the scaled vari- able µI/m0 compared to results obtained using the tree- level potential as well as to the results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [54] together with the LQCD results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Notice that the one-loop improved calculation is close to the NJL one up to µI ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='5 m0 but the latter does a better job describing the LQCD results for µI ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='5 m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' However, it is fair to say that neither the current calculation nor the NJL result reproduce the change of curvature that 6 Tree Level One loop SU(2) NJL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='4 μI/m0 nI/m0 3 Figure 3: Normalized isospin density as a function of the scaled variable µI/m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Shown are the tree-level and one-loop fermion improved effective potentials compared to a recent SU(2) NJL calculation [54] and the LQCD results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' seems to be present in the LQCD result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Figure 4 shows the normalized energy density, ǫ/m4 0, as a function of the scaled variable µI/m0, compared to the results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [54] together with the LQCD results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Although the change in curvature shown by the LQCD results is not described by the present cal- culation, it is fair to say that neither the NJL calculation captures such trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The one-loop improved calculation does a better average description of the LQCD result al- though deviations appear for µI ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='5 m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Figure 5 shows the equation of state, pressure vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' en- ergy density, compared to the results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [54] to- gether with the LQCD results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Notice that for the latter, the vacuum pion mass is taken as m0 = 135 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' As can be seen, the initial increasing trend of LQCD results is properly described by the low- energy models considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Given that the accuracy of our results is limited to the low µI domain the NJL cal- culation does a better description of the LQCD results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Figure 6 shows the square of the speed of sound, c2 s, as a function of the scaled variable µI/m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Shown are the one-loop results compared to the results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [54] to- gether with the LQCD results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The appar- ent peak in the LQCD results is not reproduced by any model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' However, notice that for the range of shown µI values, the one-loop improved result is above, although closer to the conformal bound, shown as a horizontal line at c2 s = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' SUMMARY AND CONCLUSIONS In this work we have used the LSMq, with two quark flavors, to study the phase structure of isospin asymmet- LQCD 24×48 LQCD 32×48 Tree Level One loop SU(2) NJL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='5 μI/m0 ϵ/m0 4 Figure 4: Normalized energy density as a function of the scaled variable µI/m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Shown are the tree-level and one-loop fermion improved effective potentials compared to the results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [54] together with the LQCD results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' ric matter at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The meson degrees of freedom are taken as providing the mean field on top of which we include quantum quark fluctuations at one-loop order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' We have used the renormalization of the LSMq to absorb the ultraviolet divergences with the addition of counter-terms that respect the original structure of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' An interesting aspect of the method is that it al- lows the proper handling of the disturbing µI-dependent ultraviolet divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The one-loop quark contributions are treated in the approximation whereby µ2 I is taken as small compared to g2(v2 +∆2) and working up to O(µ2 I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' After determining the model parameters, we have stud- ied the evolution of the chiral and isospin condensates as well as the pressure, energy and isospin densities and the sound velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' We have compared the model results with a recent NJL calculation of the same quantities and with LQCD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The model does a good description for µI ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='5 m0, except perhaps for the sound veloc- ity for which it does not reproduce the peak seemingly appearing in the LQCD calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The results are encouraging and set the stage to ex- plore whether the method can be used to incorporate the effect of meson fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The method also lends itself to include in the description higher powers of µ2 I as well as finite temperature effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' We are currently exploring these avenues and will report on the findings elsewhere in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors are grateful to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Endr¨odi and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Brandt for kindly sharing their LQCD data in tabular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Support for this work was received in part by LQCD 24×48 LQCD32×487 LQCD 24×48 LQCD 32×48 Tree level One loop SU(2) NJL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='5 PN/m0 4 ϵ/m0 4 Figure 5: Equation of state, pressure vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Shown are the tree-level and one-loop fermion improved effective potentials compared to the results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [54] together with the LQCD results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' For the latter, the vacuum pion mass is taken as m0 = 135 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' UNAM-PAPIIT IG100322 and by Consejo Nacional de Ciencia y Tecnolog´ıa grant number A1-S-7655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' acknowledges support from a PAPIIT-DGAPA-UNAM fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' This work was partially supported by Con- selho Nacional de Desenvolvimento Cient´ıfico e Tecno- l´ogico (CNPq), Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' 309598/2020-6 (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Funda¸c˜ao de Amparo `a Pesquisa do Estado do Rio Grande do Sul (FAPERGS), Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' 19/2551- 0000690-0 and 19/2551-0001948-3 (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' ac- knowledges the support from the Alexander von Hum- boldt Foundation postdoctoral research fellowship in Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Appendix A: One-loop quark contribution to the effective potential The thermodynamic potential accounting for the quark contribution at one-loop order is given by V 1 f = iV −1 ln � Z1 f � , (A1) where ln (Z1 f ) = ln � det �� S−1 mf ��� , (A2) and V is the space-time volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Also here, S−1 mf is the inverse propagator of the two light-quark species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' There- fore, we are bound to compute the determinant of a ma- trix M of the form M = � A B C D � , (A3) where A, B, C, D can be thought of as p×p, p×q, q ×p, and q × q complex matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' When A, and NJL SU(2) Conformal Bound LQCD 24×48 LQCD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='6 μI/m0 cs 2 Figure 6: Square of the speed of sound as a function of the scaled variable µI/m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Shown are the tree-level and one-loop fermion improved effective potentials compared to a recent SU(2) NJL calculation [54] and the LQCD results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' D, are invertible, the determinant of M is given by det{(M)} = det{(A)} det � (D − CA−1B) � , (A4) det{(M)} = det{(D)} det � (A − BD−1C) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A5) Equation (A4) can be written as det{(M)} = det{(A)} det � (D − CA−1B) � = det{(A)} det � (C−1C) � det � (D − CA−1B) � = det � (−C2A−1BC−1A + CDC−1A) � , (A6) whereas Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A5) as det{(M)} = det{(D)} det � (A − BD−1C) � = det{(D)} det � (C−1C) � det � (A − BD−1C) � = det � (−CB + CAC−1D) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A7) For our purposes, B = C = ig∆γ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Thus, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A6) and (A7), we obtain det{(M)} = det � (−C2 + CDC−1A) � , (A8) det{(M)} = det � (−C2 + CAC−1D) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A9) We explicitly compute both expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Fist, we use that the standard spin projectors Λ± satisfy γ0Λ±γ0 = ˜Λ∓, (A10) and γ5Λ±γ5 = ˜Λ±, (A11) LQCD32X48 Tree Level One Loop8 with the projectors ˜Λ± defined as ˜Λ± = 1 2 � 1 ± γ0(⃗γ · ⃗k − gv) Ek � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A12) Next, we notice that A = S−1 u and D = S−1 d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' There- fore, working first in the absence of an isospin chemical potential, for which S−1 u = S−1 d = k0γ0 − ⃗γ · ⃗k − gv, (A13) D1 ≡ −C2 + CDC−1A = g2∆2 + (ig∆γ5)S−1 d � 1 ig∆γ5 � S−1 u = g2∆2 − � k2 0 − (Eu k )2� Λ− − � k0 − � Ed k �2� Λ+, (A14) and D2 ≡ −C2 + CAC−1D = g2∆2 + γ5S−1 u γ5S−1 d = g2∆2 − � k2 0 − � Ed k �2� Λ− − � k2 0 − (Eu k )2� Λ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A15) Thus, using that Λ+ + Λ− = 11 and defining Eq ∆ = � (Eq k)2 + g2∆2, we have D1 = − � k2 0 − (Eu ∆)2� Λ− − � k0 − � Ed ∆ �2� Λ+, (A16) D2 = − � k2 0 − � Ed ∆ �2� Λ− − � k2 0 − (Eu ∆)2� Λ+, (A17) and det � (S−1 mf ) � = det{(D1)} = det{(D2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A18) Note that ln (Z1 f ) = ln � det �� S−1 mf ��� = 1 2 ln � det �� S−1 mf �2�� = 1 2 ln (det{(D1D2)}) = 1 2Tr [ln (D1D2)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A19) and since the product D1D2 is given by D1D2 = � k2 0 − (Eu ∆)2� � k2 0 − � Ed ∆ �2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A20) we get ln (Z1 f ) = 1 2 � q=u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='d Tr � ln � k2 0 − (Eq ∆)2 �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A21) where the trace is taken in Dirac,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' color (factors of 4 and Nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' respectively),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' and in coordinate spaces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' ln (Z1 f ) = 2Nc � q=u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='d � d4x � x ��� ln � k2 0 − (Eq ∆)2 ����x � = 2Nc � q=u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='d � d4x � d4k (2π)4 ln � k2 0 − (Eq ∆)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A22) Therefore ln (Z1 f) = 2V Nc � q=u,d � d4k (2π)4 ln � k2 0 − (Eq ∆)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A23) In order to obtain a more compact expression, we inte- grate and differentiate with respect to Eq ∆ as follows ln (Z1 f) = 2V Nc � q=u,d � d4k (2π)4 � dEq ∆ Eq ∆ k2 0 − (Eq ∆)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A24) Performing a Wick rotation k0 → ik4, we obtain ln (Z1 f) = 4iV Nc � q=u,d � d4kE (2π)4 � dEq ∆ Eq ∆ k2 0 − (Eq ∆)2 , (A25) and integrating over k4 and Eq ∆, in this order, we get ln (Z1 f) = 2iV Nc � q=u,d � d3k (2π)3 Eq ∆, (A26) with Re[(Eq ∆)2] ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Therefore, the quark contribution to the effective potential at one-loop order is given by V 1 f = iV −1 ln (Z1 f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A27) Thus, V 1 f = −2Nc � q=u,d � d3k (2π)3 Eq ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A28) In the presence of an isospin chemical potential for which S−1 u = (k0 + µI)γ0 − ⃗γ · ⃗k − gv, S−1 d = (k0 − µI)γ0 − ⃗γ · ⃗k − gv, (A29) and repeating the steps starting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A14), we ob- tain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A28), with the energies Eu ∆ and Ed ∆ given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' We now proceed to the explicit computation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' In the limit where µ2 I/[g2(v2 + ∆2)] is small, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A28) can be written as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' We use dimen- sional regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' The first of the integrals on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (17) is expressed as � d3k (2π)3 � k2 + g2v2 + g2∆2 → Λ3−d Γ � − 1 2 − d 2 � (4π) d 2 Γ � − 1 2 � × � 1 g2v2 + g2∆2 �− 1 2 − d 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A30) 9 Taking d → 3 − 2ǫ and working in the MS scheme Λ2 → Λ2eγE 4π , (A31) where γE is the Euler-Mascheroni constant, we get � d3k (2π)3 � k2 + g2v2 + g2∆2 → −(g2v2 + g2∆2)2 2(4π)2 �1 ǫ + 3 2 + ln � Λ2 g2v2 + g2∆2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A32) The second of the integrals on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (17) is expressed as � d3k (2π)3 1 (k2 + g2v2 + g2∆2)3/2 → Λ3−d Γ � 3 2 − d 2 � (4π) d 2 Γ � 3 2 � � 1 g2v2 + g2∆2 � 3 2 − d 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (A33) Taking d → 3 − 2ǫ and working in the MS scheme we get � d3k (2π)3 1 (k2 + g2v2 + g2∆2)3/2 → 2 (4π)2 �1 ǫ + ln � Λ2 g2v2 + g2∆2 �� , (A34) from where the result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (18) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Agarwal (CBM), (2022), arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='14585 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [2] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Abgaryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' (MPD), Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' A 58, 140 (2022), arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='08970 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='ins-det].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [3] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Fukushima and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Hatsuda, Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' 74, 014001 (2011), arXiv:1005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='4814 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Alford, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Schmitt, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Rajagopal, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Sch¨afer, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' 80, 1455 (2008), arXiv:0709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='4635 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [5] NUPECC, “Long range plan,” http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='nupecc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='org/lrp2016/Documents/lrp2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='pdf (2017), accessed: 2021-01-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [6] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Karsch, Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Notes Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' 583, 209 (2002), arXiv:hep-lat/0106019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Muroya, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Nakamura, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Nonaka, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Takaishi, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' 110, 615 (2003), arXiv:hep-lat/0306031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Son and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Stephanov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' A 74, 013614 (2006), arXiv:cond-mat/0507586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [9] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Mu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='-y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' He, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='-x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Liu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' D 82, 056006 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Kogut and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Sinclair, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' D 66, 014508 (2002), arXiv:hep-lat/0201017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [12] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Brandt and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Endrodi, PoS LATTICE2016, 039 (2016), arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='06758 [hep-lat].' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Farias, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' D 100, 116002 (2019), arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='09880 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' [55] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Raya, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Tejeda-Yeomans, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content=' A 58, 87 (2022), arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} 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PoS LATTICE2022, 144 (2023), arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} +page_content='01431 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFRT4oBgHgl3EQfvDhO/content/2301.13633v1.pdf'} diff --git a/BtE2T4oBgHgl3EQfRwcX/content/tmp_files/2301.03783v1.pdf.txt b/BtE2T4oBgHgl3EQfRwcX/content/tmp_files/2301.03783v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5352f4192c031ff43cdef0c825a3af35bca97f13 --- /dev/null +++ b/BtE2T4oBgHgl3EQfRwcX/content/tmp_files/2301.03783v1.pdf.txt @@ -0,0 +1,3502 @@ +Divergence-Conforming Isogeometric Collocation Methods for the +Incompressible Navier-Stokes Equations +Ryan M. Aronsona,∗, John A. Evansb +aStanford University, 94305, Stanford, CA, USA +bUniversity of Colorado Boulder, 80309, Boulder, CO, USA +Abstract +We develop two isogeometric divergence-conforming collocation schemes for incompress- +ible flow. The first is based on the standard, velocity-pressure formulation of the Navier- +Stokes equations, while the second is based on the rotational form and includes the vorticity +as an unknown in addition to the velocity and pressure. We describe the process of discretiz- +ing each unknown using B-splines that conform to a discrete de Rham complex and collocat- +ing each governing equation at the Greville abcissae corresponding to each discrete space. +Results on complex domains are obtained by mapping the equations back to a parametric do- +main using structure-preserving transformations. Numerical results show the promise of the +method, including accelerated convergence rates of the three field, vorticity-velocity-pressure +scheme when compared to the two field, velocity-pressure scheme. +Keywords: +Isogeometric analysis, Collocation, Incompressible flow, Divergence-conforming +discretizations, Velocity-pressure formulation, Vorticity-velocity-pressure formulation +1. Introduction +Isogeometric Analysis (IGA) is a technology [1, 2] which replaces the standard polynomial +basis functions used in traditional Finite Element Analysis (FEA) with B-splines, Non- +Uniform Rational B-splines (NURBS), and other classes of splines in an aim to reduce the +gap between geometry and analysis. IGA has a distinct advantage over traditional FEA +due to its ability to exactly represent the geometries commonly seen in Computer-Aided +Design (CAD). Moreover, the basis functions used in IGA are globally more smooth than +those of FEA and it has been shown that IGA can exhibit improved accuracy and robustness +over FEA. For example, higher continuity splines are shown to have optimal approximating +power in the sense of Kolmogorov n-widths [3] and spline discretizations have more favorable +dissipation and dispersion properties than standard, high-order FEA discretizations [2]. +To improve on the complexity and implementation details of IGA, the feasibility of isoge- +ometric collocation methods has been explored [4, 5]. In Galerkin IGA methods, the discrete +system of equations is formed by integrating the PDE residual against the test function space. +∗Corresponding author +Email address: rmaronso@stanford.edu (Ryan M. Aronson) +Preprint submitted to Elsevier +January 11, 2023 +arXiv:2301.03783v1 [math.NA] 10 Jan 2023 + +This requires numerical integration, which renders system assembly quite expensive. Collo- +cation, on the other hand, forms the discrete system by simply requiring that the residual +of the governing equations vanish at a set of discrete locations in the domain. +Many recent studies have shown the efficacy of isogeometric collocation methods in both +static and dynamic solid mechanics problems [6, 7, 8], and detailed comparisons between iso- +geometric Galerkin and collocation methods have been made [9]. In addition, recent studies +have also investigated the use of mixed collocation methods for use in nearly incompressible +elasticity [10, 11]. Isogeometric collocation has even been used for acoustic problems [12], +computing Karhunen-Loeve expansions [13, 14], and introduced into physics-informed neural +networks [15]. Finally, a method was recently introduced for IGA collocation in immersed +domains by combining with the finite cell method near the boundaries [16]. These results +all suggest that isogeometric collocation methods retain some of the improved qualities of +standard IGA, while reducing the computational cost and improving sparsity of the discrete +systems. +Isogeometric collocation methods have not been as well explored in the context of fluid +mechanics, though the idea of using B-spline collocation to solve incompressible fluid me- +chanics problems has been investigated in the past [17, 18]. In addition, spline collocation +has been employed in fundamental Direct Numerical Simulation (DNS) studies of turbulent +flows [19]. However, the methods previously introduced are typically limited to simple ge- +ometries and are not divergence-conforming, meaning that the discrete velocity field does +not exactly satisfy the continuity equation in incompressible flow. +In addition, a mixed +isogeometric collocation method has recently been proposed for use in poromechanics [20], +though the preliminary results were limited to one-dimensional problems. +In the context of incompressible fluid mechanics, divergence-conforming Galerkin meth- +ods based on B-spline basis functions have been developed for both the Stokes and the +Navier-Stokes equations [21, 22, 23]. These methods are provably inf-sup stable and yield +discrete velocity fields that are exactly pointwise divergence free, among other desirable qual- +ities such as pressure robustness. An excellent summary of divergence-conforming methods +is given by [24]. These discretizations have prospered in areas such as turbulent flow simula- +tion [25, 26, 27] and fluid-structure interaction [28]. Moreover, efficient multigrid solvers have +been developed based on these discretizations [29]. Divergence-conforming Galerkin meth- +ods have also been developed for more advanced spline discretizations, such as hierarchical +B-splines [30] and LR B-splines [31]. +In this paper we develop similar divergence-conforming methodologies for incompressible +flow using collocation. In particular, we introduce two collocation methods, one based on +the standard velocity-pressure form of the steady Navier-Stokes equations, and one based on +a three field (velocity-vorticity-pressure) form of the steady Navier-Stokes equations. The +latter form of the Navier-Stokes equations has recently been used to develop alternative +structure-preserving finite element discretizations [32, 33, 34] and we find that collocation +methods based on the resulting system of first order differential-algebraic equations returns +improved convergence rates compared to the rates obtained using collocation in conjunction +with the standard velocity-pressure form of the equations. In our collocation schemes, each +unknown is discretized with compatible B-spline spaces that preserve the structure of the +governing equations. Both collocation methods in this paper are shown to return velocity +fields which are still exactly pointwise divergence free, similar to the Galerkin methods +2 + +mentioned above. +We lay out this paper as follows: In Section 2 we describe the steady form of the Navier- +Stokes equations using velocity and pressure unknowns as well as vorticity, velocity, and +pressure unknowns. This is followed in Section 3 by a discussion of the de Rham complex and +isogeometric discrete differential forms, the tools used to develop a divergence-conforming +method. Section 4 describes the collocation schemes for square domains in two dimensions. +Then results are presented in the two dimensional setting in Section 5 which detail the +high-order convergence rates of the methods as well as agreement with standard benchmark +problems. +We then discuss the necessary changes to make the methods work for cubic +domains in three dimensions and illustrate that the methods performs similarly in this setting +in Sections 6 and 7. Finally, we return to 2D in Section 8 and consider the Stokes equations +in more complicated domains. We show that by mapping the equations and unknowns via +divergence and integral preserving transformations we can also obtain results for flow in +complex geometries. Section 9 summarizes these results. +2. Velocity-Pressure and Vorticity-Velocity-Pressure Forms of the Navier-Stokes +Equations +In this paper we consider the steady, incompressible Navier-Stokes equations on a Lips- +chitz open set Ω of points in either R2 or R3 when subjected to Dirichlet boundary conditions. +The standard form of this problem with d = 2, 3 is stated as follows: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Given ν ∈ R+, f : Ω → Rd, and g : ∂Ω → Rd, find u : Ω → Rd and +p : Ω → R such that: +−ν∆u + u · ∇u + ∇p = f +in +Ω +(1) +∇ · u = 0 +in +Ω +(2) +u = g +on +∂Ω. +(3) +Equations (1) and (2) represent the standard forms of the momentum and mass conservation +equations, while Equation (3) sets the Dirichlet boundary values. Above, we denote the +velocity field by u, the kinematic pressure field by p, the constant kinematic viscosity by ν, +the applied forcing as f, and the prescribed Dirichlet boundary values as g. +For the purposes of this paper we will not only work with this set of equations, but also +introduce vorticity ω as a separate unknown variable and introduce ω − ∇ × u = 0 as a +constitutive relation. Substituting the two vector calculus identities: +∆u = ∇(∇ · u) − ∇ × (∇ × u) = −∇ × ω, +(4) +u · ∇u = (∇ × u) × u + 1 +2∇(u · u) = ω × u + 1 +2∇(u · u), +(5) +into Equation (1) when d = 3, we arrive at the vorticity-velocity-pressure formulation of the +problem in 3D: +3 + +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Given ν ∈ R+, f : Ω → R3, and g : ∂Ω → R3, find u : Ω → R3, +P : Ω → R, and ω : Ω → R3 such that: +ν∇ × ω + ω × u + ∇P = f +in +Ω +(6) +∇ · u = 0 +in +Ω +(7) +ω − ∇ × u = 0 +in +Ω +(8) +u = g +on +∂Ω. +(9) +Note that in the above formulation we have replaced the kinematic pressure p with the total +pressure P, which are related via P = p + 1 +2u · u. +For later sections in this paper it is useful to employ the component forms of the vec- +tor equations above describing the vorticity-velocity-pressure formulation. When explicitly +broken into its components, the momentum conservation equation, given by Equation (6), +becomes: +ν(∂ωz +∂y − ∂ωy +∂z ) + (ωyuz − ωzuy) + ∂P +∂x = fx, +(10) +ν(∂ωx +∂z − ∂ωz +∂x ) + (ωzux − ωxuz) + ∂P +∂y = fy, +(11) +ν(∂ωy +∂x − ∂ωx +∂y ) + (ωxuy − ωyux) + ∂P +∂z = fz, +(12) +and the constitutive relation given by Equation (8) reads: +ωx − (∂uz +∂y − ∂uy +∂z ) = 0, +(13) +ωy − (∂ux +∂z − ∂uz +∂x ) = 0, +(14) +ωz − (∂uy +∂x − ∂ux +∂y ) = 0. +(15) +The above component form of the equations is also useful for considering 2D problems in +the three field formulation, as we do not need to redefine operations such as cross products +when the vorticity reduces to a scalar unknown. Thus we can arrive at the problem statement +for 2D domains by simply removing any terms involving ωx, ωy, or any derivatives in the z +direction. In full, the 2D problem reads: +4 + +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Given ν ∈ R+, f : Ω → R2, and g : ∂Ω → R2, find u : Ω → R2, +P : Ω → R, and ω : Ω → R such that: +ν ∂ω +∂y − ωuy + ∂P +∂x = fx +in +Ω +(16) +−ν ∂ω +∂x + ωux + ∂P +∂y = fy +in +Ω +(17) +∇ · u = 0 +in +Ω +(18) +ω − (∂uy +∂x − ∂ux +∂y ) = 0 +in +Ω +(19) +u = g +on +∂Ω. +(20) +With the governing equations fully defined we can move to a more in-depth description +of the collocation scheme, starting with the definition of discrete approximation spaces in +the following section. +3. The de Rham Complex and Isogeometric Discrete Differential Forms +The first step in creating a collocation scheme is to define the sets of basis functions used +to approximate the unknown variables. To construct the collocation methods presented in +this paper, we leverage the de Rham complex which aids in the development of exactly +divergence-conforming finite element spaces. After recalling the de Rham complex, we de- +scribe the process of constructing B-spline basis functions and conclude with the definition +of B-spline spaces that conform to the de Rham complex. +3.1. The de Rham Complex +The de Rham complex is a cochain complex that is often used as a starting point for +developing mixed finite element methods which preserve topological properties of the con- +tinuous problem and are typically more stable in practice [24]. In 3D, it is typically written +as: +R +Φ +Ψ +V +Q +0, +∇ +∇× +∇· +(21) +where: +Φ := H 1(Ω), +(22) +Ψ := H(curl, Ω), +(23) +V := H(div, Ω), +(24) +Q := L2(Ω). +(25) +In the context of fluid flow, these infinite dimensional spaces can be interpreted as the +spaces of scalar potential fields (Φ), vector potential fields (Ψ), velocity fields (V), and +5 + +pressure fields (Q). This complex is exact for simply connected domains, meaning that the +range of each map is the same as the null space of the following map. +For completeness, we also state the rotated 2D de Rham complex: +R +Ψ +V +Q +0, +∇⊥ +∇· +(26) +where: +Ψ := H1(Ω), +(27) +V := H(div, Ω), +(28) +Q := L2(Ω), +(29) +and the rotor operator ∇⊥ acting on a scalar function ω is defined as ∇⊥ω = ( ∂ω +∂y , − ∂ω +∂x). +Not only is the topological structure of the incompressible Navier-Stokes equations em- +bedded in the de Rham complex, but stability conditions result from the complex as well. By +creating approximation spaces for the unknown variables that conform to discrete analogs of +Equations (21) and (26) we can generate numerical methods which inherit these properties. +More concretely, for 3D problems let the space Φh contain the discrete scalar potentials, Ψh +contain the discrete vector potentials as well as the discrete vorticity, Vh contain the dis- +crete velocity, and Qh contain the discrete pressure. Then if there exist projection operators +ΠΦ : Φ → Φh, ΠΨ : Ψ → Ψh, ΠV : V → Vh, and ΠQ : Q → Qh such that the following +commuting diagram holds +R −−−→ Φ +∇ +−−−→ Ψ +∇× +−−−→ V +∇· +−−−→ Q −−−→ 0 +���ΠΦ +���ΠΨ +���ΠV +���ΠQ +R −−−→ Φh +∇ +−−−→ Ψh +∇× +−−−→ Vh +∇· +−−−→ Qh −−−→ 0, +(30) +a Galerkin finite element method employing Vh for the discrete velocity space and Qh for the +discrete pressure space will be inf-sup stable and will yield discrete velocity approximations +that are divergence free almost everywhere [35]. We shall prove later on that the divergence- +conforming property is maintained if we utilize these spaces in our collocation scheme. +The same holds in 2D, where we instead let Ψh define the discrete space to which the +vorticity belongs (as well as the streamfunction), Vh define the discrete velocity space, and +Qh define the discrete pressure space. The required commuting diagram in this case is +R −−−→ Ψ +∇⊥ +−−−→ V +∇· +−−−→ Q −−−→ 0 +���ΠΨ +���ΠV +���ΠQ +R −−−→ Ψh +∇⊥ +−−−→ Vh +∇· +−−−→ Qh −−−→ 0. +(31) +Of course, we have yet to define the specifics of how to construct discrete spaces such +that these discrete complexes hold. For the purposes of this paper we will use compatible +B-spline spaces, and the following section is devoted to introducing the basics of B-spline +basis functions. +6 + +3.2. Univariate and Multivariate B-Splines +The construction of a B-spline basis in one dimension requires two objects: the de- +gree of the basis (denoted k) and a series of numbers called the knot vector (denoted +Ξ = {ξ1, ...ξn+k+1}). The knots ξi are non-decreasing and denote the locations in paramet- +ric space where the parametrization can change, similar to element boundaries in standard +FEA. The number n in the previous relation represents the total number of functions in the +basis. The basis functions themselves are defined through the Cox-de Boor recursion: The +k = 0 basis functions are built as +Ni,0(ξ) = +� +1 +ξi ≤ ξ ≤ ξi+1 +0 +otherwise, +(32) +and higher-order bases are defined through +Ni,k(ξ) = +ξ − ξi +ξi+k − ξi +Ni,k−1(ξ) + +ξi+k+1 − ξ +ξi+k+1 − ξi+1 +Ni+1,k−1(ξ). +(33) +Note that in the above relations, we must utilize the convention that any occurrence of zero +divided by zero is equal to zero. +In higher dimensions (two or three for the purposes of this paper), B-spline basis functions +are constructed by simply taking the tensor product of one dimensional B-spline bases in +each parametric direction. Note that different polynomial degrees and knot vectors can be +used in each direction. +B-spline basis functions possess a number of useful properties for numerical method +development. +In particular, the smoothness of the global basis at the knot locations is +controlled by the repetition of the knot value in Ξ. The basis at these locations is Ck−r, +where r is the multiplicity of the knot. Compared to a standard finite element basis, this +basis has improved global continuity, which enables the use of collocation as more classical +derivatives of the functions are well defined. Note that if the first and last entries in the knot +vector are repeated k +1 times, the spline basis will become interpolatory at those locations. +Such knot vectors are referred to as open knot vectors, and allow for easy specification of +Dirichlet boundary conditions. For the results within this paper, all other entries in the +knot vector have multiplicity one, meaning we are utilizing a B-spline basis with the highest +possible order of continuity. Further, let us define the so-called regularity vector α for a +basis. The size of this vector is equal to the number of distinct knots, with entries equal +to the polynomial degree of the basis minus the multiplicity of the corresponding knot. In +terms of the regularity vector, the global basis functions are Cαj-continuous across the αj +unique knot. To simplify notation, the space of functions spanned by a 1D B-spline basis of +degree k and a provided knot vector is denoted as: +Sk +α = span{Ni,k}n +i=1. +(34) +We extend this notation to higher dimensions by adding extra sub- and superscripts, repre- +senting the polynomial degrees and regularities in each spatial direction. +7 + +3.3. Isogeometric Discrete Differential Forms +Using the basics of B-spine functions above allows us to develop discrete approximation +spaces for the vorticity, velocity, and pressure. These results are built upon work in the +area of isogeometric discrete differential forms [36, 37], which we will not fully develop here. +The construction of these types of spaces in the context of Galerkin approximation for the +Navier-Stokes equations can also be found in [22, 23]. For 3D problems, the B-spline spaces +used to discretize our unknown fields are given by: +Φh := {φh ∈ Sk1,k2,k3 +α1,α2,α3}, +(35) +Ψh := {ψh ∈ Sk1−1,k2,k3 +α1−1,α2,α3 × Sk1,k2−1,k3 +α1,α2−1,α3 × Sk1,k2,k3−1 +α1,α2,α3−1}, +(36) +Vh := {wh ∈ Sk1,k2−1,k3−1 +α1,α2−1,α3−1 × Sk1−1,k2,k3−1 +α1−1,α2,α3−1 × Sk1−1,k2−1,k3 +α1−1,α2−1,α3}, +(37) +Qh := {qh ∈ Sk1−1,k2−1,k3−1 +α1−1,α2−1,α3−1}. +(38) +It can be shown that these spaces satisfy the discrete complex in Equation (30). +In practice we usually define k1 = k2 = k3, and thus we can define the polynomial degree +of the spline bases constructed in the above manner using a single number k′ = k1 − 1 = +k2−1 = k3−1. This indicates that the pressure space Qh will have degree equal to k′ in each +direction. Then, according to the above, each velocity component will have degree k′ + 1 in +one direction and degree k′ in the other two. Similarly, the vorticity components will have +degree k′ + 1 in two directions and degree k in the last. +In 2D, we define the following spline spaces: +Ψh := {ψh ∈ Sk1,k2 +α1,α2}, +(39) +Vh := {wh ∈ Sk1,k2−1 +α1,α2−1 × Sk1−1,k2 +α1−1,α2}, +(40) +Qh := {qh ∈ Sk1−1,k2−1 +α1−1,α2−1}. +(41) +Similar to the 3D setting, these spaces are related as in Equation (31). +4. Collocation Methods on Square Domains +Using the discrete spaces developed above, this section focuses on the construction of +collocation methods for the Navier-Stokes equations using divergence-conforming bases. Here +we develop methods based on the velocity-pressure form of the Navier-Stokes equations as +well as the vorticity-velocity-pressure form. As the vorticity changes between a scalar in the +2D case and a vector in the 3D case, we start by considering only square domains in 2D. +This selection also lends itself to easier visualization of the methods. After briefly reviewing +the form of a typical divergence-conforming isogeometric Galerkin method, we define the +collocation grids for each unknown and then describe the imposition of boundary conditions. +The section concludes by summarizing the form of the discrete system. +8 + +4.1. Review of Galerkin Methods +We start by reviewing the form of the divergence-conforming isogeometric Galerkin meth- +ods which inspired our collocation schemes. Let us consider a problem with Dirichlet bound- +ary conditions on the velocity for concreteness. Then we define the discrete test and trial +function spaces for velocity as Vh,0 and Vh,g, which are defined as the same Vh from Equa- +tion (40) with either no penetration boundary conditions strongly enforced (for the test +space Vh,0) or with the normal velocity prescribed as given by the boundary data g at spec- +ified collocation points (for the trial space Vh,g). Similarly, define the test and trial space +for pressure as Qh,0, where Qh is the same space as in Equation (41) but with the added +condition that the pressure must have zero integral. Then the Galerkin formulation for the +velocity-pressure form would read +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Given ν ∈ R+, f : Ω → R2, and g : ∂Ω → R2, find uh ∈ Vh,g and ph ∈ Qh,0 such that, +∀(wh, qh) ∈ (Vh,0, Qh,0): +� +Ω +(ν∇wh · ∇uh + wh · (uh · ∇uh) − ph∇ · wh)dΩ +− ν +� +∂Ω +((∇uh · n) · wh − Cpen +h uh · wh)dΓ = +� +Ω +wh · fdΩ + ν +� +∂Ω +Cpen +h g · whdΓ +(42) +� +Ω +qh(∇ · uh)dΩ = 0. +(43) +Note that in the above we have used a Nitsche approach to enforce the tangential bound- +ary conditions in the momentum equations. This Galerkin formulation is valid if and only if +the minimum entry in the regularity vectors satisfy min{α1 − 1} ≥ 0 and min{α2 − 1} ≥ 0, +where α1 and α2 are the regularity vectors from Equations (39) - (41). We can write a +similar Galerkin form of the vorticity-velocity-pressure form of the Navier-Stokes equations, +which would yield +9 + +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Given ν ∈ R+, f : Ω → R2, and g : ∂Ω → R2, find uh ∈ Vh,g, P h ∈ Qh,0, and ωh ∈ Ψh +such that, ∀(wh, qh, ψh) ∈ (Vh,0, Qh,0, Ψh): +� +Ω +ν ∂ωh +∂y wh +xdΩ − +� +Ω +ωhuh +ywh +xdΩ − +� +Ω +∂wh +x +∂x P hdΩ + +� +Γ +P hwh +xdy = +� +ω +fxwh +xdΩ +(44) +− +� +Ω +ν ∂ωh +∂x wh +ydΩ + +� +Ω +ωhuh +xwh +ydΩ − +� +Ω +∂wh +y +∂y P hdΩ + +� +Γ +P hwh +ydx = +� +ω +fywh +ydΩ (45) +� +Ω +(∇ · uh)qhdΩ = 0 +(46) +� +Ω +ωhψhdΩ + +� +Ω +(∂ψh +∂x uh +y − ∂ψh +∂y uh +x)dΩ = +� +∂Ω +ψh(g · s)dΓ. +(47) +In this case the tangential velocity boundary conditions appear as natural boundary condi- +tions in the weak form of the constitutive equation, with s representing the unit tangent on +the boundary (oriented counter-clockwise). In contrast with the velocity-pressure Galerkin +formulation, this three field Galerkin formulation is valid if and only if the minimum regu- +larity of the discrete spaces satisfies min{α1 − 1} ≥ −1 and min{α2 − 1} ≥ −1 due to the +reduced differential order of the strong form. +However, we wish to pursue a collocation method inspired by the divergence-free mixed +finite element form, which we describe below. +Collocation imposes additional regularity +requirements on the spaces of unknowns, as the unknown fields and their derivatives will be +evaluated at points rather than integrated over the domain. The spaces developed above are +not only divergence-conforming, meaning discrete velocity approximations will be pointwise +divergence-free, but are also regular enough to use in collocation provided the polynomial +degree is sufficiently large and the global basis is sufficiently regular. +4.2. Collocation Grids +Similarly to the Galerkin setting, each discrete unknown in the collocation schemes is +assumed to lie in the corresponding space from above: The discrete velocity uh ∈ Vh,g, the +discrete pressure ph ∈ Qh,0, and, when applicable, the discrete vorticity ωh ∈ Ψh. For now we +ignore any boundary conditions; these will be discussed in the following section. To generate +the system of equations needed to solve for the coefficients for each basis function, we define +sets of collocation points of the form τ i for i = 1, ..., N for each of the governing equations. +Note that the total number of collocation points should be equal to the total number of +degrees of freedom in the discretization. The full discrete system is formed by requiring the +strong form of the governing equations to hold at each of the collocation points. +We choose the Greville abscissae of a B-spline space as collocation points. In one dimen- +sion, the Greville abscissae are defined by +ˆξi = ξi+1 + ... + ξi+p +p +, +(48) +10 + +A First Collocation Attempt for Stokes: +The Unit Square + +ˆξi = ξi+1 +…+ξi+p +p +Greville abscissae: += First Momentum Collocation Pt. += Second Momentum Collocation Pt. += Continuity Collocation Pt. +k = 2, 4 x 4 elements +(a) Before strong enforcement of normal ve- +locity boundary conditions +A First Collocation Attempt for Stokes: +The Unit Square +k = 2, 4 x 4 elements +Boundary Conditions: +We enforce no-penetration BCs strongly and +remove the corresponding collocation points. +We enforce no-slip BCs weakly by modifying +the momentum equations at the boundary +with a suitable penalty term. + +−div +! grad +" ˆu +( ) +( +)+ grad +" ˆp +( )− ˆf +( +) + + Cpen +2 +h2 +ˆu − ˆuBC +( +) = 0 +(b) After strong enforcement of normal ve- +locity boundary conditions +Figure 1: Example of collocation grid for k′ = 2, 4 x 4 elements for the velocity-pressure scheme. Horizontal +triangles represent the collocation points for the first momentum equation, vertical triangles represent the +points for the second momentum equation, and squares are collocation points for the continuity equation. +and in higher dimensions we simply take the tensor product of the Greville abscissae in each +direction. By construction there will be the same number of Greville points as there are basis +functions in the considered space. There are choices for collocation points other than the +Greville abscissae, such as the Cauchy-Galerkin points [38, 39] or the Demko abscissae [40]. +However, the Greville points are very easy to compute and have already been demonstrated +to give satisfactory results in practical applications (see for example [6, 18]). +As each of the discrete unknowns lie in a different B-spline space, each of the governing +equations will be collocated at a different set of Greville points. In particular, for both +the velocity-pressure formulation and the vorticity-velocity-pressure formulation we use the +Greville abscissae associated with the basis of the x-velocity (Sk1,k2−1 +α1,α2−1) as collocation points +for the x-momentum equation, the Greville abscissae associated with the basis of the y- +velocity (Sk1−1,k2 +α1−1,α2) as collocation points for the y-momentum equation, and the Greville +abscissae associated with the basis of the pressure (Sk1−1,k2−1 +α1−1,α2−1) as collocation points for the +continuity equation. The constitutive relation within the three field formulation is collocated +at the Greville abscissae for the vorticity basis (Sk1,k2 +α1,α2). The left of Figure 1 details an +example of this construction for the velocity-pressure scheme, while the left of Figure 2 +shows example grids for the vorticity-velocity-pressure scheme. +4.3. Boundary Condition Enforcement +The last unspecified aspect of the method is enforcement of the Dirichlet boundary con- +ditions. The enforcement of the normal boundary condition is done strongly and collocation +points along a boundary for the velocity component orthogonal to that boundary are re- +moved, as the boundary condition specifies the value of the solution at these points. This +is shown on the right of Figure 1 and Figure 2, which depict the same scenarios as their +counterparts but with normal boundary conditions enforced. +11 + +A Second Collocation Attempt for Stokes: +The Unit Square + +ˆξi = ξi+1 +…+ξi+p +p +Greville abscissae: += First Momentum Collocation Pt. += Second Momentum Collocation Pt. += Continuity Collocation Pt. +k = 2, 4 x 4 elements += Constitutive Collocation Pt. +(a) Before strong enforcement of normal ve- +locity boundary conditions +A Second Collocation Attempt for Stokes: +The Unit Square +k = 2, 4 x 4 elements +Boundary Conditions: +We enforce no-penetration BCs strongly and +remove the corresponding collocation points. +We enforce no-slip BCs weakly by modifying +the constitutive equations at the boundary +with a suitable penalty term. + +ˆω ˆx, ˆy +( +)− curl +! ˆu ˆx, ˆy +( +) +( +) +( +) + + Cpen +h +ˆu⋅ ˆs − ˆuBC ⋅ ˆs +( +) = 0 +An exact expression for the penalty +constant is obtained by appealing to +isogeometric finite volumes. +(b) After strong enforcement of normal ve- +locity boundary conditions +Figure 2: Example of collocation grid for k′ = 2, 4 x 4 elements for the vorticity-velocity-pressure scheme. +Horizontal triangles represent the collocation points for the first momentum equation, vertical triangles +represent the points for the second momentum equation, squares are collocation points for the continuity +equation, and circles are the points for the constitutive relation. +Enforcement of the tangential boundary condition is slightly more subtle. Recall that +in Equation (42) we utilized Nitsche’s method to enforce this boundary condition. This +motivates the enforcement in the velocity-pressure collocation scheme. Indeed if we take this +equation and undo the integration by parts, the consistency term vanishes by construction +and we are left with just the penalty terms. +If we approximate the integral of the test +function as done in [41] the collocated momentum equations will be of the form +− ν∆uh + uh · ∇uh + ∇ph + C2 +pen +h2 (uh − g) = f, +(49) +where Cpen is a penalty constant and h is the Greville mesh size perpendicular to the bound- +ary. Note that because this construction is used to only enforce the tangential boundary +conditions, this penalty term only appears in the equation for the momentum balance along +the tangential direction of each boundary. +In the vorticity-velocity-pressure scheme we do not use the same Nitsche-based approach. +The method utilized here is directly inspired by the Enhanced Collocation method for en- +forcing Neumann boundary conditions in isogeometric collocation schemes [41]. +We start by considering the weak form of the constitutive relation given by Equation +(47). The final term on the left hand side is the boundary term which would be used to +enforce natural boundary conditions in a Galerkin method. In a similar vein to the Enhanced +Collocation approach, we can undo the integration by parts to arrive at +� +Ω +ψh(ωh − (∂uh +y +∂x − ∂uh +x +∂y ))dΩ + +� +∂Ω +ψh(uh · s − g · s)ds = 0. +(50) +By approximating the integrals of the test functions as done in [41, 38] we arrive at a +modified strong form statement of the constitutive relation which can be collocated along +12 + +the boundaries +ωh − (∂uh +y +∂x − ∂uh +x +∂y ) + Cpen +h (uh · s − g · s) = 0, +(51) +where again Cpen is a penalty constant and h is the Greville mesh size perpendicular to the +boundary. +4.4. Final Collocated Equations +Finally, the results of the previous sections are collected and we present the final form of +the discrete equations used to solve for the discrete unknowns. The velocity-pressure scheme +is considered first. Let us define τ ux +ℓ +for ℓ = 1, ..., M ux to be the set of Greville points for +Sk1,k2−1 +α1,α2−1 with the points corresponding to no-penetration boundaries removed as discussed +previously. Define in a similar manner τ uy +ℓ +for ℓ = 1, ..., M uy, which are the Greville points +of Sk1−1,k2 +α1−1,α2 with no-penetration boundary points removed. Lastly, τ p +ℓ for k = 1, ..., N p are +the Greville points of Qh. Then the discrete 2D problem reads: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Find uh ∈ Vh,g and P h ∈ Qh,0 such that: +� +−ν ∂2uh +x +∂x2 − ν ∂2uh +x +∂y2 + uh +x +∂uh +x +∂x + uh +y +∂uh +x +∂y + ∂ph +∂x +� +(τ ux +ℓ ) = fx(τ ux +ℓ ) +∀τ ux +ℓ +∈ Ω +(52) +� +−ν ∂2uh +x +∂x2 − ν ∂2uh +x +∂y2 + uh +x +∂uh +x +∂x + uh +y +∂uh +x +∂y + ∂ph +∂x + C2 +pen +h2 (uh +x − gx) +� +(τ ux +ℓ ) += fx(τ ux +ℓ ) +∀τ ux +ℓ +∈ ∂Ω +(53) +� +−ν ∂2uh +y +∂x2 − ν ∂2uh +y +∂y2 + uh +x +∂uh +y +∂x + uh +y +∂uh +y +∂y + ∂ph +∂y +� +(τ uy +ℓ ) = fy(τ uy +ℓ ) +∀τ uy +ℓ +∈ Ω +(54) +� +−ν ∂2uh +y +∂x2 − ν ∂2uh +y +∂y2 + uh +x +∂uh +y +∂x + uh +y +∂uh +y +∂y + ∂ph +∂y + C2 +pen +h2 (uh +y − gy) +� +(τ uy +ℓ ) += fy(τ uy +ℓ ) +∀τ uy +ℓ +∈ ∂Ω +(55) +� +∂uh +x +∂x + ∂uh +y +∂y +� +(τ p +ℓ) = 0 +∀τ p +ℓ ∈ Ω ∪ ∂Ω. +(56) +In the above we have split the momentum equations into expressions valid on the interior +collocation points (Equations (52) and (54)) and expressions valid on the remaining boundary +collocation points (Equations (53) and (55)). +Similarly, for the vorticity-velocity-pressure scheme we also define τ ω +ℓ for ℓ = 1, ..., N ω as +the Greville points of Ψh. With this scheme the discrete 2D problem reads: +13 + +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Find uh ∈ Vh,g, P h ∈ Qh,0, and ωh ∈ Ψh such that: +� +ν ∂ωh +∂y − ωhuh +y + ∂P h +∂x +� +(τ ux +ℓ ) = fx(τ ux +ℓ ) +∀τ ux +ℓ +∈ Ω ∪ ∂Ω +(57) +� +−ν ∂ωh +∂x + ωhuh +x + ∂P h +∂y +� +(τ uy +ℓ ) = fy(τ uy +ℓ ) +∀τ uy +ℓ +∈ Ω ∪ ∂Ω +(58) +� +∂uh +x +∂x + ∂uh +y +∂y +� +(τ p +ℓ) = 0 +∀τ p +ℓ ∈ Ω ∪ ∂Ω +(59) +� +ωh − ∂uh +y +∂x − ∂uh +x +∂y +� +(τ ω +ℓ ) = 0 +∀τ ω +ℓ ∈ Ω +(60) +� +ωh − ∂uh +y +∂x − ∂uh +x +∂y + Cpen +h (uh · s − g · s) +� +(τ ω +ℓ ) = 0 +∀τ ω +ℓ ∈ ∂Ω. +(61) +In the three field formulation we split the constitutive law into an expression for the +interior collocation points (Equation (60)) and another expression for boundary collocation +points (Equation (61)). +Resulting from these equations are nonlinear systems of equations which we can use to +solve for the unknown coefficients of velocity, pressure, and vorticity using a Newton-Raphson +method. +4.5. Proof of Divergence Conforming Property +From the commuting diagrams our spaces form, shown in Equations (30) and (31), it is +simple to show that both of the resulting collocation methods return an exactly pointwise +divergence free velocity field. The commuting diagrams reveal that the divergence of the +discrete velocity lies within the discrete pressure space, ∇·uh ∈ Qh. We can thus equivalently +write the divergence of the velocity as a linear combination of the pressure basis functions: +∇ · uh = +Np +� +i=0 +ciqi, +(62) +where qi ∈ Qh are the basis functions for the pressure and ci ∈ R are the coefficients. As +part of the collocation scheme, we strongly enforce that the velocity field has zero divergence +at a number of collocation points equal to the dimension of the discrete pressure space. This +condition can be written as a system of linear equations +Mc = 0, +(63) +where M is a square matrix whose entries are the pressure basis functions evaluated at each +collocation point and c is the vector of coefficients. +14 + +If the choice of collocation points yields a set of linearly independent equations, that is +to say M is invertible, then we know that the solution to Equation (63) is c = 0, and thus +the velocity field is exactly divergence free pointwise. +5. Numerical Results on Square Domains +The developed schemes are now tested on multiple 2D problems on the unit square. First, +a manufactured vortex problem is considered to experimentally compute the convergence +rates of the error and test for pressure and Reynolds number robustness. Then, the classical +lid-driven cavity problem is considered at a variety of Reynolds numbers. +5.1. Two-Dimensional Manufactured Solution +As a first numerical experiment, we consider a vortex problem on the unit square con- +structed using the method of manufactured solutions. This solution was originally developed +in [21] and employs the following velocity and pressure fields: +˜u = +� +2ex(−1 + x)2x2(y2 − y)(−1 + 2y) +(−ex(−1 + x)x(−2 + x(3 + x))(−1 + y)2y2) +� +, +(64) +˜p += +(−424 + 156e + (y2 − y)(−456 + ex(456 + x2(228 − 5(y2 − y))+ +2x(−228 + (y2 − y)) + 2x3(−36 + (y2 − y)) + x4(12 + (y2 − y))))). +(65) +For the velocity-pressure scheme we define the forcing term to be +f = −ν∆˜u + ˜u · ∇˜u + ∇˜p, +(66) +while for the vorticity-velocity-pressure formulation we define the forcing term in the mo- +mentum equations to be: +f = ν∇⊥˜ω + ˜ω × ˜u + ∇ ˜P, +(67) +with ˜ω = ∇ × ˜u and ˜P = ˜p + 1 +2(˜u · ˜u). Enforcing homogeneous boundary conditions and +requiring that the integral of pressure is zero, it is clear to see that the velocity and kinematic +pressure solutions should be equal to ˜u and ˜p. +To understand the accuracy of this collocation method, we test the convergence rates on +a variety of grids and with different degree B-spline bases. For this test we set the Reynolds +number to Re = 1 +ν = 1. We measure error using the L2 norm as well as the H1 semi-norm. +Figure 3 details the convergence rates of velocity and pressure when using the two field +formulation. In this case the errors in both velocity and pressure converge at a rate of k′ +when k′ is even and k′ − 1 for odd k′. These are the standard, expected rates that have been +seen in other studies of isogeometric collocation, and are one and two orders suboptimal in +L2 for odd and even k′. +Figure 4 details the convergence of velocity, kinematic pressure, and vorticity as we refine +the meshes in the three field scheme. Using this scheme, all of the unknowns converge in the +L2 norm at a rate of approximately k′ for even values of k′ and at a rate of k′ + 1 for odd +values of k′. These rates match the rates achieved using even k′ in the two field formulation, +15 + +10-2 +10-1 +10-10 +10-5 +(a) Velocity L2 error +10-2 +10-1 +10-10 +10-5 +(b) Velocity H1 error +10-2 +10-1 +10-10 +10-5 +(c) Pressure L2 error +10-2 +10-1 +10-10 +10-5 +100 +(d) Pressure H1 error +Figure 3: Errors in 2D manufactured vortex solution for velocity-pressure formulation +16 + +and these rates are two orders faster for odd k′. In fact, this formulation recovers optimal +convergence rates in the L2 norm for odd k′. +In the H1 semi-norm, we see convergence rates of k′ for all polynomial degrees for the +velocity and pressure. These rates are optimal in the H1 semi-norm for all degrees and again +are as fast or better than the corresponding velocity-pressure scheme results. Interestingly, +the H1 convergence of vorticity seems to be at a rate of k′ + 1 for odd k′ and a rate of k′ for +even values. +To further study our new collocation schemes, we can also directly compare the errors +produced with divergence-conforming Galerkin schemes of the same orders. Figure 5 shows +the L2 norm and H1 semi-norm errors in velocity as well as the L2 errors in pressure for both +collocation schemes along with the Galerkin results for the same problem from [22]. This +comparison highlights the severe suboptimality of the velocity-pressure results with odd k′. +We also note that the H1 errors obtained with the three field formulation nearly match the +Galerkin results. +5.2. Pressure Robustness +Next we perform some ancillary tests related to the manufactured solution to test some +secondary robustness properties of the method. The first test relates to so-called pressure +robustness [24]. +In particular, we take the kinematic pressure ˜p from the manufactured +solution and multiply it by a scalar σ. Thus the pressure term in the forcing function f +will also be multiplied by σ, and the exact solution to which our numerical solution should +converge has the same velocity as before but with a scaled kinematic pressure field. +For a pressure robust method this increase in the pressure magnitude, and thus the +pressure approximation errors, will not affect the velocity approximation error. Conversely, +a non-pressure robust method will see its velocity errors increase as the pressure is scaled +larger. Figure 6 shows the convergence of the velocity errors for the two field scheme with +k′ = 2 and increasing values of the scalar σ, while Figure 7 shows the same for the three field +formulation. Clearly the velocity error increases in both schemes as σ increases, meaning +the method is not pressure robust. This is interesting as the divergence-conforming Galerkin +method upon which this work is based is pressure robust. +5.3. Reynolds Robustness +Similar to pressure robustness, we also want to test how the errors in the solution behave +as the Reynolds number is increased. We increase the Reynolds number by decreasing the +viscosity ν. This affects the viscous term in the forcing vector f, but the exact solution to +the problem is identical to the original manufactured solution. +Figures 8 and 9 detail the convergence of the velocity errors as the Reynolds number +increases, again for k′ = 2, in the two and three field schemes. Once again, the error in the +velocity field increases as we increase the Reynolds number, in contrast to the divergence- +conforming Galerkin setting, where the velocity error is agnostic to increasing Reynolds +number [22]. +5.4. Two-Dimensional Lid-Driven Cavity Flow +The next 2D numerical test problem that we consider is the square lid-driven cavity flow. +The left, right, and bottom walls of the cavity remain fixed while the top wall slides in the +17 + +10-2 +10-1 +10-10 +10-5 +(a) Velocity L2 error +10-2 +10-1 +10-10 +10-5 +100 +(b) Velocity H1 error +10-2 +10-1 +10-10 +10-5 +100 +(c) Pressure L2 error +10-2 +10-1 +10-8 +10-6 +10-4 +10-2 +100 +(d) Pressure H1 error +10-2 +10-1 +10-10 +10-5 +(e) Vorticity L2 error +10-2 +10-1 +10-8 +10-6 +10-4 +10-2 +100 +(f) Vorticity H1 error +Figure 4: Errors in 2D manufactured vortex solution for vorticity-velocity-pressure formulation +18 + +10-2 +10-1 +10-10 +10-9 +10-8 +10-7 +10-6 +10-5 +10-4 +10-3 +10-2 +(a) Velocity L2 error +10-2 +10-1 +10-7 +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +(b) Velocity H1 error +10-2 +10-1 +10-9 +10-8 +10-7 +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +(c) Pressure L2 error +Figure 5: Errors in 2D manufactured vortex solution comparison +19 + +10-2 +10-1 +10-6 +10-4 +10-2 +(a) Velocity L2 error +10-2 +10-1 +10-5 +100 +(b) Velocity H1 error +Figure 6: Errors in 2D manufactured vortex solution with varying pressure scaling for velocity-pressure +formulation +10-2 +10-1 +10-6 +10-4 +10-2 +(a) Velocity L2 error +10-2 +10-1 +10-5 +100 +(b) Velocity H1 error +Figure 7: Errors in 2D manufactured vortex solution with varying pressure scaling for vorticity-velocity- +pressure formulation +20 + +10-2 +10-1 +10-6 +10-4 +10-2 +(a) Velocity L2 error +10-2 +10-1 +10-5 +100 +(b) Velocity H1 error +Figure 8: Errors in 2D manufactured vortex solution with varying Reynolds number for velocity-pressure +formulation +10-2 +10-1 +10-6 +10-4 +10-2 +(a) Velocity L2 error +10-2 +10-1 +10-5 +100 +(b) Velocity H1 error +Figure 9: Errors in 2D manufactured vortex solution with varying Reynolds number for vorticity-velocity- +pressure formulation +21 + +positive x direction, causing vortices to develop within the domain. Due to the inconsistency +in the boundary conditions, pressure singularities exist in the corners of the domain, making +this a challenging test case for a numerical scheme to properly capture. +For our simulations, we set both the speed of the top wall U = 1 and the wall lengths +H = 1. The kinematic viscosity ν defines the Reynolds number through Re = UH +ν += 1 +ν. In +particular, we consider the flows produced with Re = 100, Re = 400, and Re = 1000. To +validate our results, we compare the centerline velocity profiles at each Reynolds number +with the results from Ghia et al [42]. +Figure 10 details the two field formulation results across the three considered Reynolds +numbers and two mesh sizes: a 32 element stretched mesh and a 64 element stretched mesh. +The stretched mesh is formed by setting the interior knots of the knot vectors defining the +bases in each direction as +ξi = 1 +2 +� +1 + tanh(4ih − 2) +tanh(2) +� +∀ξi ∈ Ξ, +(68) +where h is the mesh size in each direction. Figure 11 shows the same results for the three field +formulation. The collocation results from both schemes agree very well with the reference +data in all cases, and we see that the results are converging with increasing resolution. At a +Reynolds number of 100, all of our results show that the maximum and minimum values of +the vertical velocity are larger in magnitude than those of Ghia et al. This is similar to the +behavior seen in the Galerkin method [22], and we note that there are some inaccuracies in +the Ghia data for this low Reynolds number case [22, 43]. For Reynolds number 400, the +two field formulation predicts extrema in velocity that are slightly smaller than the three +field predictions, which match the corresponding Galerkin results very well. This trend is +also valid at a Reynolds number of 1000. Moreover, while we have used stretched meshes +here, the results with a non-stretched mesh are similar. +As a more quantitative comparison, we compute the minimum horizontal velocity along +the vertical centerline as well as the maximum and minimum vertical velocities along the +horizontal centerline for each of simulations presented above. These results are shown for a +Reynolds number of 100 in Table 1, along with the values from [42] and [17]. These results +show the inadequacy of the Ghia results at this Reynolds number, and for the most part the +k′ = 2 collocation results outperform the Ghia data when compared to the pseudospectral +results. To highlight the potential possibilities of the collocation methods, we also compute +results using an unstretched mesh of 8 elements in each direction and k′ = 20 for both the +two and three field formulations. While this would be essentially infeasible with a Galerkin +method, as the quadrature would be prohibitively expensive, it is handled with ease by the +collocation schemes. We see that these results match the pseudospectral results, even on the +utilized coarse meshes. +6. Collocation Methods on Cubic Domains +The previous two sections detailed the construction of the divergence-conforming colloca- +tion methods in 2D and tested their behavior numerically. In the following, we will highlight +the required modifications to the methods to solve problems in 3D cubic domains. +22 + +0 +0.2 +0.4 +0.6 +0.8 +1 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +(a) Re = 100 velocities with h = 1/32 +0 +0.2 +0.4 +0.6 +0.8 +1 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +(b) Re = 100 velocities with h = 1/64 +0 +0.2 +0.4 +0.6 +0.8 +1 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +(c) Re = 400 velocities with h = 1/32 +0 +0.2 +0.4 +0.6 +0.8 +1 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +(d) Re = 400 velocities with h = 1/64 +0 +0.2 +0.4 +0.6 +0.8 +1 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +(e) Re = 1000 velocities with h = 1/32 +0 +0.2 +0.4 +0.6 +0.8 +1 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +(f) Re = 1000 velocities with h = 1/64 +Figure 10: Centerline velocity profiles for 2D lid-driven cavity with velocity-pressure formulation, k′ = 2. +Red curves and axes represent the vertical velocity along the horizontal centerline, while blue curves and +axes represent the horizontal velocity along the vertical centerline. +23 + +0 +0.2 +0.4 +0.6 +0.8 +1 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +(a) Re = 100 velocities with h = 1/32 +0 +0.2 +0.4 +0.6 +0.8 +1 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +(b) Re = 100 velocities with h = 1/64 +0 +0.2 +0.4 +0.6 +0.8 +1 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +(c) Re = 400 velocities with h = 1/32 +0 +0.2 +0.4 +0.6 +0.8 +1 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +(d) Re = 400 velocities with h = 1/64 +0 +0.2 +0.4 +0.6 +0.8 +1 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +(e) Re = 1000 velocities with h = 1/32 +0 +0.2 +0.4 +0.6 +0.8 +1 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +(f) Re = 1000 velocities with h = 1/64 +Figure 11: Centerline velocity profiles for 2D lid-driven cavity with vorticity-velocity-pressure formulation, +k′ = 2. Red curves and axes represent the vertical velocity along the horizontal centerline, while blue curves +and axes represent the horizontal velocity along the vertical centerline. +24 + +Table 1: Velocity extrema for 2D lid-driven cavity at Re = 100 +Method +ux,min +uy,max +uy,min +Collocation, 2 field formulation, k′ = 2 and h = 1/32 +−0.21348 +0.17941 +−0.25307 +Collocation, 2 field formulation, k′ = 2 and h = 1/64 +−0.21389 +0.17953 +−0.25358 +Collocation, 2 field formulation, k′ = 20 and h = 1/8 +−0.21404 +0.17957 +−0.25380 +Collocation, 3 field formulation, k′ = 2 and h = 1/32 +−0.21800 +0.18392 +−0.25908 +Collocation, 3 field formulation, k′ = 2 and h = 1/64 +−0.21511 +0.18075 +−0.25521 +Collocation, 3 field formulation, k′ = 20 and h = 1/8 +−0.21404 +0.17957 +−0.25380 +Pseudospectral (Ref. [43]) +−0.21404 +0.17957 +−0.25380 +Ghia et al. (Ref. [42]) +−0.21090 +0.17527 +−0.24533 +6.1. Review of Galerkin Methods +Similar to 2D, we start by reviewing the form of the divergence-conforming isogeometric +Galerkin methods for 3D problems. Again assume the velocity is subject to Dirichlet bound- +ary conditions along the entire boundary. We then define the discrete test and trial function +spaces for velocity as Vh,0 and Vh,g, which are defined as the same Vh from Equation (37) +with either no penetration boundary conditions strongly enforced (for the test space Vh,0) +or with the normal velocity prescribed at collocation points as given by the boundary data g +(for the trial space Vh,g). We also define the test and trial space for pressure as Qh,0, where +Qh is the same space as in Equation (38) but with the added condition that the pressure +must have zero integral. Then the Galerkin formulation for the velocity-pressure form would +read +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Given ν ∈ R+, f : Ω → R3, and g : ∂Ω → R3, find uh ∈ Vh,g and ph ∈ Qh,0 such that, +∀(wh, qh) ∈ (Vh,0, Qh,0): +� +Ω +(ν∇wh · ∇uh + wh · (uh · ∇uh) − ph∇ · wh)dΩ +− ν +� +∂Ω +(∇uh · n) · wh − Cpen +h uh · whdA = +� +Ω +wh · fdΩ + ν +� +∂Ω +Cpen +h g · whdA +(69) +� +Ω +qh(∇ · uh)dΩ = 0. +(70) +This weak form is essentially unchanged from the 2D case, with the only major difference +being that the velocity has 3 components. The vorticity-velocity-pressure Galerkin form, +however, is more different. In this case the discrete problem reads +25 + +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Given ν ∈ R+, f : Ω → R3, and g : ∂Ω → R3, find uh ∈ Vh,g, P h ∈ Qh,0, and +ωh ∈ Ψh such that, ∀(wh, qh, ψh) ∈ (Vh,0, Qh,0, Ψh): +� +Ω +(ν∇ × ωh) · vhdΩ + +� +Ω +(ωh × uh) · vhdΩ − +� +Ω +P h(∇ · vh)dΩ = +� +Ω +f · vhdΩ +(71) +� +Ω +(∇ · uh)qhdΩ = 0 +(72) +� +Ω +(ωh · ψh)dΩ + +� +Ω +uh · (∇ × ψh)dΩ − +� +∂Ω +(ψh × g) · ndA = 0. +(73) +Again the no-slip velocity boundary conditions appear as natural boundary conditions in +the weak form of the constitutive equation. +Within the collocation schemes, the unknowns are selected to reside in the same spaces as +the corresponding Galerkin scheme, as in 2D. In the following we highlight the main changes +to the method for 3D problems with regards to the choice of collocation grids and boundary +condition enforcement before again summarizing the final form of the discrete equations. +6.2. Collocation Grids +Much like the two dimensional case, in 3D we choose to collocate at Greville abscissae +and the grids are different for each of the governing equations. For both formulations the +schemes for the momentum and pressure equations are essentially unchanged; each momen- +tum equation component is collocated at the Greville abscissae of the corresponding discrete +velocity component space, and the continuity equation is collocated at the Greville abscissae +of the discrete pressure space. Thus the velocity-pressure formulation extends fairly trivially +to 3D. +The constitutive equation in the vorticity-velocity-pressure formulation, on the other +hand, is now split into components much like how the momentum equations are treated. +We choose to collocate the x component of the constitutive equation at the Greville ab- +scissae associated with the discrete x vorticity space (Sk1−1,k2,k3 +α1−1,α2,α3), the y component of the +constitutive equation at the Greville abscissae associated with the discrete y vorticity space +(Sk1,k2−1,k3 +α1,α2−1,α3), and the z component of the constitutive equation at the Greville abscissae +associated with the discrete z vorticity space (Sk1,k2,k3−1 +α1,α2,α3−1). +6.3. Boundary Condition Enforcement +The no-penetration boundary condition is enforced identically to 2D case: We strongly +enforce the normal velocity on face collocation points corresponding to the normal velocity +component and remove these points from the set used to collocate the momentum equations. +The no-slip boundary condition in the velocity-pressure scheme is also essentially enforced +identically to the 2D case and again leads to equations of the form +26 + +− ν∆uh + uh · ∇uh + ∇ph + C2 +pen +h2 (uh − g) = f. +(74) +As the constitutive law relating velocity and vorticity is a vector relation in 3D, the weak +enforcement of no-slip boundary conditions is slightly altered in the three field formulation. +We again start by considering the weak form shown above, particularly Equation (73). The +last term in this equation represents the boundary term which would be used to enforce +boundary conditions by replacing terms with their prescribed values. Following the Enhanced +Collocation method of [41], the equation can be integrated by parts once again, to arrive at +a strong form representation given by: +� +Ω +ψ · (ω − ∇ × u)dΩ + +� +∂Ω +(ψ × u − ψ × g) · ndA = 0. +Using the properties of the scalar triple product, we can re-write this as: +� +Ω +ψ · (ω − ∇ × u)dΩ + +� +∂Ω +(u × n − g × n) · ψdA = 0. +By approximating these integrals as is done in [41, 38], we arrive at a strong form state- +ment including boundary conditions suitable for collocation: +ωh − ∇ × uh + Cpen +h (uh × n − g × n) = 0. +(75) +6.4. Final Collocated Equations +Once again the entire collocation scheme based on the velocity-pressure formulation is +summarized first. Let us again define τ ux +ℓ +for ℓ = 1, ..., M ux to be the set of Greville points +for the basis of the x velocity component (Sk1,k2−1,k3−1 +α1,α2−1,α3−1) with the points corresponding to +no-penetration boundaries removed as discussed previously. Define in a similar manner τ uy +ℓ +for ℓ = 1, ..., M uy and τ uz +ℓ +for ℓ = 1, ..., M uz. The pressure Greville points are defined as τ p +ℓ +for ℓ = 1, ..., N p. For this formulation the discrete 3D problem reads: +27 + +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Find uh ∈ Vh,g and P h ∈ Qh,0 such that: +� +−ν ∂2uh +x +∂x2 − ν ∂2uh +x +∂y2 − ν ∂2uh +x +∂z2 + uh +x +∂uh +x +∂x + uh +y +∂uh +x +∂y + uh +z +∂uh +x +∂z + ∂ph +∂x +� +(τ ux +ℓ ) += fx(τ ux +ℓ ) +∀τ ux +ℓ +∈ Ω +(76) +� +−ν ∂2uh +x +∂x2 − ν ∂2uh +x +∂y2 − ν ∂2uh +x +∂z2 + uh +x +∂uh +x +∂x + uh +y +∂uh +x +∂y + uh +z +∂uh +x +∂z + ∂ph +∂x + +C2 +pen +h2 (uh +x − gx) +� +(τ ux +ℓ ) = fx(τ ux +ℓ ) +∀τ ux +ℓ +∈ ∂Ω +(77) +� +−ν ∂2uh +y +∂x2 − ν ∂2uh +y +∂y2 − ν ∂2uh +y +∂z2 + uh +x +∂uh +y +∂x + uh +y +∂uh +y +∂y + uh +z +∂uh +y +∂z + ∂ph +∂y +� +(τ uy +ℓ ) += fy(τ uy +ℓ ) +∀τ uy +ℓ +∈ Ω +(78) +� +−ν ∂2uh +y +∂x2 − ν ∂2uh +y +∂y2 − ν ∂2uh +y +∂z2 + uh +x +∂uh +y +∂x + uh +y +∂uh +y +∂y + uh +z +∂uh +y +∂z + ∂ph +∂y + +C2 +pen +h2 (uh +y − gy) +� +(τ uy +ℓ ) = fy(τ uy +ℓ ) +∀τ uy +ℓ +∈ ∂Ω +(79) +� +−ν ∂2uh +z +∂x2 − ν ∂2uh +z +∂y2 − ν ∂2uh +z +∂z2 + uh +x +∂uh +z +∂x + uh +y +∂uh +z +∂y + uh +z +∂uh +z +∂z + ∂ph +∂z +� +(τ uz +ℓ ) += fz(τ uz +ℓ ) +∀τ uz +ℓ ∈ Ω +(80) +� +−ν ∂2uh +z +∂x2 − ν ∂2uh +z +∂y2 − ν ∂2uh +z +∂z2 + uh +x +∂uh +z +∂x + uh +y +∂uh +z +∂y + uh +z +∂uh +z +∂z + ∂ph +∂z + +C2 +pen +h2 (uh +z − gz) +� +(τ uz +ℓ ) = fz(τ uz +ℓ ) +∀τ uz +ℓ ∈ ∂Ω +(81) +� +∂uh +x +∂x + ∂uh +y +∂y + ∂uh +z +∂z +� +(τ p +ℓ) = 0 +∀τ p +ℓ ∈ Ω ∪ ∂Ω. +(82) +Similarly to the velocity, in the three field formulation we also define collocation points +for the vorticity component-wise. In particular, let τ ωx +ℓ +for ℓ = 1, ..., N ωx be the Greville +points for the x component of the vorticity, and define τ ωy +ℓ +for ℓ = 1, ..., N ωy and τ ωz +ℓ +for +ℓ = 1, ..., N ωz similarly. The final, discrete 3D problem for the vorticity-velocity-pressure +collocation scheme reads as: +28 + +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Find uh ∈ Vh,g, P h ∈ Qh,0, and ωh ∈ Ψh such that: +� +ν(∂ωh +z +∂y − ∂ωh +y +∂z ) + ωh +yuh +z − ωh +z uh +y + ∂P h +∂x +� +(τ ux +ℓ ) = fx(τ ux +ℓ ) +∀τ ux +ℓ +∈ Ω ∪ ∂Ω +(83) +� +ν(∂ωh +x +∂z − ∂ωh +z +∂x ) + ωh +z uh +x − ωh +xuh +z + ∂P h +∂y +� +(τ uy +ℓ ) = fy(τ uy +ℓ ) +∀τ uy +ℓ +∈ Ω ∪ ∂Ω +(84) +� +ν(∂ωh +y +∂x − ∂ωh +x +∂y + ωh +xuh +y − ωh +yuh +x + ∂P h +∂z +� +(τ uz +ℓ ) = fz(τ uz +ℓ ) +∀τ uz +ℓ ∈ Ω ∪ ∂Ω +(85) +� +∂uh +x +∂x + ∂uh +y +∂y + ∂uh +z +∂z +� +(τ p +ℓ) = 0 +∀τ p +ℓ ∈ Ω ∪ ∂Ω +(86) +� +ωh +x − (∂uh +z +∂y − ∂uh +y +∂z ) +� +(τ ωx +ℓ ) = 0 +∀τ ωx +ℓ +∈ Ω +(87) +� +ωh +x − (∂uh +z +∂y − ∂uh +y +∂z )+ +Cpen +h ((uh +y − gy)nz − (uh +z − gz)ny) +� +(τ ωx +ℓ ) = 0 +∀τ ωx +ℓ +∈ ∂Ω +(88) +� +ωh +y − (∂uh +x +∂z − ∂uh +z +∂x ) +� +(τ ωy +ℓ ) = 0 +∀τ ωy +ℓ +∈ Ω +(89) +� +ωh +y − (∂uh +x +∂z − ∂uh +z +∂x )+ +Cpen +h ((uh +z − gz)nx − (uh +x − gx)nz) +� +(τ ωy +ℓ ) = 0 +∀τ ωy +ℓ +∈ ∂Ω +(90) +� +ωh +z − (∂uh +y +∂x − ∂uh +x +∂y ) +� +(τ ωz +ℓ ) = 0 +∀τ ωz +ℓ +∈ Ω +(91) +� +ωh +z − (∂uh +y +∂x − ∂uh +x +∂y )+ +Cpen +h ((uh +x − gx)ny − (uh +y − gy)nx) +� +(τ ωz +ℓ ) = 0 +∀τ ωz +ℓ +∈ ∂Ω. +(92) +7. Numerical Results on Cubic Domains +To verify that the schemes properly extend into 3D, two sample problems are considered. +First, a manufactured solution gives even more insight into the convergence properties of +29 + +the methods. Then the three-dimensional lid-driven cavity problem is considered and the +results are compared with established literature. +7.1. Three-Dimensional Manufactured Solution +In 3D, we also start our numerical studies by considering a manufactured solution. In +this case, the exact solution represents the flow around a single vortex filament within the +unit cube. We define a potential function as +˜φ = +� +� +x(x − 1)y2(y − 1)2z2(z − 1)2 +0 +x2(x − 1)2y2(y − 1)2z(z − 1) +� +� , +(93) +through which we can define the velocity field as +˜u = ∇ × ˜φ, +(94) +and the vorticity as +˜ω = ∇ × ˜u. +(95) +Finally, we specify the pressure field as +˜p = sin(πx) sin(πy) − 4 +π2. +(96) +For the velocity-pressure scheme we define the forcing term on the right hand sign of the +momentum equations as +f = −ν∆˜u + ˜u · ∇˜u + ∇˜p, +(97) +while for the vorticity-velocity-pressure scheme the forcing term is given by +f = −ν∆˜u + ˜ω × ˜u + ∇ ˜P. +(98) +Once again we enforce homogeneous Dirichlet boundary conditions everywhere and require +that the kinematic pressure field has zero average. With these conditions the discrete solution +should again converge to the quantities above with mesh refinement. +Similar to the 2D case, we set Re = 1 +ν = 1 and measure the errors produced on a variety of +grids in the L2 norm and H1 semi-norm. Figure 12 shows the results for the velocity-pressure +scheme while Figure 13 details the errors for the vorticity-velocity-pressure scheme. +We start by noting that when k′ < 3 in both cases, everything behaves in the same +manner as in the 2D setting. Once k′ ≥ 3 we start to see very fast convergence rates and +some pre-asymptotic type behavior in the velocity errors produced by both schemes. This +can be explained by talking a closer look at the exact velocity field for this problem. In fact, +the exact velocity field is given by a quartic polynomial in each direction and this solution +is actually contained within the discrete velocity approximation space for k′ ≥ 3. If we +were using a pressure robust Galerkin method, the velocity error would be zero. Since the +collocation scheme is not pressure robust, we obtain superconvergence rather than exactly +zero error. +30 + +10-1 +10-10 +10-5 +(a) Velocity L2 error +10-1 +10-10 +10-5 +(b) Velocity H1 error +10-1 +10-8 +10-6 +10-4 +10-2 +(c) Pressure L2 error +10-1 +10-6 +10-4 +10-2 +100 +(d) Pressure H1 error +Figure 12: Errors in 3D manufactured vortex solution for velocity-pressure formulation +31 + +10-1 +10-10 +10-5 +(a) Velocity L2 error +10-1 +10-10 +10-5 +100 +(b) Velocity H1 error +10-1 +10-8 +10-6 +10-4 +10-2 +100 +(c) Pressure L2 error +10-1 +10-6 +10-4 +10-2 +100 +(d) Pressure H1 error +10-1 +10-10 +10-5 +100 +(e) Vorticity L2 error +10-1 +10-10 +10-5 +100 +(f) Vorticity H1 error +Figure 13: Errors in 3D manufactured vortex solution for vorticity-velocity-pressure formulation +32 + +0 +0.2 +0.4 +0.6 +0.8 +1 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +(a) Velocity-pressure formulation +0 +0.2 +0.4 +0.6 +0.8 +1 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +(b) Vorticity-velocity-pressure formulation +Figure 14: Centerline velocity profile for 3D lid-driven cavity using both formulations, k′ = 2. Red curves +and axes represent the vertical velocity along the horizontal centerline, while blue curves and axes represent +the horizontal velocity along the vertical centerline. +The pressure convergence results also show some interesting behavior. While the vorticity- +velocity-pressure scheme seems to behave in the same manner as in 2D, the velocity-pressure +scheme seems to be recovering the faster rates seen in the three field scheme. We believe +that this is a consequence of the superconvergence of velocity. +7.2. Three-Dimensional Lid-Driven Cavity +The next numerical study that we perform is on the 3D lid-driven cavity flow. Consider +again the cavity setup describing the 2D flow, but now extend the square cavity by unit +length in the out-of-page direction, thus making it a cube. The point singularities of the 2D +case now extend along the top edges of the cube and we expect to see more influence of 3D +boundary effects [44]. +In our tests we again set the wall speed U = 1, the side length H = 1, and consider +Re = UH +1 += 100. We use an unstretched mesh with 32 elements per side and k′ = 2, and +compare the x velocity along the vertical centerline and the y velocity along the horizontal +centerline with the pseudospectral results from [44]. Figure 14 shows the results with each +formulation. Once again the results match very well with the literature, and it seems as +though the results from the three field formulation match with the reference results slightly +better that the two field results. +8. Collocation Methods on Mapped Domains +As the last main component of this paper we shift our focus to problems posed on more +complicated domains. We will present some theory for both 2D and 3D problems, but for +simplicity we will focus the development of numerical schemes for the 2D, linear Stokes +equations. However, the results would generalize to the nonlinear, 3D setting as well. We +will also focus on the rotational form of the equations, as the first order nature enables easier +mappings between domains. +33 + +The main idea of this section is the mapping back to a parametric reference domain, +i.e. a square in 2D or a cube in 3D. The previous sections detail how to develop collocation +schemes on these simple geometries, thus simply pulling the equations and unknowns back to +the reference domain, collocating as before, and pushing the results forward to the physical +domain gives our numerical solution. +Let ˆΩ be the parametric domain (the unit square in 2D or the unit cube in 3D), and let +Ω be the physical domain. We define the function F as mapping from ˆΩ to Ω. Let DF be +the Jacobian of the parametric mapping, and define +J = Det(DF), +(99) +C = (DF)T(DF). +(100) +Next we can define the pull-back operators in 3D as +ιΦ(φ) = (φ ◦ F), +(101) +ιω(ψ) = (DF)T(ψ ◦ F), +(102) +ιu(v) = J(DF)−1(v ◦ F), +(103) +ιp(q) = J(q ◦ F). +(104) +We define the pulled-back unknowns on the reference domain via the ι maps, specifically +ˆu = ιu(u), ˆp = ιp(p), and ˆω = ιω(ω). These are the unknowns for which we solve us- +ing collocation, and the physical domain solution is then obtained via the corresponding +push-forward. Importantly, the push-forward of velocity as defined above maps divergences +to divergences and preserves nullity of normal components. Similarly the push-forward of +pressure preserves the nullity of the integral operator. These facts imply that the following +commuting diagram exists: +R −−−→ Φ +∇ +−−−→ Ψ +∇× +−−−→ V +∇· +−−−→ Q −−−→ 0 +���ιΦ +���ιω +���ιu +���ιp +R −−−→ Φ +∇ +−−−→ ˆΨ +∇× +−−−→ ˆV +∇· +−−−→ ˆQ −−−→ 0, +(105) +where now the hat spaces correspond to the ones defined over the parametric domain, and +are identical to the ones used in the previous sections of this paper. Moreover, by composing +the ι maps with the projectors from the de Rham complex in the square domain setting, we +arrive at a new commuting diagram between the physical domain continuous spaces and the +discrete spaces in the physical domain defined by the push-forward of the discrete spaces +chosen for the unit square. +For completeness we also define the 2D pull-back operators +ιω(ψ) = ψ ◦ F, +(106) +ιu(v) = J(DF)−1(v ◦ F), +(107) +ιp(q) = J(q ◦ F). +(108) +34 + +In 2D a commuting diagram also exists: +R −−−→ Ψ +∇⊥ +−−−→ V +∇· +−−−→ Q −−−→ 0 +���ιω +���ιu +���ιp +R −−−→ ˆΨ +∇⊥ +−−−→ ˆV +∇· +−−−→ ˆQ −−−→ 0. +(109) +Next we begin the process of mapping the governing equations back to the reference +domain. We start with Equations (6) - (8) for the rotational form of the 3D Navier-Stokes +equations. The Stokes equations are recovered by simply removing the nonlinear term in the +momentum equation, Equation (6), and noting now that the pressure becomes the standard +kinematic pressure p. In the momentum equation, the viscous term is mapped back to the +reference domain via +(∇ × ω) ◦ F = J−1(DF)( ˆ∇ × ιω(ω)) = J−1(DF)( ˆ∇ × ˆω), +(110) +and the pressure term is mapped to +(∇p) ◦ F = (DF)−T ˆ∇(ιΦ(p)) = (DF)−T ˆ∇(J−1ιp(p)) = (DF)−T ˆ∇(J−1ˆp). +(111) +Within the continuity equation, Equation (7), the divergence is mapped via +(∇ · u) ◦ F = J−1 ˆ∇ · (ιu(u)) = J−1 ˆ∇ · ˆu. +(112) +Finally, in the constitutive law, Equation (8), the curl term is mapped similarly to the +viscous momentum term +(∇ × u) ◦ F = J−1(DF)( ˆ∇ × ιω(u)) = J−1(DF)( ˆ∇ × ((DF)Tu)) += J−1(DF)( ˆ∇ × ((DF)T(J−1(DF)ˆu))) += J−1(DF)( ˆ∇ × (J−1Cˆu)). +(113) +Now we pull each equation back to the reference domain via the corresponding ι map, +so the momentum equations are pulled back via ιu, the continuity equation is pulled back +with ιp and the constitutive law is pulled back with ιω. For brevity, we will not state the +full form of the mapped equations in 3D, but instead state just the 2D form. This arises in +a similar way as the 2D rotational form of the Navier-Stokes equations was generated from +the 3D equations. In particular we can simply write the equations out component-wise and +note that z velocities as well as derivatives in the z direction are zero. This yields: +35 + +(a) Before strong enforcement of no pene- +tration conditions +(b) After strong enforcement of no penetra- +tion conditions +Figure 15: Example of collocation grid on a mapped domain for vorticity-velocity-pressure scheme +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Given ν ∈ R+, ˆf : ˆΩ → R2, and ˆg : ∂ ˆΩ → R2, find ˆu : ˆΩ → R2, ˆp : ˆΩ → R, and +ˆω : ˆΩ → R such that: +ν ∂ˆω +∂ˆy + JC−1 +11 +∂(J−1ˆp) +∂ˆx ++ JC−1 +12 +∂(J−1ˆp) +∂ˆy += ˆf1 +in +ˆΩ +(114) +− ν ∂ˆω +∂ˆx + JC−1 +21 +∂(J−1ˆp) +∂ˆx ++ JC−1 +22 +∂(J−1ˆp) +∂ˆy += ˆf2 +in +ˆΩ +(115) +ˆ∇ · ˆu = 0 +in +ˆΩ +(116) +ˆω − J−1( ∂ +∂ˆx(J−1(C21ˆux + C22ˆuy)) − ∂ +∂ˆy(J−1(C11ˆux + C12ˆuy))) = 0 +in +ˆΩ +(117) +ˆu = ˆg +on +∂ ˆΩ, +(118) +where ˆf = ιu(f) and ˆg = ιu(g). +We collocate these equations in the same manner as in the previous sections to solve +for the parametric domain variables ˆu, ˆp, and ˆω. The collocation points are chosen as the +Greville abscissae in the parametric domain, and an example of the resulting points pushed +forward into the physical domain is shown in Figure 15. No penetration boundary conditions +are enforced strongly and no slip boundary conditions are enforced weakly with a suitable +penalty term. For brevity we omit the full statement of the discrete problem and simply +note that it leads to a linear system of equations (as we are focused in this section on Stokes +flow). +36 + +9. Numerical Results on Mapped Domains +In this penultimate section we verify the performance of the vorticity-velocity-pressure +collocation scheme on non-square domains. We first consider linear Couette flow to confirm +that the expected convergence rates are maintained and then move on to modified lid-driven +cavity flows in non-square setups. +9.1. Cylindrical Couette Flow +The first problem posed on a mapped domain that we consider is Couette flow. This +models the behavior of a fluid between 2 concentric cylinders, with the outer fixed and the +inner rotating at a constant rate. We solve the problem over a quarter circle domain as shown +in Figure 15, enforcing homogeneous Dirichlet boundary conditions on the outer cylindrical +wall, zero normal and unit tangential velocity on the inner cylindrical wall, and zero pressure +gradient on the horizontal and vertical boundaries. The last Neumann boundary condition +is enforced using the Enhanced Collocation approach [41]. +The exact velocity field is given in polar coordinates as: +¯u = +� (Ar + B/r) sin θ +(Ar + B/r) sin θ +� +, +(119) +with A = −Ωin +δ2 +1−δ2, B = Ωin +r2 +in +1−δ2, Ωin = +U +rin, δ = +rin +rout, rin = 1 is the radius of the inner +cylinder, rout = 2 is the radius of the outer cylinder, and the velocity of the inner cylinder +has magnitude U = 1. The exact pressure field is zero everywhere, and the exact vorticity +is a constant equal to 2A. We use a polar mapping to map between the parametric and +physical domains: +F(ξ1, ξ2) = +� ((rout − rin)ξ2 + rin) sin(2πξ1) +((rout − rin)ξ2 + rin) cos(2πξ1) +� +. +(120) +In solving this problem with this mapping one can show analytically that the collocation +approximation to the exact solution ˆu is a function of ˆy only, the collocation approximation +to ˆv is zero, the collocation approximation to ˆp is zero, and the collocation approximation +to ˆω is a constant. However, we assemble and solve the full linear system without utilizing +this structure. +Figure 16 shows the errors in the solution as a function of resolution. For the L2 norm +and H1 semi-norm errors of velocity we recover the same rates are in the square domain +setting. The collocation scheme also captures the zero pressure up to finite precision on the +coarsest mesh as both the L2 and H1 errors are essentially zero. As the mesh is refined we see +this error increase, which we attribute to worsening matrix conditioning and roundoff error +effects. We also see the same rates as in square domains for the L2 convergence of vorticity. +Note that a constant vorticity is also recovered even on the coarsest mesh, as evidenced by +the numerically zero H1 semi-norm error. Like the pressure errors the H1 error grows with +mesh refinement, and we believe the explanation is the same. +37 + +10-2 +10-1 +10-10 +10-8 +10-6 +10-4 +10-2 +(a) Velocity L2 error +10-2 +10-1 +10-10 +10-5 +100 +(b) Velocity H1 error +10-2 +10-1 +4 +6 +8 +10 +12 +14 +16 +10-16 +(c) Pressure L2 error +10-2 +10-1 +10-15 +10-14 +10-13 +(d) Pressure H1 error +10-2 +10-1 +10-10 +10-5 +(e) Vorticity L2 error +10-2 +10-1 +10-15 +10-14 +10-13 +(f) Vorticity H1 error +Figure 16: Errors in Couette flow solution for vorticity-velocity-pressure formulation +38 + +x +y +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +x +y +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.1 +0.2 +x +y +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.1 +0.2 +Lid-Driven Cavity Flow Over a Wavy Wall +Figure 17: Mapped Stokes results for lid-driven cavity with varying numbers of bumps +9.2. Lid-Driven Cavity Over Wavy Wall +Our final numerical test case concerns the Stokes flow in a 2D lid-driven cavity, similar +to the square domain examples, but now with a non-flat bottom surface of the cavity. In +particular, the mapping from parametric to physical domain is given by +F(ξ1, ξ2) = +� +ξ1 +A(B(1 − ξ2) sin(Cπξ1) + ξ2) +� +, +(121) +where A, B, and C are constants which control the shape of the domain. We use three +combinations in this paper, in particular A = 1, B = 0.75, and C = 1 gives a domain with +one bump, A = 0.25, B = 0.3, and C = 3 gives a domain with two bumps, and A = 0.25, +B = 0.3, and C = 5 gives a domain with three bumps. +Figure 17 shows the streamfunctions obtained with 64 elements and k′ = 2. Clearly we +are able to recover symmetric fields in all cases which are appropriate for Stokes flow. +39 + +10. Conclusions +In this paper, two divergence-conforming collocation methodologies have been presented +for solution of the steady, incompressible Navier-Stokes equations using a velocity-pressure +formulation and a vorticity-velocity-pressure formulation. +By employing B-spline spaces +that conform to the de Rham complex, these methods produce velocity fields which are +exactly pointwise divergence free. Moreover, by the nature of collocation methods, these +methods are much less computationally expensive than traditional Galerkin finite element +formulations as no costly numerical integrations are required. By applying the discretizations +to benchmark problems in two and three dimensions we have shown that the methods retain +a high order of accuracy. Moreover, we have seen that by re-writing the equations in the +vorticity-velocity-pressure form many convergence rates are improved compared to those +obtained with a velocity-pressure scheme. However, useful properties of the corresponding +divergence-conforming B-spline Galerkin method, such as pressure and Reynolds robustness, +are not maintained in these collocation schemes. Finally, methods for problems posed in +more complicated domains were created by mapping unknowns and equations between the +physical and reference domains using structure-preserving transformations. +There are many interesting directions for future work. Collocation schemes that do retain +pressure and Reynolds robustness properties would be useful, as would developing a strategy +for stabilization of these types of collocation schemes in advection-dominated flow regimes. +The schemes proposed in this paper could also be extended to the multi-patch setting to +allow for simulations posed on even more complicated domains. The use of locally adaptive +splines would also aid in maximizing the ratio of accuracy to cost in which collocation already +excels. Finally, while collocation improves upon the cost of numerical integration, unsteady, +incompressible Navier-Stokes solution strategies will still likely involve the solution of linear +systems during each time step, and thus reducing cost of linear system solution is also very +important. +Acknowledgements +This material is based upon work supported by the National Science Foundation Graduate +Research Fellowship Program under Grant No. DGE-1656518. Any opinions, findings, and +conclusions or recommendations expressed in this material are those of the authors and do +not necessarily reflect the views of the National Science Foundation. +References +[1] T. J. Hughes, J. A. Cottrell, Y. Bazilevs, Isogeometric analysis: CAD, finite elements, +NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechan- +ics and Engineering 194 (39-41) (2005) 4135–4195. +[2] J. A. Cottrell, T. J. Hughes, Y. 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Taylor, A pseudospectral method for solution of the three- +dimensional incompressible Navier-Stokes equations, Journal of Computational Physics +70 (2) (1987) 439–462. +44 + diff --git a/BtE2T4oBgHgl3EQfRwcX/content/tmp_files/load_file.txt b/BtE2T4oBgHgl3EQfRwcX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a4a906a43a727024fc80c4a0dc9c598b6ca162f9 --- /dev/null +++ b/BtE2T4oBgHgl3EQfRwcX/content/tmp_files/load_file.txt @@ -0,0 +1,1865 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf,len=1864 +page_content='Divergence-Conforming Isogeometric Collocation Methods for the Incompressible Navier-Stokes Equations Ryan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Aronsona,∗, John A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Evansb aStanford University, 94305, Stanford, CA, USA bUniversity of Colorado Boulder, 80309, Boulder, CO, USA Abstract We develop two isogeometric divergence-conforming collocation schemes for incompress- ible flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The first is based on the standard, velocity-pressure formulation of the Navier- Stokes equations, while the second is based on the rotational form and includes the vorticity as an unknown in addition to the velocity and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We describe the process of discretiz- ing each unknown using B-splines that conform to a discrete de Rham complex and collocat- ing each governing equation at the Greville abcissae corresponding to each discrete space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Results on complex domains are obtained by mapping the equations back to a parametric do- main using structure-preserving transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Numerical results show the promise of the method, including accelerated convergence rates of the three field, vorticity-velocity-pressure scheme when compared to the two field, velocity-pressure scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Keywords: Isogeometric analysis, Collocation, Incompressible flow, Divergence-conforming discretizations, Velocity-pressure formulation, Vorticity-velocity-pressure formulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Introduction Isogeometric Analysis (IGA) is a technology [1, 2] which replaces the standard polynomial basis functions used in traditional Finite Element Analysis (FEA) with B-splines, Non- Uniform Rational B-splines (NURBS), and other classes of splines in an aim to reduce the gap between geometry and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' IGA has a distinct advantage over traditional FEA due to its ability to exactly represent the geometries commonly seen in Computer-Aided Design (CAD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Moreover, the basis functions used in IGA are globally more smooth than those of FEA and it has been shown that IGA can exhibit improved accuracy and robustness over FEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' For example, higher continuity splines are shown to have optimal approximating power in the sense of Kolmogorov n-widths [3] and spline discretizations have more favorable dissipation and dispersion properties than standard, high-order FEA discretizations [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' To improve on the complexity and implementation details of IGA, the feasibility of isoge- ometric collocation methods has been explored [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In Galerkin IGA methods, the discrete system of equations is formed by integrating the PDE residual against the test function space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' ∗Corresponding author Email address: rmaronso@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='edu (Ryan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Aronson) Preprint submitted to Elsevier January 11, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='03783v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='NA] 10 Jan 2023 This requires numerical integration, which renders system assembly quite expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Collo- cation, on the other hand, forms the discrete system by simply requiring that the residual of the governing equations vanish at a set of discrete locations in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Many recent studies have shown the efficacy of isogeometric collocation methods in both static and dynamic solid mechanics problems [6, 7, 8], and detailed comparisons between iso- geometric Galerkin and collocation methods have been made [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In addition, recent studies have also investigated the use of mixed collocation methods for use in nearly incompressible elasticity [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Isogeometric collocation has even been used for acoustic problems [12], computing Karhunen-Loeve expansions [13, 14], and introduced into physics-informed neural networks [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Finally, a method was recently introduced for IGA collocation in immersed domains by combining with the finite cell method near the boundaries [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' These results all suggest that isogeometric collocation methods retain some of the improved qualities of standard IGA, while reducing the computational cost and improving sparsity of the discrete systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Isogeometric collocation methods have not been as well explored in the context of fluid mechanics, though the idea of using B-spline collocation to solve incompressible fluid me- chanics problems has been investigated in the past [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In addition, spline collocation has been employed in fundamental Direct Numerical Simulation (DNS) studies of turbulent flows [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' However, the methods previously introduced are typically limited to simple ge- ometries and are not divergence-conforming, meaning that the discrete velocity field does not exactly satisfy the continuity equation in incompressible flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In addition, a mixed isogeometric collocation method has recently been proposed for use in poromechanics [20], though the preliminary results were limited to one-dimensional problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In the context of incompressible fluid mechanics, divergence-conforming Galerkin meth- ods based on B-spline basis functions have been developed for both the Stokes and the Navier-Stokes equations [21, 22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' These methods are provably inf-sup stable and yield discrete velocity fields that are exactly pointwise divergence free, among other desirable qual- ities such as pressure robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' An excellent summary of divergence-conforming methods is given by [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' These discretizations have prospered in areas such as turbulent flow simula- tion [25, 26, 27] and fluid-structure interaction [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Moreover, efficient multigrid solvers have been developed based on these discretizations [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Divergence-conforming Galerkin meth- ods have also been developed for more advanced spline discretizations, such as hierarchical B-splines [30] and LR B-splines [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In this paper we develop similar divergence-conforming methodologies for incompressible flow using collocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In particular, we introduce two collocation methods, one based on the standard velocity-pressure form of the steady Navier-Stokes equations, and one based on a three field (velocity-vorticity-pressure) form of the steady Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The latter form of the Navier-Stokes equations has recently been used to develop alternative structure-preserving finite element discretizations [32, 33, 34] and we find that collocation methods based on the resulting system of first order differential-algebraic equations returns improved convergence rates compared to the rates obtained using collocation in conjunction with the standard velocity-pressure form of the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In our collocation schemes, each unknown is discretized with compatible B-spline spaces that preserve the structure of the governing equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Both collocation methods in this paper are shown to return velocity fields which are still exactly pointwise divergence free, similar to the Galerkin methods 2 mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We lay out this paper as follows: In Section 2 we describe the steady form of the Navier- Stokes equations using velocity and pressure unknowns as well as vorticity, velocity, and pressure unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' This is followed in Section 3 by a discussion of the de Rham complex and isogeometric discrete differential forms, the tools used to develop a divergence-conforming method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Section 4 describes the collocation schemes for square domains in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Then results are presented in the two dimensional setting in Section 5 which detail the high-order convergence rates of the methods as well as agreement with standard benchmark problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We then discuss the necessary changes to make the methods work for cubic domains in three dimensions and illustrate that the methods performs similarly in this setting in Sections 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Finally, we return to 2D in Section 8 and consider the Stokes equations in more complicated domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We show that by mapping the equations and unknowns via divergence and integral preserving transformations we can also obtain results for flow in complex geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Section 9 summarizes these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Velocity-Pressure and Vorticity-Velocity-Pressure Forms of the Navier-Stokes Equations In this paper we consider the steady, incompressible Navier-Stokes equations on a Lips- chitz open set Ω of points in either R2 or R3 when subjected to Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The standard form of this problem with d = 2, 3 is stated as follows: � � � � � � � � � � � � � � � � � � � � � � � Given ν ∈ R+, f : Ω → Rd, and g : ∂Ω → Rd, find u : Ω → Rd and p : Ω → R such that: −ν∆u + u · ∇u + ∇p = f in Ω (1) ∇ · u = 0 in Ω (2) u = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (3) Equations (1) and (2) represent the standard forms of the momentum and mass conservation equations, while Equation (3) sets the Dirichlet boundary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Above, we denote the velocity field by u, the kinematic pressure field by p, the constant kinematic viscosity by ν, the applied forcing as f, and the prescribed Dirichlet boundary values as g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' For the purposes of this paper we will not only work with this set of equations, but also introduce vorticity ω as a separate unknown variable and introduce ω − ∇ × u = 0 as a constitutive relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Substituting the two vector calculus identities: ∆u = ∇(∇ · u) − ∇ × (∇ × u) = −∇ × ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (4) u · ∇u = (∇ × u) × u + 1 2∇(u · u) = ω × u + 1 2∇(u · u),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (5) into Equation (1) when d = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' we arrive at the vorticity-velocity-pressure formulation of the problem in 3D: 3 � � � � � � � � � � � � � � � � � � � � � � � � � � � Given ν ∈ R+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' f : Ω → R3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' and g : ∂Ω → R3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' find u : Ω → R3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' P : Ω → R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' and ω : Ω → R3 such that: ν∇ × ω + ω × u + ∇P = f in Ω (6) ∇ · u = 0 in Ω (7) ω − ∇ × u = 0 in Ω (8) u = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (9) Note that in the above formulation we have replaced the kinematic pressure p with the total pressure P, which are related via P = p + 1 2u · u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' For later sections in this paper it is useful to employ the component forms of the vec- tor equations above describing the vorticity-velocity-pressure formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' When explicitly broken into its components, the momentum conservation equation, given by Equation (6), becomes: ν(∂ωz ∂y − ∂ωy ∂z ) + (ωyuz − ωzuy) + ∂P ∂x = fx, (10) ν(∂ωx ∂z − ∂ωz ∂x ) + (ωzux − ωxuz) + ∂P ∂y = fy, (11) ν(∂ωy ∂x − ∂ωx ∂y ) + (ωxuy − ωyux) + ∂P ∂z = fz, (12) and the constitutive relation given by Equation (8) reads: ωx − (∂uz ∂y − ∂uy ∂z ) = 0, (13) ωy − (∂ux ∂z − ∂uz ∂x ) = 0, (14) ωz − (∂uy ∂x − ∂ux ∂y ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (15) The above component form of the equations is also useful for considering 2D problems in the three field formulation, as we do not need to redefine operations such as cross products when the vorticity reduces to a scalar unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Thus we can arrive at the problem statement for 2D domains by simply removing any terms involving ωx, ωy, or any derivatives in the z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In full, the 2D problem reads: 4 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Given ν ∈ R+, f : Ω → R2, and g : ∂Ω → R2, find u : Ω → R2, P : Ω → R, and ω : Ω → R such that: ν ∂ω ∂y − ωuy + ∂P ∂x = fx in Ω (16) −ν ∂ω ∂x + ωux + ∂P ∂y = fy in Ω (17) ∇ · u = 0 in Ω (18) ω − (∂uy ∂x − ∂ux ∂y ) = 0 in Ω (19) u = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (20) With the governing equations fully defined we can move to a more in-depth description of the collocation scheme, starting with the definition of discrete approximation spaces in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The de Rham Complex and Isogeometric Discrete Differential Forms The first step in creating a collocation scheme is to define the sets of basis functions used to approximate the unknown variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' To construct the collocation methods presented in this paper, we leverage the de Rham complex which aids in the development of exactly divergence-conforming finite element spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' After recalling the de Rham complex, we de- scribe the process of constructing B-spline basis functions and conclude with the definition of B-spline spaces that conform to the de Rham complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The de Rham Complex The de Rham complex is a cochain complex that is often used as a starting point for developing mixed finite element methods which preserve topological properties of the con- tinuous problem and are typically more stable in practice [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In 3D, it is typically written as: R Φ Ψ V Q 0, ∇ ∇× ∇· (21) where: Φ := H 1(Ω), (22) Ψ := H(curl, Ω), (23) V := H(div, Ω), (24) Q := L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (25) In the context of fluid flow, these infinite dimensional spaces can be interpreted as the spaces of scalar potential fields (Φ), vector potential fields (Ψ), velocity fields (V), and 5 pressure fields (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' This complex is exact for simply connected domains, meaning that the range of each map is the same as the null space of the following map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' For completeness, we also state the rotated 2D de Rham complex: R Ψ V Q 0, ∇⊥ ∇· (26) where: Ψ := H1(Ω), (27) V := H(div, Ω), (28) Q := L2(Ω), (29) and the rotor operator ∇⊥ acting on a scalar function ω is defined as ∇⊥ω = ( ∂ω ∂y , − ∂ω ∂x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Not only is the topological structure of the incompressible Navier-Stokes equations em- bedded in the de Rham complex, but stability conditions result from the complex as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' By creating approximation spaces for the unknown variables that conform to discrete analogs of Equations (21) and (26) we can generate numerical methods which inherit these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' More concretely, for 3D problems let the space Φh contain the discrete scalar potentials, Ψh contain the discrete vector potentials as well as the discrete vorticity, Vh contain the dis- crete velocity, and Qh contain the discrete pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Then if there exist projection operators ΠΦ : Φ → Φh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' ΠΨ : Ψ → Ψh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' ΠV : V → Vh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' and ΠQ : Q → Qh such that the following commuting diagram holds R −−−→ Φ ∇ −−−→ Ψ ∇× −−−→ V ∇· −−−→ Q −−−→ 0 ���ΠΦ ���ΠΨ ���ΠV ���ΠQ R −−−→ Φh ∇ −−−→ Ψh ∇× −−−→ Vh ∇· −−−→ Qh −−−→ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (30) a Galerkin finite element method employing Vh for the discrete velocity space and Qh for the discrete pressure space will be inf-sup stable and will yield discrete velocity approximations that are divergence free almost everywhere [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We shall prove later on that the divergence- conforming property is maintained if we utilize these spaces in our collocation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The same holds in 2D, where we instead let Ψh define the discrete space to which the vorticity belongs (as well as the streamfunction), Vh define the discrete velocity space, and Qh define the discrete pressure space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The required commuting diagram in this case is R −−−→ Ψ ∇⊥ −−−→ V ∇· −−−→ Q −−−→ 0 ���ΠΨ ���ΠV ���ΠQ R −−−→ Ψh ∇⊥ −−−→ Vh ∇· −−−→ Qh −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (31) Of course, we have yet to define the specifics of how to construct discrete spaces such that these discrete complexes hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' For the purposes of this paper we will use compatible B-spline spaces, and the following section is devoted to introducing the basics of B-spline basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Univariate and Multivariate B-Splines The construction of a B-spline basis in one dimension requires two objects: the de- gree of the basis (denoted k) and a series of numbers called the knot vector (denoted Ξ = {ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ξn+k+1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The knots ξi are non-decreasing and denote the locations in paramet- ric space where the parametrization can change, similar to element boundaries in standard FEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The number n in the previous relation represents the total number of functions in the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The basis functions themselves are defined through the Cox-de Boor recursion: The k = 0 basis functions are built as Ni,0(ξ) = � 1 ξi ≤ ξ ≤ ξi+1 0 otherwise, (32) and higher-order bases are defined through Ni,k(ξ) = ξ − ξi ξi+k − ξi Ni,k−1(ξ) + ξi+k+1 − ξ ξi+k+1 − ξi+1 Ni+1,k−1(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (33) Note that in the above relations, we must utilize the convention that any occurrence of zero divided by zero is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In higher dimensions (two or three for the purposes of this paper), B-spline basis functions are constructed by simply taking the tensor product of one dimensional B-spline bases in each parametric direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Note that different polynomial degrees and knot vectors can be used in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' B-spline basis functions possess a number of useful properties for numerical method development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In particular, the smoothness of the global basis at the knot locations is controlled by the repetition of the knot value in Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The basis at these locations is Ck−r, where r is the multiplicity of the knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Compared to a standard finite element basis, this basis has improved global continuity, which enables the use of collocation as more classical derivatives of the functions are well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Note that if the first and last entries in the knot vector are repeated k +1 times, the spline basis will become interpolatory at those locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Such knot vectors are referred to as open knot vectors, and allow for easy specification of Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' For the results within this paper, all other entries in the knot vector have multiplicity one, meaning we are utilizing a B-spline basis with the highest possible order of continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Further, let us define the so-called regularity vector α for a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The size of this vector is equal to the number of distinct knots, with entries equal to the polynomial degree of the basis minus the multiplicity of the corresponding knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In terms of the regularity vector, the global basis functions are Cαj-continuous across the αj unique knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' To simplify notation, the space of functions spanned by a 1D B-spline basis of degree k and a provided knot vector is denoted as: Sk α = span{Ni,k}n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (34) We extend this notation to higher dimensions by adding extra sub- and superscripts, repre- senting the polynomial degrees and regularities in each spatial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Isogeometric Discrete Differential Forms Using the basics of B-spine functions above allows us to develop discrete approximation spaces for the vorticity, velocity, and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' These results are built upon work in the area of isogeometric discrete differential forms [36, 37], which we will not fully develop here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The construction of these types of spaces in the context of Galerkin approximation for the Navier-Stokes equations can also be found in [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' For 3D problems, the B-spline spaces used to discretize our unknown fields are given by: Φh := {φh ∈ Sk1,k2,k3 α1,α2,α3}, (35) Ψh := {ψh ∈ Sk1−1,k2,k3 α1−1,α2,α3 × Sk1,k2−1,k3 α1,α2−1,α3 × Sk1,k2,k3−1 α1,α2,α3−1}, (36) Vh := {wh ∈ Sk1,k2−1,k3−1 α1,α2−1,α3−1 × Sk1−1,k2,k3−1 α1−1,α2,α3−1 × Sk1−1,k2−1,k3 α1−1,α2−1,α3}, (37) Qh := {qh ∈ Sk1−1,k2−1,k3−1 α1−1,α2−1,α3−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (38) It can be shown that these spaces satisfy the discrete complex in Equation (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In practice we usually define k1 = k2 = k3, and thus we can define the polynomial degree of the spline bases constructed in the above manner using a single number k′ = k1 − 1 = k2−1 = k3−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' This indicates that the pressure space Qh will have degree equal to k′ in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Then, according to the above, each velocity component will have degree k′ + 1 in one direction and degree k′ in the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Similarly, the vorticity components will have degree k′ + 1 in two directions and degree k in the last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In 2D, we define the following spline spaces: Ψh := {ψh ∈ Sk1,k2 α1,α2}, (39) Vh := {wh ∈ Sk1,k2−1 α1,α2−1 × Sk1−1,k2 α1−1,α2}, (40) Qh := {qh ∈ Sk1−1,k2−1 α1−1,α2−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (41) Similar to the 3D setting, these spaces are related as in Equation (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Collocation Methods on Square Domains Using the discrete spaces developed above, this section focuses on the construction of collocation methods for the Navier-Stokes equations using divergence-conforming bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Here we develop methods based on the velocity-pressure form of the Navier-Stokes equations as well as the vorticity-velocity-pressure form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' As the vorticity changes between a scalar in the 2D case and a vector in the 3D case, we start by considering only square domains in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' This selection also lends itself to easier visualization of the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' After briefly reviewing the form of a typical divergence-conforming isogeometric Galerkin method, we define the collocation grids for each unknown and then describe the imposition of boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The section concludes by summarizing the form of the discrete system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Review of Galerkin Methods We start by reviewing the form of the divergence-conforming isogeometric Galerkin meth- ods which inspired our collocation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Let us consider a problem with Dirichlet bound- ary conditions on the velocity for concreteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Then we define the discrete test and trial function spaces for velocity as Vh,0 and Vh,g, which are defined as the same Vh from Equa- tion (40) with either no penetration boundary conditions strongly enforced (for the test space Vh,0) or with the normal velocity prescribed as given by the boundary data g at spec- ified collocation points (for the trial space Vh,g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Similarly, define the test and trial space for pressure as Qh,0, where Qh is the same space as in Equation (41) but with the added condition that the pressure must have zero integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Then the Galerkin formulation for the velocity-pressure form would read � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Given ν ∈ R+, f : Ω → R2, and g : ∂Ω → R2, find uh ∈ Vh,g and ph ∈ Qh,0 such that, ∀(wh, qh) ∈ (Vh,0, Qh,0): � Ω (ν∇wh · ∇uh + wh · (uh · ∇uh) − ph∇ · wh)dΩ − ν � ∂Ω ((∇uh · n) · wh − Cpen h uh · wh)dΓ = � Ω wh · fdΩ + ν � ∂Ω Cpen h g · whdΓ (42) � Ω qh(∇ · uh)dΩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (43) Note that in the above we have used a Nitsche approach to enforce the tangential bound- ary conditions in the momentum equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' This Galerkin formulation is valid if and only if the minimum entry in the regularity vectors satisfy min{α1 − 1} ≥ 0 and min{α2 − 1} ≥ 0, where α1 and α2 are the regularity vectors from Equations (39) - (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We can write a similar Galerkin form of the vorticity-velocity-pressure form of the Navier-Stokes equations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' which would yield 9 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Given ν ∈ R+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' f : Ω → R2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' and g : ∂Ω → R2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' find uh ∈ Vh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' P h ∈ Qh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' and ωh ∈ Ψh such that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' ∀(wh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' qh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' ψh) ∈ (Vh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Qh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Ψh): � Ω ν ∂ωh ∂y wh xdΩ − � Ω ωhuh ywh xdΩ − � Ω ∂wh x ∂x P hdΩ + � Γ P hwh xdy = � ω fxwh xdΩ (44) − � Ω ν ∂ωh ∂x wh ydΩ + � Ω ωhuh xwh ydΩ − � Ω ∂wh y ∂y P hdΩ + � Γ P hwh ydx = � ω fywh ydΩ (45) � Ω (∇ · uh)qhdΩ = 0 (46) � Ω ωhψhdΩ + � Ω (∂ψh ∂x uh y − ∂ψh ∂y uh x)dΩ = � ∂Ω ψh(g · s)dΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (47) In this case the tangential velocity boundary conditions appear as natural boundary condi- tions in the weak form of the constitutive equation, with s representing the unit tangent on the boundary (oriented counter-clockwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In contrast with the velocity-pressure Galerkin formulation, this three field Galerkin formulation is valid if and only if the minimum regu- larity of the discrete spaces satisfies min{α1 − 1} ≥ −1 and min{α2 − 1} ≥ −1 due to the reduced differential order of the strong form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' However, we wish to pursue a collocation method inspired by the divergence-free mixed finite element form, which we describe below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Collocation imposes additional regularity requirements on the spaces of unknowns, as the unknown fields and their derivatives will be evaluated at points rather than integrated over the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The spaces developed above are not only divergence-conforming, meaning discrete velocity approximations will be pointwise divergence-free, but are also regular enough to use in collocation provided the polynomial degree is sufficiently large and the global basis is sufficiently regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Collocation Grids Similarly to the Galerkin setting, each discrete unknown in the collocation schemes is assumed to lie in the corresponding space from above: The discrete velocity uh ∈ Vh,g, the discrete pressure ph ∈ Qh,0, and, when applicable, the discrete vorticity ωh ∈ Ψh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' For now we ignore any boundary conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' these will be discussed in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' To generate the system of equations needed to solve for the coefficients for each basis function, we define sets of collocation points of the form τ i for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=', N for each of the governing equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Note that the total number of collocation points should be equal to the total number of degrees of freedom in the discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The full discrete system is formed by requiring the strong form of the governing equations to hold at each of the collocation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We choose the Greville abscissae of a B-spline space as collocation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In one dimen- sion, the Greville abscissae are defined by ˆξi = ξi+1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' + ξi+p p , (48) 10 A First Collocation Attempt for Stokes: The Unit Square ˆξi = ξi+1 +…+ξi+p p Greville abscissae: = First Momentum Collocation Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' = Second Momentum Collocation Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' = Continuity Collocation Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' k = 2, 4 x 4 elements (a) Before strong enforcement of normal ve- locity boundary conditions A First Collocation Attempt for Stokes: The Unit Square k = 2, 4 x 4 elements Boundary Conditions: We enforce no-penetration BCs strongly and remove the corresponding collocation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We enforce no-slip BCs weakly by modifying the momentum equations at the boundary with a suitable penalty term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' −div !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' grad " ˆu ( ) ( )+ grad " ˆp ( )− ˆf ( ) + Cpen 2 h2 ˆu − ˆuBC ( ) = 0 (b) After strong enforcement of normal ve- locity boundary conditions Figure 1: Example of collocation grid for k′ = 2, 4 x 4 elements for the velocity-pressure scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Horizontal triangles represent the collocation points for the first momentum equation, vertical triangles represent the points for the second momentum equation, and squares are collocation points for the continuity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' and in higher dimensions we simply take the tensor product of the Greville abscissae in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' By construction there will be the same number of Greville points as there are basis functions in the considered space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' There are choices for collocation points other than the Greville abscissae, such as the Cauchy-Galerkin points [38, 39] or the Demko abscissae [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' However, the Greville points are very easy to compute and have already been demonstrated to give satisfactory results in practical applications (see for example [6, 18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' As each of the discrete unknowns lie in a different B-spline space, each of the governing equations will be collocated at a different set of Greville points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' for both the velocity-pressure formulation and the vorticity-velocity-pressure formulation we use the Greville abscissae associated with the basis of the x-velocity (Sk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='k2−1 α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='α2−1) as collocation points for the x-momentum equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' the Greville abscissae associated with the basis of the y- velocity (Sk1−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='k2 α1−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='α2) as collocation points for the y-momentum equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' and the Greville abscissae associated with the basis of the pressure (Sk1−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='k2−1 α1−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='α2−1) as collocation points for the continuity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The constitutive relation within the three field formulation is collocated at the Greville abscissae for the vorticity basis (Sk1,k2 α1,α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The left of Figure 1 details an example of this construction for the velocity-pressure scheme, while the left of Figure 2 shows example grids for the vorticity-velocity-pressure scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Boundary Condition Enforcement The last unspecified aspect of the method is enforcement of the Dirichlet boundary con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The enforcement of the normal boundary condition is done strongly and collocation points along a boundary for the velocity component orthogonal to that boundary are re- moved, as the boundary condition specifies the value of the solution at these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' This is shown on the right of Figure 1 and Figure 2, which depict the same scenarios as their counterparts but with normal boundary conditions enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 11 A Second Collocation Attempt for Stokes: The Unit Square ˆξi = ξi+1 +…+ξi+p p Greville abscissae: = First Momentum Collocation Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' = Second Momentum Collocation Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' = Continuity Collocation Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' k = 2, 4 x 4 elements = Constitutive Collocation Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (a) Before strong enforcement of normal ve- locity boundary conditions A Second Collocation Attempt for Stokes: The Unit Square k = 2, 4 x 4 elements Boundary Conditions: We enforce no-penetration BCs strongly and remove the corresponding collocation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We enforce no-slip BCs weakly by modifying the constitutive equations at the boundary with a suitable penalty term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' ˆω ˆx, ˆy ( )− curl !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' ˆu ˆx, ˆy ( ) ( ) ( ) + Cpen h ˆu⋅ ˆs − ˆuBC ⋅ ˆs ( ) = 0 An exact expression for the penalty constant is obtained by appealing to isogeometric finite volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (b) After strong enforcement of normal ve- locity boundary conditions Figure 2: Example of collocation grid for k′ = 2, 4 x 4 elements for the vorticity-velocity-pressure scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Horizontal triangles represent the collocation points for the first momentum equation, vertical triangles represent the points for the second momentum equation, squares are collocation points for the continuity equation, and circles are the points for the constitutive relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Enforcement of the tangential boundary condition is slightly more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Recall that in Equation (42) we utilized Nitsche’s method to enforce this boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' This motivates the enforcement in the velocity-pressure collocation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Indeed if we take this equation and undo the integration by parts, the consistency term vanishes by construction and we are left with just the penalty terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' If we approximate the integral of the test function as done in [41] the collocated momentum equations will be of the form − ν∆uh + uh · ∇uh + ∇ph + C2 pen h2 (uh − g) = f, (49) where Cpen is a penalty constant and h is the Greville mesh size perpendicular to the bound- ary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Note that because this construction is used to only enforce the tangential boundary conditions, this penalty term only appears in the equation for the momentum balance along the tangential direction of each boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In the vorticity-velocity-pressure scheme we do not use the same Nitsche-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The method utilized here is directly inspired by the Enhanced Collocation method for en- forcing Neumann boundary conditions in isogeometric collocation schemes [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We start by considering the weak form of the constitutive relation given by Equation (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The final term on the left hand side is the boundary term which would be used to enforce natural boundary conditions in a Galerkin method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In a similar vein to the Enhanced Collocation approach, we can undo the integration by parts to arrive at � Ω ψh(ωh − (∂uh y ∂x − ∂uh x ∂y ))dΩ + � ∂Ω ψh(uh · s − g · s)ds = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (50) By approximating the integrals of the test functions as done in [41, 38] we arrive at a modified strong form statement of the constitutive relation which can be collocated along 12 the boundaries ωh − (∂uh y ∂x − ∂uh x ∂y ) + Cpen h (uh · s − g · s) = 0, (51) where again Cpen is a penalty constant and h is the Greville mesh size perpendicular to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Final Collocated Equations Finally, the results of the previous sections are collected and we present the final form of the discrete equations used to solve for the discrete unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The velocity-pressure scheme is considered first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Let us define τ ux ℓ for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=', M ux to be the set of Greville points for Sk1,k2−1 α1,α2−1 with the points corresponding to no-penetration boundaries removed as discussed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Define in a similar manner τ uy ℓ for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=', M uy, which are the Greville points of Sk1−1,k2 α1−1,α2 with no-penetration boundary points removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Lastly, τ p ℓ for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=', N p are the Greville points of Qh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Then the discrete 2D problem reads: � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Find uh ∈ Vh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='g and P h ∈ Qh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='0 such that: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='−ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x2 − ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y2 + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y + ∂ph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ ux ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) = fx(τ ux ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ ux ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∈ Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(52) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='−ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x2 − ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y2 + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y + ∂ph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x + C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='pen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='h2 (uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x − gx) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ ux ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='= fx(τ ux ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ ux ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∈ ∂Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(53) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='−ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x2 − ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y2 + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y + ∂ph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ uy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) = fy(τ uy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ uy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∈ Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(54) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='−ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x2 − ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y2 + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y + ∂ph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y + C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='pen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='h2 (uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y − gy) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ uy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='= fy(τ uy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ uy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∈ ∂Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(55) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x + ∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ∈ Ω ∪ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (56) In the above we have split the momentum equations into expressions valid on the interior collocation points (Equations (52) and (54)) and expressions valid on the remaining boundary collocation points (Equations (53) and (55)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Similarly, for the vorticity-velocity-pressure scheme we also define τ ω ℓ for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=', N ω as the Greville points of Ψh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' With this scheme the discrete 2D problem reads: 13 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Find uh ∈ Vh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' P h ∈ Qh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' and ωh ∈ Ψh such that: � ν ∂ωh ∂y − ωhuh y + ∂P h ∂x � (τ ux ℓ ) = fx(τ ux ℓ ) ∀τ ux ℓ ∈ Ω ∪ ∂Ω (57) � −ν ∂ωh ∂x + ωhuh x + ∂P h ∂y � (τ uy ℓ ) = fy(τ uy ℓ ) ∀τ uy ℓ ∈ Ω ∪ ∂Ω (58) � ∂uh x ∂x + ∂uh y ∂y � (τ p ℓ) = 0 ∀τ p ℓ ∈ Ω ∪ ∂Ω (59) � ωh − ∂uh y ∂x − ∂uh x ∂y � (τ ω ℓ ) = 0 ∀τ ω ℓ ∈ Ω (60) � ωh − ∂uh y ∂x − ∂uh x ∂y + Cpen h (uh · s − g · s) � (τ ω ℓ ) = 0 ∀τ ω ℓ ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (61) In the three field formulation we split the constitutive law into an expression for the interior collocation points (Equation (60)) and another expression for boundary collocation points (Equation (61)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Resulting from these equations are nonlinear systems of equations which we can use to solve for the unknown coefficients of velocity, pressure, and vorticity using a Newton-Raphson method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Proof of Divergence Conforming Property From the commuting diagrams our spaces form, shown in Equations (30) and (31), it is simple to show that both of the resulting collocation methods return an exactly pointwise divergence free velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The commuting diagrams reveal that the divergence of the discrete velocity lies within the discrete pressure space, ∇·uh ∈ Qh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We can thus equivalently write the divergence of the velocity as a linear combination of the pressure basis functions: ∇ · uh = Np � i=0 ciqi, (62) where qi ∈ Qh are the basis functions for the pressure and ci ∈ R are the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' As part of the collocation scheme, we strongly enforce that the velocity field has zero divergence at a number of collocation points equal to the dimension of the discrete pressure space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' This condition can be written as a system of linear equations Mc = 0, (63) where M is a square matrix whose entries are the pressure basis functions evaluated at each collocation point and c is the vector of coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 14 If the choice of collocation points yields a set of linearly independent equations, that is to say M is invertible, then we know that the solution to Equation (63) is c = 0, and thus the velocity field is exactly divergence free pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Numerical Results on Square Domains The developed schemes are now tested on multiple 2D problems on the unit square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' First, a manufactured vortex problem is considered to experimentally compute the convergence rates of the error and test for pressure and Reynolds number robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Then, the classical lid-driven cavity problem is considered at a variety of Reynolds numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Two-Dimensional Manufactured Solution As a first numerical experiment, we consider a vortex problem on the unit square con- structed using the method of manufactured solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' This solution was originally developed in [21] and employs the following velocity and pressure fields: ˜u = � 2ex(−1 + x)2x2(y2 − y)(−1 + 2y) (−ex(−1 + x)x(−2 + x(3 + x))(−1 + y)2y2) � , (64) ˜p = (−424 + 156e + (y2 − y)(−456 + ex(456 + x2(228 − 5(y2 − y))+ 2x(−228 + (y2 − y)) + 2x3(−36 + (y2 − y)) + x4(12 + (y2 − y))))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (65) For the velocity-pressure scheme we define the forcing term to be f = −ν∆˜u + ˜u · ∇˜u + ∇˜p, (66) while for the vorticity-velocity-pressure formulation we define the forcing term in the mo- mentum equations to be: f = ν∇⊥˜ω + ˜ω × ˜u + ∇ ˜P, (67) with ˜ω = ∇ × ˜u and ˜P = ˜p + 1 2(˜u · ˜u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Enforcing homogeneous boundary conditions and requiring that the integral of pressure is zero, it is clear to see that the velocity and kinematic pressure solutions should be equal to ˜u and ˜p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' To understand the accuracy of this collocation method, we test the convergence rates on a variety of grids and with different degree B-spline bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' For this test we set the Reynolds number to Re = 1 ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We measure error using the L2 norm as well as the H1 semi-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Figure 3 details the convergence rates of velocity and pressure when using the two field formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In this case the errors in both velocity and pressure converge at a rate of k′ when k′ is even and k′ − 1 for odd k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' These are the standard, expected rates that have been seen in other studies of isogeometric collocation, and are one and two orders suboptimal in L2 for odd and even k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Figure 4 details the convergence of velocity, kinematic pressure, and vorticity as we refine the meshes in the three field scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Using this scheme, all of the unknowns converge in the L2 norm at a rate of approximately k′ for even values of k′ and at a rate of k′ + 1 for odd values of k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' These rates match the rates achieved using even k′ in the two field formulation, 15 10-2 10-1 10-10 10-5 (a) Velocity L2 error 10-2 10-1 10-10 10-5 (b) Velocity H1 error 10-2 10-1 10-10 10-5 (c) Pressure L2 error 10-2 10-1 10-10 10-5 100 (d) Pressure H1 error Figure 3: Errors in 2D manufactured vortex solution for velocity-pressure formulation 16 and these rates are two orders faster for odd k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In fact, this formulation recovers optimal convergence rates in the L2 norm for odd k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In the H1 semi-norm, we see convergence rates of k′ for all polynomial degrees for the velocity and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' These rates are optimal in the H1 semi-norm for all degrees and again are as fast or better than the corresponding velocity-pressure scheme results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Interestingly, the H1 convergence of vorticity seems to be at a rate of k′ + 1 for odd k′ and a rate of k′ for even values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' To further study our new collocation schemes, we can also directly compare the errors produced with divergence-conforming Galerkin schemes of the same orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Figure 5 shows the L2 norm and H1 semi-norm errors in velocity as well as the L2 errors in pressure for both collocation schemes along with the Galerkin results for the same problem from [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' This comparison highlights the severe suboptimality of the velocity-pressure results with odd k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We also note that the H1 errors obtained with the three field formulation nearly match the Galerkin results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Pressure Robustness Next we perform some ancillary tests related to the manufactured solution to test some secondary robustness properties of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The first test relates to so-called pressure robustness [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In particular, we take the kinematic pressure ˜p from the manufactured solution and multiply it by a scalar σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Thus the pressure term in the forcing function f will also be multiplied by σ, and the exact solution to which our numerical solution should converge has the same velocity as before but with a scaled kinematic pressure field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' For a pressure robust method this increase in the pressure magnitude, and thus the pressure approximation errors, will not affect the velocity approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Conversely, a non-pressure robust method will see its velocity errors increase as the pressure is scaled larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Figure 6 shows the convergence of the velocity errors for the two field scheme with k′ = 2 and increasing values of the scalar σ, while Figure 7 shows the same for the three field formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Clearly the velocity error increases in both schemes as σ increases, meaning the method is not pressure robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' This is interesting as the divergence-conforming Galerkin method upon which this work is based is pressure robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Reynolds Robustness Similar to pressure robustness, we also want to test how the errors in the solution behave as the Reynolds number is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We increase the Reynolds number by decreasing the viscosity ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' This affects the viscous term in the forcing vector f, but the exact solution to the problem is identical to the original manufactured solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Figures 8 and 9 detail the convergence of the velocity errors as the Reynolds number increases, again for k′ = 2, in the two and three field schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Once again, the error in the velocity field increases as we increase the Reynolds number, in contrast to the divergence- conforming Galerkin setting, where the velocity error is agnostic to increasing Reynolds number [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Two-Dimensional Lid-Driven Cavity Flow The next 2D numerical test problem that we consider is the square lid-driven cavity flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The left,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' right,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' and bottom walls of the cavity remain fixed while the top wall slides in the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(a) Velocity L2 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(b) Velocity H1 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(c) Pressure L2 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(d) Pressure H1 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(e) Vorticity L2 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(f) Vorticity H1 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='Figure 4: Errors in 2D manufactured vortex solution for vorticity-velocity-pressure formulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(a) Velocity L2 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(b) Velocity H1 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(c) Pressure L2 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='Figure 5: Errors in 2D manufactured vortex solution comparison ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(a) Velocity L2 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(b) Velocity H1 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='Figure 6: Errors in 2D manufactured vortex solution with varying pressure scaling for velocity-pressure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='formulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(a) Velocity L2 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(b) Velocity H1 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='Figure 7: Errors in 2D manufactured vortex solution with varying pressure scaling for vorticity-velocity- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='pressure formulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(a) Velocity L2 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(b) Velocity H1 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='Figure 8: Errors in 2D manufactured vortex solution with varying Reynolds number for velocity-pressure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='formulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(a) Velocity L2 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(b) Velocity H1 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='Figure 9: Errors in 2D manufactured vortex solution with varying Reynolds number for vorticity-velocity- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='pressure formulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='positive x direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' causing vortices to develop within the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Due to the inconsistency in the boundary conditions, pressure singularities exist in the corners of the domain, making this a challenging test case for a numerical scheme to properly capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' For our simulations, we set both the speed of the top wall U = 1 and the wall lengths H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The kinematic viscosity ν defines the Reynolds number through Re = UH ν = 1 ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In particular, we consider the flows produced with Re = 100, Re = 400, and Re = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' To validate our results, we compare the centerline velocity profiles at each Reynolds number with the results from Ghia et al [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Figure 10 details the two field formulation results across the three considered Reynolds numbers and two mesh sizes: a 32 element stretched mesh and a 64 element stretched mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The stretched mesh is formed by setting the interior knots of the knot vectors defining the bases in each direction as ξi = 1 2 � 1 + tanh(4ih − 2) tanh(2) � ∀ξi ∈ Ξ, (68) where h is the mesh size in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Figure 11 shows the same results for the three field formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The collocation results from both schemes agree very well with the reference data in all cases, and we see that the results are converging with increasing resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' At a Reynolds number of 100, all of our results show that the maximum and minimum values of the vertical velocity are larger in magnitude than those of Ghia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' This is similar to the behavior seen in the Galerkin method [22], and we note that there are some inaccuracies in the Ghia data for this low Reynolds number case [22, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' For Reynolds number 400, the two field formulation predicts extrema in velocity that are slightly smaller than the three field predictions, which match the corresponding Galerkin results very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' This trend is also valid at a Reynolds number of 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Moreover, while we have used stretched meshes here, the results with a non-stretched mesh are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' As a more quantitative comparison, we compute the minimum horizontal velocity along the vertical centerline as well as the maximum and minimum vertical velocities along the horizontal centerline for each of simulations presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' These results are shown for a Reynolds number of 100 in Table 1, along with the values from [42] and [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' These results show the inadequacy of the Ghia results at this Reynolds number, and for the most part the k′ = 2 collocation results outperform the Ghia data when compared to the pseudospectral results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' To highlight the potential possibilities of the collocation methods, we also compute results using an unstretched mesh of 8 elements in each direction and k′ = 20 for both the two and three field formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' While this would be essentially infeasible with a Galerkin method, as the quadrature would be prohibitively expensive, it is handled with ease by the collocation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We see that these results match the pseudospectral results, even on the utilized coarse meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Collocation Methods on Cubic Domains The previous two sections detailed the construction of the divergence-conforming colloca- tion methods in 2D and tested their behavior numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In the following, we will highlight the required modifications to the methods to solve problems in 3D cubic domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 22 0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 (a) Re = 100 velocities with h = 1/32 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 (b) Re = 100 velocities with h = 1/64 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 (c) Re = 400 velocities with h = 1/32 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 (d) Re = 400 velocities with h = 1/64 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 (e) Re = 1000 velocities with h = 1/32 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 (f) Re = 1000 velocities with h = 1/64 Figure 10: Centerline velocity profiles for 2D lid-driven cavity with velocity-pressure formulation, k′ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Red curves and axes represent the vertical velocity along the horizontal centerline, while blue curves and axes represent the horizontal velocity along the vertical centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 23 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 (a) Re = 100 velocities with h = 1/32 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 (b) Re = 100 velocities with h = 1/64 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 (c) Re = 400 velocities with h = 1/32 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 (d) Re = 400 velocities with h = 1/64 0 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 (e) Re = 1000 velocities with h = 1/32 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 (f) Re = 1000 velocities with h = 1/64 Figure 11: Centerline velocity profiles for 2D lid-driven cavity with vorticity-velocity-pressure formulation, k′ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Red curves and axes represent the vertical velocity along the horizontal centerline, while blue curves and axes represent the horizontal velocity along the vertical centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 24 Table 1: Velocity extrema for 2D lid-driven cavity at Re = 100 Method ux,min uy,max uy,min Collocation, 2 field formulation, k′ = 2 and h = 1/32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='21348 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='17941 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='25307 Collocation, 2 field formulation, k′ = 2 and h = 1/64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='21389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='17953 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='25358 Collocation, 2 field formulation, k′ = 20 and h = 1/8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='21404 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='17957 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='25380 Collocation, 3 field formulation, k′ = 2 and h = 1/32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='21800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='18392 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='25908 Collocation, 3 field formulation, k′ = 2 and h = 1/64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='21511 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='18075 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='25521 Collocation, 3 field formulation, k′ = 20 and h = 1/8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='21404 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='17957 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='25380 Pseudospectral (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' [43]) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='21404 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='17957 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='25380 Ghia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' [42]) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='21090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='17527 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='24533 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Review of Galerkin Methods Similar to 2D, we start by reviewing the form of the divergence-conforming isogeometric Galerkin methods for 3D problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Again assume the velocity is subject to Dirichlet bound- ary conditions along the entire boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We then define the discrete test and trial function spaces for velocity as Vh,0 and Vh,g, which are defined as the same Vh from Equation (37) with either no penetration boundary conditions strongly enforced (for the test space Vh,0) or with the normal velocity prescribed at collocation points as given by the boundary data g (for the trial space Vh,g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We also define the test and trial space for pressure as Qh,0, where Qh is the same space as in Equation (38) but with the added condition that the pressure must have zero integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Then the Galerkin formulation for the velocity-pressure form would read � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Given ν ∈ R+, f : Ω → R3, and g : ∂Ω → R3, find uh ∈ Vh,g and ph ∈ Qh,0 such that, ∀(wh, qh) ∈ (Vh,0, Qh,0): � Ω (ν∇wh · ∇uh + wh · (uh · ∇uh) − ph∇ · wh)dΩ − ν � ∂Ω (∇uh · n) · wh − Cpen h uh · whdA = � Ω wh · fdΩ + ν � ∂Ω Cpen h g · whdA (69) � Ω qh(∇ · uh)dΩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (70) This weak form is essentially unchanged from the 2D case, with the only major difference being that the velocity has 3 components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The vorticity-velocity-pressure Galerkin form, however, is more different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In this case the discrete problem reads 25 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Given ν ∈ R+, f : Ω → R3, and g : ∂Ω → R3, find uh ∈ Vh,g, P h ∈ Qh,0, and ωh ∈ Ψh such that, ∀(wh, qh, ψh) ∈ (Vh,0, Qh,0, Ψh): � Ω (ν∇ × ωh) · vhdΩ + � Ω (ωh × uh) · vhdΩ − � Ω P h(∇ · vh)dΩ = � Ω f · vhdΩ (71) � Ω (∇ · uh)qhdΩ = 0 (72) � Ω (ωh · ψh)dΩ + � Ω uh · (∇ × ψh)dΩ − � ∂Ω (ψh × g) · ndA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (73) Again the no-slip velocity boundary conditions appear as natural boundary conditions in the weak form of the constitutive equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Within the collocation schemes, the unknowns are selected to reside in the same spaces as the corresponding Galerkin scheme, as in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In the following we highlight the main changes to the method for 3D problems with regards to the choice of collocation grids and boundary condition enforcement before again summarizing the final form of the discrete equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Collocation Grids Much like the two dimensional case, in 3D we choose to collocate at Greville abscissae and the grids are different for each of the governing equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' For both formulations the schemes for the momentum and pressure equations are essentially unchanged;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' each momen- tum equation component is collocated at the Greville abscissae of the corresponding discrete velocity component space, and the continuity equation is collocated at the Greville abscissae of the discrete pressure space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Thus the velocity-pressure formulation extends fairly trivially to 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The constitutive equation in the vorticity-velocity-pressure formulation, on the other hand, is now split into components much like how the momentum equations are treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We choose to collocate the x component of the constitutive equation at the Greville ab- scissae associated with the discrete x vorticity space (Sk1−1,k2,k3 α1−1,α2,α3), the y component of the constitutive equation at the Greville abscissae associated with the discrete y vorticity space (Sk1,k2−1,k3 α1,α2−1,α3), and the z component of the constitutive equation at the Greville abscissae associated with the discrete z vorticity space (Sk1,k2,k3−1 α1,α2,α3−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Boundary Condition Enforcement The no-penetration boundary condition is enforced identically to 2D case: We strongly enforce the normal velocity on face collocation points corresponding to the normal velocity component and remove these points from the set used to collocate the momentum equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The no-slip boundary condition in the velocity-pressure scheme is also essentially enforced identically to the 2D case and again leads to equations of the form 26 − ν∆uh + uh · ∇uh + ∇ph + C2 pen h2 (uh − g) = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (74) As the constitutive law relating velocity and vorticity is a vector relation in 3D, the weak enforcement of no-slip boundary conditions is slightly altered in the three field formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We again start by considering the weak form shown above, particularly Equation (73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The last term in this equation represents the boundary term which would be used to enforce boundary conditions by replacing terms with their prescribed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Following the Enhanced Collocation method of [41], the equation can be integrated by parts once again, to arrive at a strong form representation given by: � Ω ψ · (ω − ∇ × u)dΩ + � ∂Ω (ψ × u − ψ × g) · ndA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Using the properties of the scalar triple product, we can re-write this as: � Ω ψ · (ω − ∇ × u)dΩ + � ∂Ω (u × n − g × n) · ψdA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' By approximating these integrals as is done in [41, 38], we arrive at a strong form state- ment including boundary conditions suitable for collocation: ωh − ∇ × uh + Cpen h (uh × n − g × n) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (75) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Final Collocated Equations Once again the entire collocation scheme based on the velocity-pressure formulation is summarized first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Let us again define τ ux ℓ for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=', M ux to be the set of Greville points for the basis of the x velocity component (Sk1,k2−1,k3−1 α1,α2−1,α3−1) with the points corresponding to no-penetration boundaries removed as discussed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Define in a similar manner τ uy ℓ for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=', M uy and τ uz ℓ for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=', M uz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The pressure Greville points are defined as τ p ℓ for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=', N p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' For this formulation the discrete 3D problem reads: 27 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Find uh ∈ Vh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='g and P h ∈ Qh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='0 such that: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='−ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x2 − ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y2 − ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z2 + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z + ∂ph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ ux ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='= fx(τ ux ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ ux ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∈ Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(76) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='−ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x2 − ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y2 − ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z2 + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z + ∂ph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='pen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='h2 (uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x − gx) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ ux ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) = fx(τ ux ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ ux ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∈ ∂Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(77) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='−ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x2 − ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y2 − ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z2 + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z + ∂ph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ uy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='= fy(τ uy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ uy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∈ Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(78) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='−ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x2 − ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y2 − ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z2 + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z + ∂ph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='pen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='h2 (uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y − gy) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ uy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) = fy(τ uy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ uy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∈ ∂Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(79) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='−ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x2 − ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y2 − ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z2 + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z + ∂ph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ uz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='= fz(τ uz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ uz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ∈ Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(80) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='−ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x2 − ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y2 − ν ∂2uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z2 + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y + uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z + ∂ph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='pen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='h2 (uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z − gz) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ uz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) = fz(τ uz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ uz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ∈ ∂Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(81) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x + ∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y + ∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ∈ Ω ∪ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (82) Similarly to the velocity, in the three field formulation we also define collocation points for the vorticity component-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In particular, let τ ωx ℓ for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=', N ωx be the Greville points for the x component of the vorticity, and define τ ωy ℓ for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=', N ωy and τ ωz ℓ for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=', N ωz similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The final,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' discrete 3D problem for the vorticity-velocity-pressure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='collocation scheme reads as: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='Find uh ∈ Vh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' P h ∈ Qh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' and ωh ∈ Ψh such that: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ν(∂ωh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y − ∂ωh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z ) + ωh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='yuh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z − ωh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y + ∂P h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ ux ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) = fx(τ ux ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ ux ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∈ Ω ∪ ∂Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(83) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ν(∂ωh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z − ∂ωh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x ) + ωh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x − ωh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='xuh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z + ∂P h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ uy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) = fy(τ uy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ uy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∈ Ω ∪ ∂Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(84) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ν(∂ωh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x − ∂ωh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y + ωh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='xuh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y − ωh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='yuh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x + ∂P h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ uz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) = fz(τ uz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ uz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ∈ Ω ∪ ∂Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(85) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x + ∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y + ∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ∈ Ω ∪ ∂Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(86) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ωh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x − (∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y − ∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ ωx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ ωx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∈ Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(87) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ωh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x − (∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y − ∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z )+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='Cpen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='h ((uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y − gy)nz − (uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z − gz)ny) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ ωx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ ωx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∈ ∂Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(88) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ωh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y − (∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z − ∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ ωy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ ωy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∈ Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(89) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ωh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y − (∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂z − ∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x )+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='Cpen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='h ((uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z − gz)nx − (uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x − gx)nz) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ ωy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ ωy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∈ ∂Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(90) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ωh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z − (∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x − ∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ ωz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ ωz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∈ Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(91) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ωh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='z − (∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂x − ∂uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∂y )+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='Cpen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='h ((uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='x − gx)ny − (uh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='y − gy)nx) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(τ ωz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∀τ ωz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (92) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Numerical Results on Cubic Domains To verify that the schemes properly extend into 3D, two sample problems are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' First, a manufactured solution gives even more insight into the convergence properties of 29 the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Then the three-dimensional lid-driven cavity problem is considered and the results are compared with established literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Three-Dimensional Manufactured Solution In 3D, we also start our numerical studies by considering a manufactured solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In this case, the exact solution represents the flow around a single vortex filament within the unit cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We define a potential function as ˜φ = � � x(x − 1)y2(y − 1)2z2(z − 1)2 0 x2(x − 1)2y2(y − 1)2z(z − 1) � � , (93) through which we can define the velocity field as ˜u = ∇ × ˜φ, (94) and the vorticity as ˜ω = ∇ × ˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (95) Finally, we specify the pressure field as ˜p = sin(πx) sin(πy) − 4 π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (96) For the velocity-pressure scheme we define the forcing term on the right hand sign of the momentum equations as f = −ν∆˜u + ˜u · ∇˜u + ∇˜p, (97) while for the vorticity-velocity-pressure scheme the forcing term is given by f = −ν∆˜u + ˜ω × ˜u + ∇ ˜P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (98) Once again we enforce homogeneous Dirichlet boundary conditions everywhere and require that the kinematic pressure field has zero average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' With these conditions the discrete solution should again converge to the quantities above with mesh refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Similar to the 2D case, we set Re = 1 ν = 1 and measure the errors produced on a variety of grids in the L2 norm and H1 semi-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Figure 12 shows the results for the velocity-pressure scheme while Figure 13 details the errors for the vorticity-velocity-pressure scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We start by noting that when k′ < 3 in both cases, everything behaves in the same manner as in the 2D setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Once k′ ≥ 3 we start to see very fast convergence rates and some pre-asymptotic type behavior in the velocity errors produced by both schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' This can be explained by talking a closer look at the exact velocity field for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In fact, the exact velocity field is given by a quartic polynomial in each direction and this solution is actually contained within the discrete velocity approximation space for k′ ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' If we were using a pressure robust Galerkin method, the velocity error would be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Since the collocation scheme is not pressure robust, we obtain superconvergence rather than exactly zero error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(a) Velocity L2 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(b) Velocity H1 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(c) Pressure L2 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(d) Pressure H1 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='Figure 12: Errors in 3D manufactured vortex solution for velocity-pressure formulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(a) Velocity L2 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(b) Velocity H1 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(c) Pressure L2 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(d) Pressure H1 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(e) Vorticity L2 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='10-5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='(f) Vorticity H1 error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='Figure 13: Errors in 3D manufactured vortex solution for vorticity-velocity-pressure formulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 (a) Velocity-pressure formulation 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 (b) Vorticity-velocity-pressure formulation Figure 14: Centerline velocity profile for 3D lid-driven cavity using both formulations, k′ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Red curves and axes represent the vertical velocity along the horizontal centerline, while blue curves and axes represent the horizontal velocity along the vertical centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The pressure convergence results also show some interesting behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' While the vorticity- velocity-pressure scheme seems to behave in the same manner as in 2D, the velocity-pressure scheme seems to be recovering the faster rates seen in the three field scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We believe that this is a consequence of the superconvergence of velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Three-Dimensional Lid-Driven Cavity The next numerical study that we perform is on the 3D lid-driven cavity flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Consider again the cavity setup describing the 2D flow, but now extend the square cavity by unit length in the out-of-page direction, thus making it a cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The point singularities of the 2D case now extend along the top edges of the cube and we expect to see more influence of 3D boundary effects [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In our tests we again set the wall speed U = 1, the side length H = 1, and consider Re = UH 1 = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We use an unstretched mesh with 32 elements per side and k′ = 2, and compare the x velocity along the vertical centerline and the y velocity along the horizontal centerline with the pseudospectral results from [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Figure 14 shows the results with each formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Once again the results match very well with the literature, and it seems as though the results from the three field formulation match with the reference results slightly better that the two field results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Collocation Methods on Mapped Domains As the last main component of this paper we shift our focus to problems posed on more complicated domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We will present some theory for both 2D and 3D problems, but for simplicity we will focus the development of numerical schemes for the 2D, linear Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' However, the results would generalize to the nonlinear, 3D setting as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We will also focus on the rotational form of the equations, as the first order nature enables easier mappings between domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 33 The main idea of this section is the mapping back to a parametric reference domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' a square in 2D or a cube in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The previous sections detail how to develop collocation schemes on these simple geometries, thus simply pulling the equations and unknowns back to the reference domain, collocating as before, and pushing the results forward to the physical domain gives our numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Let ˆΩ be the parametric domain (the unit square in 2D or the unit cube in 3D), and let Ω be the physical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We define the function F as mapping from ˆΩ to Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Let DF be the Jacobian of the parametric mapping, and define J = Det(DF), (99) C = (DF)T(DF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (100) Next we can define the pull-back operators in 3D as ιΦ(φ) = (φ ◦ F), (101) ιω(ψ) = (DF)T(ψ ◦ F), (102) ιu(v) = J(DF)−1(v ◦ F), (103) ιp(q) = J(q ◦ F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (104) We define the pulled-back unknowns on the reference domain via the ι maps, specifically ˆu = ιu(u), ˆp = ιp(p), and ˆω = ιω(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' These are the unknowns for which we solve us- ing collocation, and the physical domain solution is then obtained via the corresponding push-forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Importantly, the push-forward of velocity as defined above maps divergences to divergences and preserves nullity of normal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Similarly the push-forward of pressure preserves the nullity of the integral operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' These facts imply that the following commuting diagram exists: R −−−→ Φ ∇ −−−→ Ψ ∇× −−−→ V ∇· −−−→ Q −−−→ 0 ���ιΦ ���ιω ���ιu ���ιp R −−−→ Φ ∇ −−−→ ˆΨ ∇× −−−→ ˆV ∇· −−−→ ˆQ −−−→ 0, (105) where now the hat spaces correspond to the ones defined over the parametric domain, and are identical to the ones used in the previous sections of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Moreover, by composing the ι maps with the projectors from the de Rham complex in the square domain setting, we arrive at a new commuting diagram between the physical domain continuous spaces and the discrete spaces in the physical domain defined by the push-forward of the discrete spaces chosen for the unit square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' For completeness we also define the 2D pull-back operators ιω(ψ) = ψ ◦ F, (106) ιu(v) = J(DF)−1(v ◦ F), (107) ιp(q) = J(q ◦ F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (108) 34 In 2D a commuting diagram also exists: R −−−→ Ψ ∇⊥ −−−→ V ∇· −−−→ Q −−−→ 0 ���ιω ���ιu ���ιp R −−−→ ˆΨ ∇⊥ −−−→ ˆV ∇· −−−→ ˆQ −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (109) Next we begin the process of mapping the governing equations back to the reference domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We start with Equations (6) - (8) for the rotational form of the 3D Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The Stokes equations are recovered by simply removing the nonlinear term in the momentum equation, Equation (6), and noting now that the pressure becomes the standard kinematic pressure p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In the momentum equation, the viscous term is mapped back to the reference domain via (∇ × ω) ◦ F = J−1(DF)( ˆ∇ × ιω(ω)) = J−1(DF)( ˆ∇ × ˆω), (110) and the pressure term is mapped to (∇p) ◦ F = (DF)−T ˆ∇(ιΦ(p)) = (DF)−T ˆ∇(J−1ιp(p)) = (DF)−T ˆ∇(J−1ˆp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (111) Within the continuity equation, Equation (7), the divergence is mapped via (∇ · u) ◦ F = J−1 ˆ∇ · (ιu(u)) = J−1 ˆ∇ · ˆu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (112) Finally, in the constitutive law, Equation (8), the curl term is mapped similarly to the viscous momentum term (∇ × u) ◦ F = J−1(DF)( ˆ∇ × ιω(u)) = J−1(DF)( ˆ∇ × ((DF)Tu)) = J−1(DF)( ˆ∇ × ((DF)T(J−1(DF)ˆu))) = J−1(DF)( ˆ∇ × (J−1Cˆu)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (113) Now we pull each equation back to the reference domain via the corresponding ι map, so the momentum equations are pulled back via ιu, the continuity equation is pulled back with ιp and the constitutive law is pulled back with ιω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' For brevity, we will not state the full form of the mapped equations in 3D, but instead state just the 2D form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' This arises in a similar way as the 2D rotational form of the Navier-Stokes equations was generated from the 3D equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In particular we can simply write the equations out component-wise and note that z velocities as well as derivatives in the z direction are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' This yields: 35 (a) Before strong enforcement of no pene- tration conditions (b) After strong enforcement of no penetra- tion conditions Figure 15: Example of collocation grid on a mapped domain for vorticity-velocity-pressure scheme � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Given ν ∈ R+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' ˆf : ˆΩ → R2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' and ˆg : ∂ ˆΩ → R2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' find ˆu : ˆΩ → R2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' ˆp : ˆΩ → R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' and ˆω : ˆΩ → R such that: ν ∂ˆω ∂ˆy + JC−1 11 ∂(J−1ˆp) ∂ˆx + JC−1 12 ∂(J−1ˆp) ∂ˆy = ˆf1 in ˆΩ (114) − ν ∂ˆω ∂ˆx + JC−1 21 ∂(J−1ˆp) ∂ˆx + JC−1 22 ∂(J−1ˆp) ∂ˆy = ˆf2 in ˆΩ (115) ˆ∇ · ˆu = 0 in ˆΩ (116) ˆω − J−1( ∂ ∂ˆx(J−1(C21ˆux + C22ˆuy)) − ∂ ∂ˆy(J−1(C11ˆux + C12ˆuy))) = 0 in ˆΩ (117) ˆu = ˆg on ∂ ˆΩ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (118) where ˆf = ιu(f) and ˆg = ιu(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We collocate these equations in the same manner as in the previous sections to solve for the parametric domain variables ˆu, ˆp, and ˆω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The collocation points are chosen as the Greville abscissae in the parametric domain, and an example of the resulting points pushed forward into the physical domain is shown in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' No penetration boundary conditions are enforced strongly and no slip boundary conditions are enforced weakly with a suitable penalty term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' For brevity we omit the full statement of the discrete problem and simply note that it leads to a linear system of equations (as we are focused in this section on Stokes flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 36 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Numerical Results on Mapped Domains In this penultimate section we verify the performance of the vorticity-velocity-pressure collocation scheme on non-square domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We first consider linear Couette flow to confirm that the expected convergence rates are maintained and then move on to modified lid-driven cavity flows in non-square setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Cylindrical Couette Flow The first problem posed on a mapped domain that we consider is Couette flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' This models the behavior of a fluid between 2 concentric cylinders, with the outer fixed and the inner rotating at a constant rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We solve the problem over a quarter circle domain as shown in Figure 15, enforcing homogeneous Dirichlet boundary conditions on the outer cylindrical wall, zero normal and unit tangential velocity on the inner cylindrical wall, and zero pressure gradient on the horizontal and vertical boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The last Neumann boundary condition is enforced using the Enhanced Collocation approach [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The exact velocity field is given in polar coordinates as: ¯u = � (Ar + B/r) sin θ (Ar + B/r) sin θ � , (119) with A = −Ωin δ2 1−δ2, B = Ωin r2 in 1−δ2, Ωin = U rin, δ = rin rout, rin = 1 is the radius of the inner cylinder, rout = 2 is the radius of the outer cylinder, and the velocity of the inner cylinder has magnitude U = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The exact pressure field is zero everywhere, and the exact vorticity is a constant equal to 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We use a polar mapping to map between the parametric and physical domains: F(ξ1, ξ2) = � ((rout − rin)ξ2 + rin) sin(2πξ1) ((rout − rin)ξ2 + rin) cos(2πξ1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' (120) In solving this problem with this mapping one can show analytically that the collocation approximation to the exact solution ˆu is a function of ˆy only, the collocation approximation to ˆv is zero, the collocation approximation to ˆp is zero, and the collocation approximation to ˆω is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' However, we assemble and solve the full linear system without utilizing this structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Figure 16 shows the errors in the solution as a function of resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' For the L2 norm and H1 semi-norm errors of velocity we recover the same rates are in the square domain setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The collocation scheme also captures the zero pressure up to finite precision on the coarsest mesh as both the L2 and H1 errors are essentially zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' As the mesh is refined we see this error increase, which we attribute to worsening matrix conditioning and roundoff error effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We also see the same rates as in square domains for the L2 convergence of vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Note that a constant vorticity is also recovered even on the coarsest mesh, as evidenced by the numerically zero H1 semi-norm error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Like the pressure errors the H1 error grows with mesh refinement, and we believe the explanation is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 37 10-2 10-1 10-10 10-8 10-6 10-4 10-2 (a) Velocity L2 error 10-2 10-1 10-10 10-5 100 (b) Velocity H1 error 10-2 10-1 4 6 8 10 12 14 16 10-16 (c) Pressure L2 error 10-2 10-1 10-15 10-14 10-13 (d) Pressure H1 error 10-2 10-1 10-10 10-5 (e) Vorticity L2 error 10-2 10-1 10-15 10-14 10-13 (f) Vorticity H1 error Figure 16: Errors in Couette flow solution for vorticity-velocity-pressure formulation 38 x y 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='9 1 x y 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 x y 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2 Lid-Driven Cavity Flow Over a Wavy Wall Figure 17: Mapped Stokes results for lid-driven cavity with varying numbers of bumps 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Lid-Driven Cavity Over Wavy Wall Our final numerical test case concerns the Stokes flow in a 2D lid-driven cavity, similar to the square domain examples, but now with a non-flat bottom surface of the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' In particular, the mapping from parametric to physical domain is given by F(ξ1, ξ2) = � ξ1 A(B(1 − ξ2) sin(Cπξ1) + ξ2) � , (121) where A, B, and C are constants which control the shape of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' We use three combinations in this paper, in particular A = 1, B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='75, and C = 1 gives a domain with one bump, A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='25, B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='3, and C = 3 gives a domain with two bumps, and A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='25, B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content='3, and C = 5 gives a domain with three bumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Figure 17 shows the streamfunctions obtained with 64 elements and k′ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Clearly we are able to recover symmetric fields in all cases which are appropriate for Stokes flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' 39 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Conclusions In this paper, two divergence-conforming collocation methodologies have been presented for solution of the steady, incompressible Navier-Stokes equations using a velocity-pressure formulation and a vorticity-velocity-pressure formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' By employing B-spline spaces that conform to the de Rham complex, these methods produce velocity fields which are exactly pointwise divergence free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Moreover, by the nature of collocation methods, these methods are much less computationally expensive than traditional Galerkin finite element formulations as no costly numerical integrations are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' By applying the discretizations to benchmark problems in two and three dimensions we have shown that the methods retain a high order of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Moreover, we have seen that by re-writing the equations in the vorticity-velocity-pressure form many convergence rates are improved compared to those obtained with a velocity-pressure scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' However, useful properties of the corresponding divergence-conforming B-spline Galerkin method, such as pressure and Reynolds robustness, are not maintained in these collocation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Finally, methods for problems posed in more complicated domains were created by mapping unknowns and equations between the physical and reference domains using structure-preserving transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' There are many interesting directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Collocation schemes that do retain pressure and Reynolds robustness properties would be useful, as would developing a strategy for stabilization of these types of collocation schemes in advection-dominated flow regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The schemes proposed in this paper could also be extended to the multi-patch setting to allow for simulations posed on even more complicated domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' The use of locally adaptive splines would also aid in maximizing the ratio of accuracy to cost in which collocation already excels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Finally, while collocation improves upon the cost of numerical integration, unsteady, incompressible Navier-Stokes solution strategies will still likely involve the solution of linear systems during each time step, and thus reducing cost of linear system solution is also very important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' Acknowledgements This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE2T4oBgHgl3EQfRwcX/content/2301.03783v1.pdf'} +page_content=' DGE-1656518.' 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Jiankun Wang, Senior Member, IEEE, +and Changchun Hua, Senior Member, IEEE, +Abstract—The efficiency of sampling-based motion planning +brings wide application in autonomous mobile robots. Conven- +tional rapidly exploring random tree (RRT) algorithm and its +variants have gained great successes, but there are still challenges +for the real-time optimal motion planning of mobile robots in +dynamic environments. In this paper, based on Bidirectional RRT +(Bi-RRT) and the use of an assisting metric (AM), we propose a +novel motion planning algorithm, namely Bi-AM-RRT*. Different +from the existing RRT-based methods, the AM is introduced +in this paper to optimize the performance of robot motion +planning in dynamic environments with obstacles. On this basis, +the bidirectional search sampling strategy is employed, in order to +increase the planning efficiency. Further, we present an improved +rewiring method to shorten path lengths. The effectiveness and +efficiency of the proposed Bi-AM-RRT* are proved through com- +parative experiments in different environments. Experimental +results show that the Bi-AM-RRT* algorithm can achieve better +performance in terms of path length and search time. +Note to Practitioners—The motivation of this paper is to +develop a fast and efficient motion planning algorithm for mobile +robots in dynamic environments, which is suitable for both +small simple environments and large complex environments with +static or dynamic obstacles. Sampling-based algorithms and grid- +based algorithms have been investigated in this field. Despite +considerable advances, the planner usually takes a long time +to search for a feasible trajectory, and the resulting trajectory +is not optimal. In order to improve the real-time performance +and quality of motion planning, this paper proposes a Bi-AM- +RRT* method based on the Bi-RRT and the use of AM. In +the experiment, it is found that the introduction of bidirectional +search strategy reduces the target search time of planners and +obtains real-time response by combining with the employment of +AM. Besides, the improved rewiring strategy is used to optimize +the path simultaneously, which accelerates the convergence pro- +cess of path optimization and shortens the length of navigation +path. Our proposed Bi-AM-RRT* algorithm can be applied not +only to small simple environments, but also to large complex +environments, and has achieved superior performance. +This work was supported in part by the National Natural Science Foundation +of China under Grant No. 62203378, 62073279, in part by the Hebei +Natural Science Foundation under Grant No. F2022203098, F2021203054, +in part by the Cultivation Project for Basic Research and Innovation of +Yanshan University under Grant No. 2021LGQN003, and in part by the Hebei +Innovation Capability Improvement Plan Project under Grant No. 22567619H. +(Corresponding author: Ying Zhang.) +Y. Zhang, H. Wang, M. Yin, and C. Hua are with the School of +Electrical Engineering and the Key Laboratory of Intelligent Rehabilita- +tion and Neromodulation of Hebei Province, Yanshan University, Qin- +huangdao, 066004, China. (e-mail: yzhang@ysu.edu.cn; wtk0405@163.com; +yin924431601@163.com; cch@ysu.edu.cn). +J. Wang is with the Shenzhen Key Laboratory of Robotics Perception +and Intelligence, Shenzhen 518055 China, and also with the Department of +Electronic and Electrical Engineering, Southern University of Science and +Technology, Shenzhen 518055, China (e-mail: wangjk@sustech.edu.cn). +Index Terms—Mobile robot, motion planning, sampling strat- +egy, optimization +I. INTRODUCTION +Recent advances in robotics have prompted an increasing +number of autonomous mobile robots to be used in various +fields, such as manufacturing [1], transportation [2], rescue +[3], domestic service [4], and so on. As a fundamental task +of mobile robots, motion planning aims to plan a feasible +collision-free path from the starting point to the target point for +the robot in the working environment with static or dynamic +obstacles [5]. In such context, lots of research efforts have +been conducted on the motion planning problem. For instance, +based on the grid map, the Dijkstra [6] algorithm can derive +a feasible trajectory by traversing the entire map. In order to +save computing resources, A* [7] and anytime repairing A* +(ARA*) [8] use heuristic search strategy to quickly obtain +optimal solution. However, these methods are not suitable +for high-dimensional environments or differential constraints. +D* [9] and anytime D* [10] considers dynamic obstacles +and searches for feasible solutions in dynamic environments. +The methods above are grid-based algorithms that require +discretization of the state space, which leads to an exponential +growth in terms of time consuming and memory requirements +with the increase of the state space dimension [11]. To reduce +the time cost and memory usage, diffusion map is employed +[12]. It is a non-linear dimensionality reduction technique, and +seeks for a feasible solution by transforming each state on +the map into a diffusion coordinate [13]. Nevertheless, this +treatment tends to ignore some details in the environment, +leading to poor planning performance or even getting into trap +in complex dynamic environments. +For fast and high-quality motion planning in complex dy- +namic environments, sampling-based methods have attracted +significant attention. Typically, rapidly exploring random tree +(RRT) algorithm [14] has been widely used and achieved +great success because of its efficiency and low memory +usage. To this end, many of its variants have been presented. +For example, RRT-Connect [15] shortens the search time by +exploiting goal bias and using two trees to search simultane- +ously. RRT* [16] adds a rewiring process to shorten the path +length. Extended-RRT [17] re-searches for new collision-free +path from the root when there are obstacles in the planned +trajectory. But this practice is time-consuming. RT-RRT* [18] +arXiv:2301.11816v1 [cs.RO] 27 Jan 2023 + +IN PREPARATION FOR SUBMITTING TO IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING +2 +retains information about the whole tree from the robot current +position, and uses existing branches around obstacles to locally +plan feasible paths. However, the growth of the whole tree +takes more time. In such case, AM-RRT* [19] uses the +assisting metric (AM) to guide the growth of the tree, which +not only shortens the path search time, but also ensures the +probabilistic completeness of RRT. Although the utilization of +AM can accelerate the RRT exploration process, the real-time +performance needs to be improved in dynamic environments. +In real-world scenarios, it is crucial for the motion planner +to maintain a real-time response to quickly address dynamic +obstacles. +In this paper, we propose a novel motion planning method +based on bidirectional RRT (Bi-RRT) and AM, namely Bi- +AM-RRT*, to improve the performance in dynamic environ- +ments. The presented Bi-AM-RRT* exploits the trunk infor- +mation of the reverse tree with the forward tree to efficiently +generate a feasible path to the goal position. Based on this, the +AM is used to improve the real-time performance of motion +planning in environments with obstacles. In order to optimize +the search path, an improved rewiring strategy based on the +root and goal is introduced to shorten the search time and path +length. The main contributions of this work include: +• a bidirectional search sampling approach based on the +AM to improve the motion planning performance in +dynamic environments; +• a novelly fast and efficient motion planning algorithm, +namely Bi-AM-RRT*, which can obtain real-time re- +sponse capability; +• an improved rewiring strategy to accelerates the path +optimization process and reduce the search cost; +• evaluation and discussion on comparative experiments in +different environments, which demonstrate the validity +and efficiency of Bi-AM-RRT*. +The remainder of this paper is structured as follows. Section +II presents the related work. In Section III, the problem +definition and AM-RRT* are introduced. Section IV elaborates +the proposed Bi-AM-RRT*. Section V and Section VI describe +the extensive experiments and discuss the results, respectively. +Section VII concludes this paper. +II. RELATED WORK +Robot motion planning aims at planning a feasible path for +robots, and has received significant attention over the years, +especially in dynamic environments. Many algorithms have +been proposed to address the motion planning problem. +To plan a feasible path, the artificial potential field algorithm +was introduced for robot motion planning [20], which uses the +direction of the fastest potential field decline as the moving +direction of the robot. However, when in an environment with +obstacles, such solution is prone to fall into local optimisation. +In recent years, the learning-based motion planning strategies +have been investigated. Everett et al. [21] proposed a method +that uses reinforcement learning to achieve obstacle avoidance. +Wang et al. [22] designed a local planner based on reinforce- +ment learning, which adopts the global path as the guide path +to adapt to the dynamic environment. To optimize the planner, +Pérez-Higueras et al. [23] combined inverse reinforcement +learning with RRT* to learn the cost function. Notably, these +learning-based methods need to train the model in advance and +the process is time-consuming. Moreover, in order to find the +optimal trajectory, grid-based motion planning research efforts +have been conducted extensively. For example, based on the +grid map, A* [7] was used to search for feasible solutions +and gained great success. Similarly, Koenig et al. presented +D*-lite for robot planning in unknown terrain based on the +lifelong planning A*. The performance is closely related to +the degree of discretization of the state space. Although these +grid-based approaches can always search for the optimal path +(if one exists), they do not perform well as the scale of the +problem increases, such as time-consuming and high memory +consumption. +To improve performance, sampling-based methods are con- +sidered to be the promising solution. In particular, RRT-based +algorithms are widely popular due to their ability to efficiently +search state spaces and have proven to be an effective way to +plan a feasible path for robots [24]. For instance, Kuwata et +al. [25] proposed CL-RRT for motion planning in complex +environments. This method uses the input space of a closed- +loop controller to select samples and combines effective tech- +niques to reduce the computational complexity of constructing +a state space. Based on the probabilist collision risk function, +Fulgenzi et al. [26] introduced a Risk-RRT method. In this +solution, a Gaussian prediction algorithm is used to actively +predict the moving obstacles and the sampled trajectories to +avoid collisions. To achieve real-time performance, Naderi et +al. [18] designed a RT-RRT* algorithm that interweaves path +planning with tree growth and avoids waiting for the tree to +be fully constructed by moving the tree root with the agent. +Analogously, Armstrong et al. [19] put forward an AM-RRT* +by using AM to accelerate the path planning process of RT- +RRT*. However, the real-time response of these measures can +not be well satisfied with real-world applications. In order +to improve the real-time performance, bidirectional search +sampling methods are widely employed. As an early proposed +bidirectional tree algorithm, RRT-Connect [15] usesd a greedy +heuristic to guide the growth of two trees, thereby shortening +the search time. Subsequently, other variants such as Informed +RRT*-Connect [27], B2U-RRT [28], Bi-Risk-RRT [29], etc. +were proposed, and proved the efficiency of the bidirectional +search approach. Inspired by AM-RRT* and RRT-Connect, a +novel fast and efficient motion planning approach, i.e., Bi- +AM-RRT*, is proposed in this paper to further improve the +real-time performance. +In order to guarantee the quality of planning, it is necessary +to obtain an optimal path while ensuring the speed of the +planner. To address the problem of path optimization, Kara- +man et al. [30] proposed RRT* by using newly generated +nodes to reconnect adjacent vertices to ensure asymptotic +optimality. But the convergence speed is slow. In order to +accelerate the convergence speed, Yi et al. [31] suggested a +sampling-based planning method with Markov Chain Monte +Carlo for asymptotically-optimal motion planning. Chen et +al. [32] designed DT-RRT to abandon the rewire process +and add re-search parent based on the shortcut principle. + +IN PREPARATION FOR SUBMITTING TO IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING +3 +Although this approach can speed up the convergence process, +it tends to produce suboptimal paths. Analogously, Wang et al. +[33] presented a P-RRT*-connect arithmetic to accelerate the +convergence of RRT* using an artificial potential field method. +Besides, based on the path optimization and intelligent sam- +pling techniques, Islam et al. [34] proposed RRT*-Smart, +which aims to obtain an optimum or near optimum solution. +Gammell et al. [35] investigated the optimal sampling-based +path planning with a focused sampling method and presented +Informe RRT* to improve the covergence of RRT*. However, +these methods cannot ensure the planning quality in dynamic +environments. +Based on Bi-RRT and the employment of AM, this paper +proposes a novel motion planning method, namely Bi-AM- +RRT*, to optimize the path and ensure real-time performance +in dynamic environments. +III. PRELIMINARIES +In this section, the problem definition of motion planning is +introduced first, and then the AM-based sampling algorithm, +AM-RRT*, is described. +A. Motion Planning Problem Definition +Let us define the state space as X ∈ Rd. xobs denotes +obstacles in the state space Xobs ∈ X, while xfree = X/Xobs +is the free space without obstacles. xagent ∈ Xfree is defined +as the state of the mobile robot in the space, and the goal state +is represented as xgoal ∈ Xfree. In this paper, a search tree +T ∈ Xfree is used to generate a feasible collision-free path +(i.e., point on the path xi ∈ T) from the start point xinit to +the goal point xgoal During exploration, the proposed Bi-AM- +RRT* can grow both forward tree and reverse tree, which are +denoted Tf and Tr, respectively. In addition, there are user- +defined the maximum edge length emax and the maximum +number of nodes nmax in the circular domain with radius emax +to control the growth state of T. Let texp be the tree growth +time. Meanwhile, the root rewiring time and goal rewiring +time are denoted as troot and tgoal, respectively. Together, they +guarantee the real-time performance. To optimize the path, +when the Euler distance is less than σ and there is no obstacle, +two trees can be joined as one tree, where σ represents the +connecting distance of two trees. +In the presented method, AM dA can be the Eulerian metric, +the diffusion metric [13], and the geodesic metric [36], which +are indicated as dE , dD , and dG, respectively. These AMs +are used to calculate the distance information of two states in +the state space. The Euclidean distance is expressed as +dE(xa, xb) = ∥xa, xb∥. +(1) +The Euclidean distance between the two points is obtained +based on the L2 norm of xa and xb. The diffusion distance is +yielded by calculating the Euclidean distance of the approxi- +mate diffusion coordinates corresponding to each of the two +states, and is described as +dD(xa, xb) = dE(h(g(xa)), h(g(xb))). +(2) +where g(x) refers to mapping the state x in the grid to the +nearest point, and h(x) is mapping x to the diffuse coordinates. +dD can provide a good approximation when an obstacle is +present. The geodesic distance is the use of the Dijkstra [6] +method to generate a distance matrix from the connection +matrix of discretization state space. It has the advantage of +high precision, but time-consuming. +Next, the procedures on which the algorithm depends are +described [19]. Add node(T, xnew, xparent) is to add the node +xnew to T and connect the existing node xparent. Cost(T, x) +refers to the length of the path from the root to x in T based on +Eulerian distance. Path(T, x) refers to returning the sequence +of path nodes from the root to x in T. SetRoot(T, x) updates +the root of T to x. Free path(xa, xb) returns true when there +are no obstacles between xa and xb. Nearest(T, x) returns the +nearest neighbor to x if there is no obstacle between x and its +Euclidean neighbour. Steer(xint, xend) returns a point closer to +xend when there is an obstacle between xint and xend with the +help of an AM. Rewire ellipse(T, xgoal) is to return the state +set within the reconnected ellipse [35]. Enqueue(Q, x) is the +addition of x to the end of Q. MultiEnqueue(Q, X) means that +Enqueue(Q, x) is called repeatedly in sequence for each x in +X. Dequeue(Q) refers to removing and returning the first item +in Q. Push(S, x) is to add x to the front of S. MultiPush(S, X) +means that Push(S, x) is called repeatedly in sequence order +for each x in X. Pop(S) refers to deleting and returning the first +item in S. First(S) means that the first item in S is returned +but not removed, while Second(S) means that the second +item in S is returned but not removed. Update edge(T, xnew, +xchild) replaces the edge(xparent, xchild) with(xnew, xchild) in +T, where xparent is the parent of xchild in T. Len(X) returns the +queue length of X. Sort(Xs, x) means that the elements in Xs +are returned, and they are sorted according to the increasing the +distance to x. Reverse(X) is the reverse sequence that returns +X. Sample state(T, xgoal) returns the sampling set Xs, which +is defined as +Xs = +� +� +� +{xgoal} +p > 0.7 and xgoal /∈ T +Xrandom ∈ Xfree +p < 0.5 or xgoal /∈ T +Rewire ellipse(T, xgoal) +otherwise +where p ∈ [0,1). +B. AM-RRT* +AM-RRT* uses the diffusion distance as an AM, which +is derived from the diffusion map [12] and is also a kind +of grid map. It utilizes a dimensional collapse method to +reduce time and memory consumption significantly. Although +the performance of diffusion distance alone is poor in complex +environment, AM-RRT* algorithm, as an assisting metric of +RRT*, not only guarantees probability completeness, but also +achieves good performance. It quickly finds collision-free +paths, especially when obstacles appear. Fig. 1 shows the +obstacle avoidance performance. When the obstacle appears +on the path, AM-RRT* does not regenerate the tree, but +use the information in the whole tree for obstacle avoidance +action, especially the node information around the obstacle. +As can be viewed in Fig. 1, when obstacles appear, with + +IN PREPARATION FOR SUBMITTING TO IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING +4 +(a) +(b) +Fig. 1. +The obstacle avoidance process of AM-RRT*. (a) The branch +information (green circular area) is used for motion planning, and (b) obstacle +avoidance when encountering a dynamic obstacle, where red represents the +agent, blue represents the goal point, and black represents the obstacle. +the help of diffusion metric, a feasible path can be quickly +drawn based on node rewiring by using the branch information +of a tree below the original path to ensure the real-time +performance. During agent movement, the tree is maintained +in real time. At the same time, the planned path is also rewired +for optimization, and the path lengths are made approximately +optimal by successive iterations. The whole process of tree +growth is similar to RRT and its variants. In this process, the +diffusion map only plays a leading role in guiding the tree +to find the goal point quickly and cover the full space faster +while maintaining its probabilistic completeness in a complex +environment [19]. +Our proposed Bi-AM-RRT* can further shorten the time +of feasible path generation and improve the real-time perfor- +mance of the system. When the path is generated, the reverse +tree stops exploring and initializes, and is then maintained sep- +arately by the forward tree. Besides, a novel rewiring method +is proposed in this paper to accelerate the path optimization +process and shorten the path length, thereby achieving high- +quality motion planning. +IV. PROPOSED METHOD +In this paper, we propose the Bi-AM-RRT* for real-time +optimal motion planning of mobile robots in dynamic environ- +ments. Generally, our method extends AM-RRT*, which uses +both forward and reverse trees for searching and accelerates +the path optimization by a new rewiring process. Fig. 2 +illustrates the bidirectional tree growth rewiring process. First, +the two trees grow simultaneously, and when the two trees are +close enough to connect into one tree and the complete path +to the goal point is generated, the reverse tree stops growing +and initializes. The forward tree continues to grow to the full +map. In this process, the branch information of the tree is +used for obstacle avoidance and path optimization (see Fig. +1). Besides, a path optimization strategy is proposed in this +paper to further improve the tree optimization efficiency and +shorten the path length, as depicted in Fig. 3. In the following, +the Bi-AM-RRT* process and the use of optimization strategy +are described. +(a) +(b) +Fig. 2. Bidirectional tree growth rewiring process. (a) The forward tree (in +blue) and the reverse tree (in green) grow at the same time. (b) When the two +trees are close enough to connect into one tree, the reverse tree stops growing +and initializes. +Fig. 3. The path optimization process. The tree path is further optimized to +A-B on the right when there is a less costly proximity point a around point D. +Although the A-C path is better when there is a less costly proximity point b +around point E, the path is not optimized due to the obstacle (black square) +blocking it. +A. Bi-AM-RRT* +Algorithm 1 describes the whole process of Bi-AM-RRT*. +First, the forward tree and reverse tree information are initial- +ized and the map information is loaded (Lines 1∼2). Then, +the goal points are set and the Xfree and Xobs information +is continuously updated. The root state of the forward tree +follows the position state of the agent, and the goal state is +provided by someone. The root state of the reverse tree is set +to the goal state, while the goal state is set to the initial state +of the agent position and does not change as the agent position +moves (Lines 3∼9). To limit the growth time of the tree, texp +is used. When two trees are not connected (i.e., xgoal ∈ Tr, +xgoal /∈ Tf), they grow simultaneously, and each tree generates +a path from the point closest to the goal point to the root +node (Lines 11∼16). When the Euler distance from the farthest +point to the respective root node is less than σ and there are +no obstacles in the two paths, two trees are connected as one +tree. Here the information from the reverse path is merged +into the forward tree (xgoal /∈ Tf), and the reverse tree stops +growing and initializes (Lines 22∼28). Whereafter, the entire +path is maintained by the forward tree until agent reaches the +goal point (Line 29). As the agent moves along the path, the +tree root is also changed with the path (Lines 30∼31). It is +notable that the agent can be moved along the path using any +control method (Line 32). In the algorithm, the update of the +root and the movement of the agent are guaranteed by limiting +the growth time of the tree, enabling real-time response. The +bidirectional tree can generate the path from the agent to the +goal point faster and make the tree grow radially along the + +L +HLE. +C +b +D +BIN PREPARATION FOR SUBMITTING TO IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING +5 +(a) +(b) +Fig. 4. Path optimization process. (a) When a bidirectional tree connection +produces a suboptimal path (red zone), (b) it can be optimized due to the +continuous growth of the tree. +whole path in full space. It optimizes the suboptimal paths +generated by the bidirectional tree connection. As shown in +Fig. 4, the impact of the suboptimal paths is almost eliminated. +The tree is grown in a way that maintains the probabilistic +completeness of random sampling while using AM for guid- +ance, which make the tree growth more aggressive and effi- +cient. The growth of the forward tree is presented in Algorithm +2. When the goal point is given (Line 1), the forward tree +actively grows toward the goal point under the guidance of +the AM, and finds the goal point when successfully connected +to the reverse tree. The uniform sampling is then continued so +that the tree can cover the entire space. Notably, sampling in +the reconnected ellipse can significantly increase the efficiency +of path optimization. When there is an obstacle between the +sampling point and the nearest point, a node is found near the +sampling point between the two points by using the AM (Lines +2∼5). In order to control the size of the tree, the number of +sampling points around xnew is limited in this paper. When the +number of sampling points is less than nmax or the distance +between the sampling point and its nearest point is greater +than emax, xnew is added to the tree and connected with the +nearest neighbor to minimize the cost (Lines 6∼14). The root +rewiring (see Algorithm 3) is performed to optimize the path +(Line 15). After finding the goal point in the forward tree, the +goal rewiring (see Algorithm 5) is executed to optimize the +path (Lines 16∼17). +In particular, the growth of the reverse tree is basically the +same as that of the forward tree. Since the reverse tree does +not need to reach its own goal point, there is no goal rewiring +step. +B. Path optimization +Algorithm 3 provides the general process of root rewiring, +which optimizes the path when the position of the root +changes. When the root queue is empty, the information of the +offset root is added and the root queue is reset (Lines 1∼2). +As shown in Line 5∼6, the nearest neighbor optimization is +maintained when the root queue length is greater than 0 and +less than or equal to 2. When the queue length is greater than +2, the optimization is performed according to Fig. 3, which +is given in Algorithm 4. This method can speed up the path +optimization. Also, in combination with the nearest neighbor +Algorithm 1: Bi-AM-RRT* +Input: Agent, Goal, Map, Qroot←[], Q rroot←[], +Qgoal←[], Sgoal←[] +Output: Path +1 Path← φ; Tf← φ; Tr← φ; +2 load()←Map; +3 while xgoal is not reached do +4 +load()←Xfree, Xobs; +5 +xagent←Agent; +6 +xroot←Agent; +7 +xgoal←Goal; +8 +xroot r←Agent; +9 +xgoal r←xroot(time=0); +10 +start←clock(); +11 +while clock() - start2. In this case, xr1 and xr2 are dequeued in +sequence and try to find the point in xr1 nearest neighbor +that can reduce the path cost. If it exists, the tree is updated +(Lines 1∼8). If Otherwise, it is added to the root queue, as +shown in Lines 9∼10. +When the goal point is in the tree, the goal rewiring method + +CLIN PREPARATION FOR SUBMITTING TO IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING +6 +Algorithm 2: Expend f +Input: Goal, Map, Qroot, Qgoal, Sgoal +Output: Tf +1 xgoal←Goal; +2 xrand←Sample state(Tf, xgoal); +3 xnearest←Nearest(Tf, xrand); +4 xnew←Steer(xnearest, xrand); +5 Xnear←Nearby(Tf, xnew); +6 if len(Xnear)emax or +xnew==xgoal then +7 +xmin=xnearest; +8 +cmin=Cost(Tf, xnearest)+dE(xnearest, xnew); +9 +for xnew in Xnear do +10 +cnew=Cost(Tf, xnear)+dE(xnear, xnew); +11 +if cnew2 then +8 +Rewire root second(T, Qroot); +Algorithm 4: Rewire root second +Input: Qroot +Output: T +1 xr1←Dequeue(Qroot); +2 xr2←Dequeue(Qroot); +3 for xnear in Nearby(T, xr2) do +4 +if Free path(xnear, xr1) then +5 +cold←Cost(T, xr1); +6 +cnew←Cost(T, xr1)+dE(xr2, xnear)+dE(xr1, +xr2); +7 +if cnew0 or len(Sgoal)>0 then +6 +Rewire goal first(T, Qgoal, Sgoal, xgoal); +7 +while tgoal2 or len(Sgoal)>2 then +9 +Rewire goal second(T, Qgoal, Sgoal, xgoal); +Algorithm 6 elaborates the second step of the optimization +in Algorithm 5. When either Sgoal and Qgoal is greater than 2 +in length, xr1 and xr2 (Lines 1∼ 6) are popped or dequeued in +turn. When xr1 is inside the reconnected ellipse (nodes inside +the ellipse are more likely to be utilized), the cost of each +node within the xr1 radius of emax is calculated. And if there +is a point with a smaller cost, the rewiring optimization is +performed (Lines 7∼14). Otherwise, it is temporarily in Xnext +and is added to Sgoal and Qgoal by sorting it differently using +AM. Moreover, if the distance from the second node at the +top in Sgoal to the goal point is greater than the sum of the +distance from xr1 to the goal point and the distance from xr1 +to xr1, the branch is discarded (Lines 15∼20). +The proposed Bi-AM-RRT* can significantly reduce plan- +ning costs in both small simple environments and large com- +plex environments with dynamic obstacles. The use of bidi- +rectional tree combined with AM can effectively shorten the +time cost of finding the feasible path. During exploration, the +suboptimal paths resulting from bidirectional tree connection +are optimized by growing the entire path radially around. +In addition, deploying the improved root rewiring and goal +rewiring methods (refer to Fig. 3) accelerate path optimization +efficiency, thereby improving the real-time performance. + +IN PREPARATION FOR SUBMITTING TO IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING +7 +Algorithm 6: Rewire goal second +Input: Qgoal, Sgoal, xgoal +Output: T +1 if len(Sgoal)>2 then +2 +xr1=Pop(Sgoal); +3 +xr2=Pop(Sgoal); +4 else +5 +xr1=Dequeue(Qgoal); +6 +xr2=Dequeue(Qgoal); +7 if xr1 ∈Rewire ellipse(xgoal) then +8 +Xnext={}; +9 +for xnear in Nearby(T, xr1) do +10 +if Free path(xnear, xr2) then +11 +cold←Cost(T, xnear); +12 +cnew←Cost(T, xr2)+dE(xr1, +xnear)+dE(xr1, xr2); +13 +if cnew1 and dA(Second(Sgoal), xgoal)>dA(xr1, +xgoal)+dA(xr1, xr2) then +20 +Sgoal=[] +V. EXPERIMENTS AND RESULTS +In order to prove the effectiveness and efficiency of the +proposed method, extensive comparative experiments are car- +ried out in different environments. This section gives the +experimental details, while the comparison and discussion of +experimental results are provided. +A. Setup +The experiments are conducted in PyCharm 2021 on top of +a Lenovo Y7000p laptop running Windows OS Intel i5-8300H +CPU at 2.3 GHz having 16 GB of RAM. To demonstrate the +validity and efficiency of the proposed method, our method +is compared with RT-RRT* [18] and AM-RRT* [19]. Fur- +ther, based on the bidirectional search sampling strategy and +improved rewiring strategy proposed in this work, extensive +comparative experiments are designed using five state-of-the- +art planners [i.e, RT-RRT, RT-RRT(D), AM-RRT(E), AM- +RRT(D) and AM-RRT(G)]1 [18], [19] to fully evaluate the +performance of the Bi-AM-RRT*. Specifically, +1These five planners are represented as RT-RRT*, RT-RRT*(D), AM- +RRT*(E), AM-RRT*(D) and AM-RRT*(G) in [18], [19], and they are written +as RT-RRT, RT-RRT(D), AM-RRT(E), AM-RRT(D) and AM-RRT(G) in this +paper for convenience. In the planners, RT-RRT(D) is RT-RRT* planner using +the diffusion distance as the metric for sampling instead of the European +distance, while AM-RRT(E), AM-RRT(D) and AM-RRT(G) refer to the AM- +RRT* planner that uses the Euclidean distance, diffusion distance and geodesic +distance as the AM, respectively. +1) based on five planners, the bidirectional search strategy is +used to design five types of planners, which are denoted +as Bi-RT-RRT, Bi-RT-RRT(D), Bi-AM-RRT(E), Bi-AM- +RRT(D) and Bi-AM-RRT(G). +2) based on five planners, the improved rewiring strategy is +used to design five types of planners, which are denoted +as RT-RRT*, RT-RRT*(D), AM-RRT*(E), AM-RRT*(D) +and AM-RRT*(G). +3) based on five planners, the bidirectional search strategy +and improved rewiring strategy are used to design five +types of planners, which are denoted as Bi-RT-RRT*, +Bi-RT-RRT*(D), Bi-AM-RRT*(E), Bi-AM-RRT*(D) and +Bi-AM-RRT*(G). +As shown in Table I, a total of 20 planners are implemented +for comparison. Moreover, experiments are carried out in three +challenging scenarios to better demonstrate the robustness and +applicability of the presented method, namely Bug trap, Maze, +and Office (see Fig. 5), where the size of Bug trap and Maze +is 100m × 100m, and the size of Office is 200m × 200m. +In the three scenarios, the parameter settings of planners are +listed in Table II. Note that the connection distance σ used in +bidirectional tree is set to 50m in the Bug trap scenario and +30m in the other scenarios. +In the experiment, each planner is tested in a typical task +where the agent needs to paln a feasible path to the goal +point G from the starting point S in different scenarios with +static obstacles, while recording the search time cost and path +length. To fairly evaluate the performance of the method, +each experiment is repeated 25 times. The average of the +25 experiments is then used for an unbiased comparison +of experimental results. In addition, we further verify the +performance of the proposed method in the environment with +dynamic obstacles. +B. Results +1) Scenario With Static Obstacles: According to the exper- +imental setup, 20 different planners were implemented in three +different scenarios. The experimental results are shown in Fig. +6. Fig. 6(a) presents the performance comparison comparison +with and without the bidirectional search sampling strategy. +Since suboptimal paths can be generated in bidirectional tree +connections (see Fig. 4), the path length of the five planners +based on bidirectional strategy increases, but only by 0.8%. +Note that the search times are significantly improved with +the use of the bidirectional search strategy. In the Bug trap, +Maze, and Office, the time costs are reduced by about 69%, +40.1%, and 41.7%, respectively. In particular, the search time +of Bi-AM-RRT(E) can be reduced by up to 75.6% in the +Bug trap scenario. Therefore, the results illustrate that the +use of bidirectional search sampling strategy is effective for +improving real-time performance. +The results of Fig. 6(b) demonstrate that the combination +of the original method and the improved rewiring strategy can +optimize the average performance of the path length and search +time in the three scenarios to a certain extent. And the path +length can be reduced by an average of about 2.2%. Especially +for AM-RRT*(D), it can still shorten the path length by 3% + +IN PREPARATION FOR SUBMITTING TO IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING +8 +TABLE I +DESCRIPTION OF DIFFERENT PLANNERS +Scheme +Planner +Original +RT-RRT +RT-RRT(D) +AM-RRT(E) +AM-RRT(D) +AM-RRT(G) +Bidirectional search-based +Bi-RT-RRT +Bi-RT-RRT(D) +Bi-AM-RRT(E) +Bi-AM-RRT(D) +Bi-AM-RRT(G) +Improved rewiring strategy-based +RT-RRT* +RT-RRT*(D) +AM-RRT*(E) +AM-RRT*(D) +AM-RRT*(G) +Bidirectional-and improved rewiring strategy-based +Bi-RT-RRT* +Bi-RT-RRT*(D) +Bi-AM-RRT*(E) +Bi-AM-RRT*(D) +Bi-AM-RRT*(G) +(a) +(b) +(c) +Fig. 5. Experimental scenario: (a) Bug trap, (b) Maze, and (c) Office, where the letters S and G represent the starting point and goal point, respectively. The +sizes of the three scenarios are 100m × 100m, 100m × 100m, and 200m × 200m, respectively. +TABLE II +PARAMETERS SETTING OF PLANNER +texp/s +troot/s +tgoal/s +emax/m +nmax +σ/m +RT-RRT +0.15 +0.003 +0.003 +5 +12 +30/50 +RT-RRT(D) +0.15 +0.003 +0.003 +5 +12 +30/50 +AM-RRT(E) +0.15 +0.002 +0.004 +5 +20 +30/50 +AM-RRT(D) +0.15 +0.002 +0.004 +5 +20 +30/50 +AM-RRT(G) +0.15 +0.002 +0.004 +5 +20 +30/50 +and reduce the search time by 5.6% even in the large scenario +(i.e., Office). Overall, the results well show the effectiveness +and generalizability of the improved rewiring method in this +paper. +Fig. 6(c) illustrates a comparison of the results between the +solution presented in this article (i.e., the strategy of fusing +bidirectional search sampling and improved rewiring) and the +original solution. It can be seen that the proposed solution +can achieve superior performance in terms of path length and +search time, except for the slight increase in path length of +Bi-RT-RRT* and Bi-AM-RRT*(E) planners in the Bug trap +scenario. The reason for this is that the bidirectional strategy +and greater connection distance reduce the search time, but +when the goal point is found, the number of nodes generated +in the tree is insufficient, resulting in a lower degree of path +optimization, which will be discussed in detail in Section VI. +Besides, on average, Bi-AM-RRT*(E) achieves the superior +optimization performance in terms of search time, which can +be reduced by 76.7%, but increased by 2.6%in terms of path +length. It is worth noting that Bi-AM-RRT*(D) obtains the +most promising performance overall. In the three scenarios, +Bi-AM-RRT*(D) optimizes search time by 24.6%, 45.2% and +44.9%, respectively, and reduces path length by 1.4%, 0.7% +and 2.8%, respectively. In addition, for Bi-RRT RRT* and Bi- +AM RRT*(E) planners, map processing time is not required. +But for Bi-RRT-RT*(D) and Bi-AM-RRT*(D) planners, the +diffusion maps are needed, while the geodesic metric is +required for Bi-AM-RRT*(G) planner. Although diffusion map +processing takes time, for Bug trap and Maze scenarios, map +processing time is about 1.5s. Even for larger Office scenario, +it only takes 5.6s. As tested in this work, the geodesic metric +for Bug trap and Maze scenarios take about 49s, while Office +scenarios take about 232s. In this context, Bi-AM RRT*(D) +outperforms other planners in terms of search time and path +length, even if the search time is not optimal. The reason +behind this is that diffusion maps are a way to use dimensional +collapse to reduce map processing time [12], [13], such that +some details are ignored when processing larger and more +complex maps. Therefore, in the Office scenario, the search +time of Bi-RRT RRT*(D) is still large, but as a AM the +impact on the planner is small. Although the comprehensive +performance of Bi-AM RRT* in small scenes is similar to +that of Bi-RT-RRT*(D), it is not suitable for larger scenarios. +In conclusion, Bi-AM RRT*(D) is an excellent planner that +further improves performance, and is suitable for both small +and large scenarios. The results, then, further demonstrate the +effectiveness and efficiency of our proposed strategy. +2) Scenario With Dynamic Obstacles: In order to test the +real-time response and obstacle avoidance performance of the +proposed method, the experiment is conducted in the Office +scenario with the dynamic obstacle. The experimental results +are depicted in Fig. 7. When the goal point is given, both trees +grow at the same time [see Fig. 7(a)]. When the distance is +close enough [refer to Fig. 7(ab)], the two paths are connected +to one path at two green dots, and the reverse tree stops +growing and initializes. The forward tree uses information +from the reverse path to grow quickly to the goal point and to +the whole map. As shown in Fig. 7(c) and (d), when there + +速 +8 +98ssIN PREPARATION FOR SUBMITTING TO IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING +9 +RT-RRT +RT-RRT(D) +AM-RRT(E) +AM-RRT(D) +AM-RRT(G) +193.02 +186.59 +178.32 +178.68 +175.19 +150 +160 +170 +180 +190 +200 +210 +Path length / m +Bug_trap +RT-RRT +RT-RRT(D) +AM-RRT(E) +AM-RRT(D) +AM-RRT(G) +131.99 +127.42 +129.79 +124.01 +122.43 +115 +120 +125 +130 +135 +Path length / m +Maze +RT-RRT +RT-RRT(D) +AM-RRT(E) +AM-RRT(D) +AM-RRT(G) +292.54 +297.83 +288.01 +285.71 +281.86 +270 +275 +280 +285 +290 +295 +300 +305 +Path length / m +Office +RT-RRT +RT-RRT(D) +AM-RRT(E) +AM-RRT(D) +AM-RRT(G) +8.17 +1.92 +18.37 +0.57 +0.47 +0 +5 +10 +15 +20 +Search time / s +RT-RRT +RT-RRT(D) +AM-RRT(E) +AM-RRT(D) +AM-RRT(G) +1.17 +1.07 +1.4 +0.31 +0.24 +0 +0.4 +0.8 +1.2 +1.6 +Search time / s +RT-RRT +RT-RRT(D) +AM-RRT(E) +AM-RRT(D) +AM-RRT(G) +30.96 +24.75 +42.56 +0.89 +0.83 +0 +10 +20 +30 +40 +50 +Search time / s +RT-RRT +RT-RRT(D) +AM-RRT(E) +AM-RRT(D) +AM-RRT(G) +193.02 +186.59 +178.32 +178.68 +175.19 +160 +165 +170 +175 +180 +185 +190 +195 +Path length / m +Bug_trap +RT-RRT +RT-RRT(D) +AM-RRT(E) +AM-RRT(D) +AM-RRT(G) +131.99 +127.42 +129.79 +124.01 +122.43 +115 +120 +125 +130 +135 +Path length / m +Maze +RT-RRT +RT-RRT(D) +AM-RRT(E) +AM-RRT(D) +AM-RRT(G) +292.54 +297.83 +288.01 +285.71 +281.86 +265 +270 +275 +280 +285 +290 +295 +300 +Path length / m +Office +RT-RRT +RT-RRT(D) +AM-RRT(E) +AM-RRT(D) +AM-RRT(G) +8.17 +1.92 +18.37 +0.57 +0.47 +0 +5 +10 +15 +20 +25 +Search time / s +RT-RRT +RT-RRT(D) +AM-RRT(E) +AM-RRT(D) +AM-RRT(G) +1.17 +1.07 +1.4 +0.31 +0.24 +0 +0.4 +0.8 +1.2 +1.6 +Search time / s +RT-RRT +RT-RRT(D) +AM-RRT(E) +AM-RRT(D) +AM-RRT(G) +30.96 +24.75 +42.56 +0.89 +0.83 +0 +10 +20 +30 +40 +50 +Search time / s +RT-RRT +RT-RRT(D) +AM-RRT(E) +AM-RRT(D) +AM-RRT(G) +193.02 +186.59 +178.32 +178.68 +175.19 +160 +170 +180 +190 +200 +Path length / m +Bug_trap +RT-RRT +RT-RRT(D) +AM-RRT(E) +AM-RRT(D) +AM-RRT(G) +131.99 +127.42 +129.79 +124.01 +122.43 +115 +120 +125 +130 +135 +Path length / m +Maze +RT-RRT +RT-RRT(D) +AM-RRT(E) +AM-RRT(D) +AM-RRT(G) +292.54 +297.83 +288.01 +285.71 +281.86 +265 +270 +275 +280 +285 +290 +295 +300 +Path length / m +Office +RT-RRT +RT-RRT(D) +AM-RRT(E) +AM-RRT(D) +AM-RRT(G) +8.17 +1.92 +18.37 +0.57 +0.47 +0 +5 +10 +15 +20 +Search time / s +RT-RRT +RT-RRT(D) +AM-RRT(E) +AM-RRT(D) +AM-RRT(G) +1.17 +1.07 +1.4 +0.31 +0.24 +0 +0.4 +0.8 +1.2 +1.6 +Search time / s +RT-RRT +RT-RRT(D) +AM-RRT(E) +AM-RRT(D) +AM-RRT(G) +30.96 +24.75 +42.56 +0.89 +0.83 +0 +10 +20 +30 +40 +50 +Search time / s +Original +198.5 +187.3 +187.03 +178.77 +175.56 +Modify +Bi-RT-RRT +Bi-RT-RRT(D) +Bi-AM-RRT(E) +Bi-AM-RRT(D) +Bi-AM-RRT(G) +132.73 +128.25 +130.19 +124.75 +122.73 +293.61 +299.87 +290.06 +286.33 +282.56 +Bi-RT-RRT +Bi-RT-RRT(D) +Bi-AM-RRT(E) +Bi-AM-RRT(D) +Bi-AM-RRT(G) +Bi-RT-RRT +Bi-RT-RRT(D) +Bi-AM-RRT(E) +Bi-AM-RRT(D) +Bi-AM-RRT(G) +Original +Modify +Original +Modify +2.94 +0.96 +4.49 +0.39 +0.36 +0.79 +0.54 +0.84 +0.17 +0.17 +17.36 +15.02 +24.97 +0.5 +0.48 +Original +Modify +Original +Modify +Modify +Original +Bi-RT-RRT +Bi-RT-RRT(D) +Bi-AM-RRT(E) +Bi-AM-RRT(D) +Bi-AM-RRT(G) +Bi-RT-RRT +Bi-RT-RRT(D) +Bi-AM-RRT(E) +Bi-AM-RRT(D) +Bi-AM-RRT(G) +Bi-RT-RRT +Bi-RT-RRT(D) +Bi-AM-RRT(E) +Bi-AM-RRT(D) +Bi-AM-RRT(G) +(a) +(b) +(c) +182.5 +181.86 +176.86 +175.68 +173.47 +128.2 +126.73 +127.46 +123.11 +121.46 +285.02 +292.06 +280.81 +277.11 +276.35 +Original +Modify +Original +Modify +Original +Modify +RT-RRT* +RT-RRT*(D) +AM-RRT*(E) +AM-RRT*(D) +AM-RRT*(G) +RT-RRT* +RT-RRT*(D) +AM-RRT*(E) +AM-RRT*(D) +AM-RRT*(G) +RT-RRT* +RT-RRT*(D) +AM-RRT*(E) +AM-RRT*(D) +AM-RRT*(G) +8.02 +1.7 +19.34 +0.53 +0.45 +1.15 +0.98 +1.45 +0.29 +0.21 +30.14 +24.6 +41.14 +0.84 +0.79 +RT-RRT* +RT-RRT*(D) +AM-RRT*(E) +AM-RRT*(D) +AM-RRT*(G) +RT-RRT* +RT-RRT*(D) +AM-RRT*(E) +AM-RRT*(D) +AM-RRT*(G) +RT-RRT* +RT-RRT*(D) +AM-RRT*(E) +AM-RRT*(D) +AM-RRT*(G) +Original +Modify +Original +Modify +Original +Modify +195.75 +182.69 +183.02 +176.24 +173.9 +128.77 +127.35 +128.78 +123.2 +121.54 +285.52 +293.06 +283.81 +277.71 +277.35 +Original +Modify +Original +Modify +Original +Modify +Bi-RT-RRT* +Bi-RT-RRT*(D) +Bi-AM-RRT*(E) +Bi-AM-RRT*(G) +Bi-AM-RRT*(D) +Bi-RT-RRT* +Bi-RT-RRT*(D) +Bi-AM-RRT*(E) +Bi-AM-RRT*(G) +Bi-AM-RRT*(D) +Bi-RT-RRT* +Bi-RT-RRT*(D) +Bi-AM-RRT*(E) +Bi-AM-RRT*(G) +Bi-AM-RRT*(D) +3.23 +0.94 +4.28 +0.43 +0.38 +0.8 +0.57 +0.82 +0.17 +0.16 +16.63 +14.68 +24.63 +0.49 +0.47 +Original +Modify +Original +Modify +Original +Modify +Bi-RT-RRT* +Bi-RT-RRT*(D) +Bi-AM-RRT*(E) +Bi-AM-RRT*(G) +Bi-AM-RRT*(D) +Bi-RT-RRT* +Bi-RT-RRT*(D) +Bi-AM-RRT*(E) +Bi-AM-RRT*(G) +Bi-AM-RRT*(D) +Bi-RT-RRT* +Bi-RT-RRT*(D) +Bi-AM-RRT*(E) +Bi-AM-RRT*(G) +Bi-AM-RRT*(D) +Fig. 6. Comparison of experimental results. The average path length and search time required by different planners to find a feasible path from the starting +point S to the goal point G in different scenarios, where (a) represents the results with (i.e., Modify) and without (i.e., Original) the bidirectional search +sampling strategy, and (b) represents the results with (i.e., Modify) and without (i.e., Original) the improved rewiring strategy, and (c) represents the results +with (i.e., Modify) and without (i.e., Original) the bidirectional search sampling strategy and improved rewiring strategy. + +IN PREPARATION FOR SUBMITTING TO IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING +10 +(a) +(b) +(c) +(d) +Fig. 7. Obstacle avoidance performance of the proposed algorithm in the Office scenario. In (a) the blue line is the forward tree path and the green line is +the reverse tree path. When two trees are close enough, they are connected into one tree through two green points (b). And when the black circle of obstacle +appears in the path (c), a feasible path is quickly planned by using the information of nearby branches (d). +is an obstacle in the path, the forward tree uses the node +information near the obstacle to quickly generate a feasible +path to avoid the obstacle, allowing the agent to move along +the planned path and safely reach the the goal point. Hence the +results show that the proposed method can address the obstacle +avoidance well and guarantee the quality of planning. +VI. DISSCUSSION +This section discusses the effect of the connection distance +σ on the obstacle avoidance performance and the number of +nodes on the path optimization. +In this paper, the values of σ are set to 30m and 50m. +This is very large for the connection distance between the two +trees. While this makes it easier to connect the two trees, it +also makes it easier to produce the suboptimal path, as shown +in Fig. 4. For example, in the Bug trap scenario, σ is set to +50m to allow the two trees of the Bi-RT-RRT* and Bi-AM- +RRT*(E) planners to connect more easily, as illustrated in Fig. +8. Due to the guidance of Euler distance, the two trees are +trapped at point A and point B in Fig. 8(a) for a long time, +and the distance is far away. This is why the search times +of these two planners are optimized by more than 50%, as +shown in Fig. 6(a) and (c). When two trees are connected by +the diffusion map or geodesic metric, the other three planners +are connected basically in the upper left area of the Bug trap +scenario (see Fig. 2). Although σ is set at 50m, it is not fully +utilized, resulting in less optimization of the search time. +Also taking Bug trap scenario as an example, it can be +seen from Fig. 6 that the search time using AM-RRT(E) is +the longest, but in terms of path length, it is shorter than +that of AM-RRT and almost the same as that of Am-RRT +(D). Since the agent starts moving when the planner finds the +goal point, AM-RRT(E) can generate more nodes in the search +time, making its search tree grow vigorously. In other words, +its path can be fully optimized by the time the agent starts +moving. Although the tree grows in real time, RT-RRT does +not have enough nodes to path optimization, as depicted in +Fig. 9. In the experiment, when a feasible path to the goal +point is found, RT-RRT generates an average of 445 nodes, +AM-RRT(E) generates an average of 1062 nodes, and AM- +RRT(D) generates only 76 nodes on average. After fusing +the bidirectional strategy, Bi-RT-RRT and Bi-AM-RRT(E) can +(a) +(b) +Fig. 8. Path planned by Bi-RT-RRT* and Bi-AM-RRT*(E) in the Bug trap +scenario. Since the Eulerian metric guides, the forward tree will be trapped +in point A, and the reverse tree will be trapped in point B in (a) for a long +time, so σ is set to 50 for optimization. Although there is a longer suboptimal +path after successful connection, but it has been gradually optimized before +the agent arrives (b). +(a) +(b) +Fig. 9. Tree optimization process of the AM-RRT*(E) (a) and RT-RRT* (b). +improve the efficiency of search time by more than 60%, +but lead to a slight increase in path length. Although Bi- +AM-RRT(G) achieves the shortest search time, it requires a +long map processing time. Bi-AM-RRT(D), on the other hand, +enables high-quality planning with fewer nodes. +In the Bug trap scenario, σ is set to 50m, and all plan- +ners can achieve obstacle avoidance performance. Although +the three planners do not take full advantage of this large +connection range, the Bi-RT-RRT* and Bi-AM-RRT*(E) can +maintain path optimization and obstacle avoidance functions. +This is due to the loopback path generated after the paths are + +L +2 +88 +28康A +BIN PREPARATION FOR SUBMITTING TO IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING +11 +(a) +(b) +Fig. 10. The obstacle avoidance of Bi-AM-RRT*(D) in the Office scenario +when σ is set to 50m. Although the two trees are successfully connected, +there are no nodes between the two points A and B for tree growth (a). And +the growth rate of the two points A and B, is not enough to maintain the path +optimization and obstacle avoidance in that connected path when the agent +arrives, resulting in the agent colliding with the obstacle (b). +connected, and the fact that there are more nodes available in +this region to grow the tree and optimize the structure of the +tree by rewiring before the agent arrives. In other scenarios, +however, setting σ to 50m does not maintain rewiring and +obstacle avoidance in some extreme connection situations, as +shown in Fig. 10. To this end, the σ setting of 30 is tested in +the Office scenario, which shows that excellent performance +can be maintained even under extreme connection conditions. +For this purpose, the σ is set to 30m in experiments. The +value of σ should be determined based on factors such as +the size of the scene map, the tree growth time texp, and the +movement speed of the agent. In this paper, the values of σ are +not universal in different scenarios, but have certain reference +value. +VII. CONCLUSION +In this paper, a novel motion planning approach, namely +Bi-AM-RRT* has been proposed based on the Bi-RRT and +the use of an AM. In the Bi-AM-RRT*, two trees grow +simultaneously when the goal point is not in the forward +tree. Then they are connected as one tree when the distance +is less than connection distance. In this case, the reverse +tree stops growing and initializes, while the whole path is +maintained by the forward tree using the improved rewiring +method. To this end, the shorter search time allows for faster +generation of agent-to-goal paths, which in turn allows for +more efficient tree growth by growing trees from points in +the path to other regions. Extensive experiments have been +carried out in three different scenarios for comparison. The +results have demonstrated the validity of our proposal, and +effectively improved the motion planning performance in the +environment with obstacles. In particular, Bi-AM-RRT*(D) +has the best comprehensive performance, while optimizing the +search time and path length. 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Provan, “A fast algorithm for computing geodesic +distances in tree space,” IEEE/ACM Transactions on Computational +Biology and Bioinformatics, vol. 8, no. 1, pp. 2–13, 2010. + diff --git a/HNFKT4oBgHgl3EQfci4v/content/tmp_files/load_file.txt b/HNFKT4oBgHgl3EQfci4v/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fcd0e4368a72b1b8e768d3762ccb3f9d9527a5be --- /dev/null +++ b/HNFKT4oBgHgl3EQfci4v/content/tmp_files/load_file.txt @@ -0,0 +1,1060 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf,len=1059 +page_content='IN PREPARATION FOR SUBMITTING TO IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING 1 Bi-AM-RRT*: A Fast and Efficient Sampling-Based Motion Planning Algorithm in Dynamic Environments Ying Zhang, Heyong Wang, Maoliang Yin, Jiankun Wang, Senior Member, IEEE, and Changchun Hua, Senior Member, IEEE, Abstract—The efficiency of sampling-based motion planning brings wide application in autonomous mobile robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Conven- tional rapidly exploring random tree (RRT) algorithm and its variants have gained great successes, but there are still challenges for the real-time optimal motion planning of mobile robots in dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In this paper, based on Bidirectional RRT (Bi-RRT) and the use of an assisting metric (AM), we propose a novel motion planning algorithm, namely Bi-AM-RRT*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Different from the existing RRT-based methods, the AM is introduced in this paper to optimize the performance of robot motion planning in dynamic environments with obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' On this basis, the bidirectional search sampling strategy is employed, in order to increase the planning efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Further, we present an improved rewiring method to shorten path lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The effectiveness and efficiency of the proposed Bi-AM-RRT* are proved through com- parative experiments in different environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Experimental results show that the Bi-AM-RRT* algorithm can achieve better performance in terms of path length and search time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Note to Practitioners—The motivation of this paper is to develop a fast and efficient motion planning algorithm for mobile robots in dynamic environments, which is suitable for both small simple environments and large complex environments with static or dynamic obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Sampling-based algorithms and grid- based algorithms have been investigated in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Despite considerable advances, the planner usually takes a long time to search for a feasible trajectory, and the resulting trajectory is not optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In order to improve the real-time performance and quality of motion planning, this paper proposes a Bi-AM- RRT* method based on the Bi-RRT and the use of AM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In the experiment, it is found that the introduction of bidirectional search strategy reduces the target search time of planners and obtains real-time response by combining with the employment of AM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Besides, the improved rewiring strategy is used to optimize the path simultaneously, which accelerates the convergence pro- cess of path optimization and shortens the length of navigation path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Our proposed Bi-AM-RRT* algorithm can be applied not only to small simple environments, but also to large complex environments, and has achieved superior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' This work was supported in part by the National Natural Science Foundation of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 62203378, 62073279, in part by the Hebei Natural Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' F2022203098, F2021203054, in part by the Cultivation Project for Basic Research and Innovation of Yanshan University under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 2021LGQN003, and in part by the Hebei Innovation Capability Improvement Plan Project under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 22567619H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' (Corresponding author: Ying Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=') Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Yin, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Hua are with the School of Electrical Engineering and the Key Laboratory of Intelligent Rehabilita- tion and Neromodulation of Hebei Province, Yanshan University, Qin- huangdao, 066004, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' (e-mail: yzhang@ysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' wtk0405@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' yin924431601@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' cch@ysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Wang is with the Shenzhen Key Laboratory of Robotics Perception and Intelligence, Shenzhen 518055 China, and also with the Department of Electronic and Electrical Engineering, Southern University of Science and Technology, Shenzhen 518055, China (e-mail: wangjk@sustech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Index Terms—Mobile robot, motion planning, sampling strat- egy, optimization I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' INTRODUCTION Recent advances in robotics have prompted an increasing number of autonomous mobile robots to be used in various fields, such as manufacturing [1], transportation [2], rescue [3], domestic service [4], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' As a fundamental task of mobile robots, motion planning aims to plan a feasible collision-free path from the starting point to the target point for the robot in the working environment with static or dynamic obstacles [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In such context, lots of research efforts have been conducted on the motion planning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' For instance, based on the grid map, the Dijkstra [6] algorithm can derive a feasible trajectory by traversing the entire map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In order to save computing resources, A* [7] and anytime repairing A* (ARA*) [8] use heuristic search strategy to quickly obtain optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' However, these methods are not suitable for high-dimensional environments or differential constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' D* [9] and anytime D* [10] considers dynamic obstacles and searches for feasible solutions in dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The methods above are grid-based algorithms that require discretization of the state space, which leads to an exponential growth in terms of time consuming and memory requirements with the increase of the state space dimension [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' To reduce the time cost and memory usage, diffusion map is employed [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' It is a non-linear dimensionality reduction technique, and seeks for a feasible solution by transforming each state on the map into a diffusion coordinate [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Nevertheless, this treatment tends to ignore some details in the environment, leading to poor planning performance or even getting into trap in complex dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' For fast and high-quality motion planning in complex dy- namic environments, sampling-based methods have attracted significant attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Typically, rapidly exploring random tree (RRT) algorithm [14] has been widely used and achieved great success because of its efficiency and low memory usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' To this end, many of its variants have been presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' For example, RRT-Connect [15] shortens the search time by exploiting goal bias and using two trees to search simultane- ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' RRT* [16] adds a rewiring process to shorten the path length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Extended-RRT [17] re-searches for new collision-free path from the root when there are obstacles in the planned trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' But this practice is time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' RT-RRT* [18] arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content='11816v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content='RO] 27 Jan 2023 IN PREPARATION FOR SUBMITTING TO IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING 2 retains information about the whole tree from the robot current position, and uses existing branches around obstacles to locally plan feasible paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' However, the growth of the whole tree takes more time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In such case, AM-RRT* [19] uses the assisting metric (AM) to guide the growth of the tree, which not only shortens the path search time, but also ensures the probabilistic completeness of RRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Although the utilization of AM can accelerate the RRT exploration process, the real-time performance needs to be improved in dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In real-world scenarios, it is crucial for the motion planner to maintain a real-time response to quickly address dynamic obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In this paper, we propose a novel motion planning method based on bidirectional RRT (Bi-RRT) and AM, namely Bi- AM-RRT*, to improve the performance in dynamic environ- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The presented Bi-AM-RRT* exploits the trunk infor- mation of the reverse tree with the forward tree to efficiently generate a feasible path to the goal position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Based on this, the AM is used to improve the real-time performance of motion planning in environments with obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In order to optimize the search path, an improved rewiring strategy based on the root and goal is introduced to shorten the search time and path length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The main contributions of this work include: a bidirectional search sampling approach based on the AM to improve the motion planning performance in dynamic environments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' a novelly fast and efficient motion planning algorithm, namely Bi-AM-RRT*, which can obtain real-time re- sponse capability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' an improved rewiring strategy to accelerates the path optimization process and reduce the search cost;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' evaluation and discussion on comparative experiments in different environments, which demonstrate the validity and efficiency of Bi-AM-RRT*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The remainder of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Section II presents the related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In Section III, the problem definition and AM-RRT* are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Section IV elaborates the proposed Bi-AM-RRT*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Section V and Section VI describe the extensive experiments and discuss the results, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Section VII concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' RELATED WORK Robot motion planning aims at planning a feasible path for robots, and has received significant attention over the years, especially in dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Many algorithms have been proposed to address the motion planning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' To plan a feasible path, the artificial potential field algorithm was introduced for robot motion planning [20], which uses the direction of the fastest potential field decline as the moving direction of the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' However, when in an environment with obstacles, such solution is prone to fall into local optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In recent years, the learning-based motion planning strategies have been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Everett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' [21] proposed a method that uses reinforcement learning to achieve obstacle avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' [22] designed a local planner based on reinforce- ment learning, which adopts the global path as the guide path to adapt to the dynamic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' To optimize the planner, Pérez-Higueras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' [23] combined inverse reinforcement learning with RRT* to learn the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Notably, these learning-based methods need to train the model in advance and the process is time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Moreover, in order to find the optimal trajectory, grid-based motion planning research efforts have been conducted extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' For example, based on the grid map, A* [7] was used to search for feasible solutions and gained great success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Similarly, Koenig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' presented D*-lite for robot planning in unknown terrain based on the lifelong planning A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The performance is closely related to the degree of discretization of the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Although these grid-based approaches can always search for the optimal path (if one exists), they do not perform well as the scale of the problem increases, such as time-consuming and high memory consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' To improve performance, sampling-based methods are con- sidered to be the promising solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In particular, RRT-based algorithms are widely popular due to their ability to efficiently search state spaces and have proven to be an effective way to plan a feasible path for robots [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' For instance, Kuwata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' [25] proposed CL-RRT for motion planning in complex environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' This method uses the input space of a closed- loop controller to select samples and combines effective tech- niques to reduce the computational complexity of constructing a state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Based on the probabilist collision risk function, Fulgenzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' [26] introduced a Risk-RRT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In this solution, a Gaussian prediction algorithm is used to actively predict the moving obstacles and the sampled trajectories to avoid collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' To achieve real-time performance, Naderi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' [18] designed a RT-RRT* algorithm that interweaves path planning with tree growth and avoids waiting for the tree to be fully constructed by moving the tree root with the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Analogously, Armstrong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' [19] put forward an AM-RRT* by using AM to accelerate the path planning process of RT- RRT*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' However, the real-time response of these measures can not be well satisfied with real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In order to improve the real-time performance, bidirectional search sampling methods are widely employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' As an early proposed bidirectional tree algorithm, RRT-Connect [15] usesd a greedy heuristic to guide the growth of two trees, thereby shortening the search time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Subsequently, other variants such as Informed RRT*-Connect [27], B2U-RRT [28], Bi-Risk-RRT [29], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' were proposed, and proved the efficiency of the bidirectional search approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Inspired by AM-RRT* and RRT-Connect, a novel fast and efficient motion planning approach, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=', Bi- AM-RRT*, is proposed in this paper to further improve the real-time performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In order to guarantee the quality of planning, it is necessary to obtain an optimal path while ensuring the speed of the planner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' To address the problem of path optimization, Kara- man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' [30] proposed RRT* by using newly generated nodes to reconnect adjacent vertices to ensure asymptotic optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' But the convergence speed is slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In order to accelerate the convergence speed, Yi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' [31] suggested a sampling-based planning method with Markov Chain Monte Carlo for asymptotically-optimal motion planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' [32] designed DT-RRT to abandon the rewire process and add re-search parent based on the shortcut principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' IN PREPARATION FOR SUBMITTING TO IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING 3 Although this approach can speed up the convergence process, it tends to produce suboptimal paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Analogously, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' [33] presented a P-RRT*-connect arithmetic to accelerate the convergence of RRT* using an artificial potential field method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Besides, based on the path optimization and intelligent sam- pling techniques, Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' [34] proposed RRT*-Smart, which aims to obtain an optimum or near optimum solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Gammell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' [35] investigated the optimal sampling-based path planning with a focused sampling method and presented Informe RRT* to improve the covergence of RRT*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' However, these methods cannot ensure the planning quality in dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Based on Bi-RRT and the employment of AM, this paper proposes a novel motion planning method, namely Bi-AM- RRT*, to optimize the path and ensure real-time performance in dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' PRELIMINARIES In this section, the problem definition of motion planning is introduced first, and then the AM-based sampling algorithm, AM-RRT*, is described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Motion Planning Problem Definition Let us define the state space as X ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' xobs denotes obstacles in the state space Xobs ∈ X, while xfree = X/Xobs is the free space without obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' xagent ∈ Xfree is defined as the state of the mobile robot in the space, and the goal state is represented as xgoal ∈ Xfree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In this paper, a search tree T ∈ Xfree is used to generate a feasible collision-free path (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=', point on the path xi ∈ T) from the start point xinit to the goal point xgoal During exploration, the proposed Bi-AM- RRT* can grow both forward tree and reverse tree, which are denoted Tf and Tr, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In addition, there are user- defined the maximum edge length emax and the maximum number of nodes nmax in the circular domain with radius emax to control the growth state of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Let texp be the tree growth time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Meanwhile, the root rewiring time and goal rewiring time are denoted as troot and tgoal, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Together, they guarantee the real-time performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' To optimize the path, when the Euler distance is less than σ and there is no obstacle, two trees can be joined as one tree, where σ represents the connecting distance of two trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In the presented method, AM dA can be the Eulerian metric, the diffusion metric [13], and the geodesic metric [36], which are indicated as dE , dD , and dG, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' These AMs are used to calculate the distance information of two states in the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The Euclidean distance is expressed as dE(xa, xb) = ∥xa, xb∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' (1) The Euclidean distance between the two points is obtained based on the L2 norm of xa and xb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The diffusion distance is yielded by calculating the Euclidean distance of the approxi- mate diffusion coordinates corresponding to each of the two states, and is described as dD(xa, xb) = dE(h(g(xa)), h(g(xb))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' (2) where g(x) refers to mapping the state x in the grid to the nearest point, and h(x) is mapping x to the diffuse coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' dD can provide a good approximation when an obstacle is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The geodesic distance is the use of the Dijkstra [6] method to generate a distance matrix from the connection matrix of discretization state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' It has the advantage of high precision, but time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Next, the procedures on which the algorithm depends are described [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Add node(T, xnew, xparent) is to add the node xnew to T and connect the existing node xparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Cost(T, x) refers to the length of the path from the root to x in T based on Eulerian distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Path(T, x) refers to returning the sequence of path nodes from the root to x in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' SetRoot(T, x) updates the root of T to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Free path(xa, xb) returns true when there are no obstacles between xa and xb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Nearest(T, x) returns the nearest neighbor to x if there is no obstacle between x and its Euclidean neighbour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Steer(xint, xend) returns a point closer to xend when there is an obstacle between xint and xend with the help of an AM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Rewire ellipse(T, xgoal) is to return the state set within the reconnected ellipse [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Enqueue(Q, x) is the addition of x to the end of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' MultiEnqueue(Q, X) means that Enqueue(Q, x) is called repeatedly in sequence for each x in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Dequeue(Q) refers to removing and returning the first item in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Push(S, x) is to add x to the front of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' MultiPush(S, X) means that Push(S, x) is called repeatedly in sequence order for each x in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Pop(S) refers to deleting and returning the first item in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' First(S) means that the first item in S is returned but not removed, while Second(S) means that the second item in S is returned but not removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Update edge(T, xnew, xchild) replaces the edge(xparent, xchild) with(xnew, xchild) in T, where xparent is the parent of xchild in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Len(X) returns the queue length of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Sort(Xs, x) means that the elements in Xs are returned, and they are sorted according to the increasing the distance to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Reverse(X) is the reverse sequence that returns X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Sample state(T, xgoal) returns the sampling set Xs, which is defined as Xs = � � � {xgoal} p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content='7 and xgoal /∈ T Xrandom ∈ Xfree p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content='5 or xgoal /∈ T Rewire ellipse(T, xgoal) otherwise where p ∈ [0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' AM-RRT* AM-RRT* uses the diffusion distance as an AM, which is derived from the diffusion map [12] and is also a kind of grid map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' It utilizes a dimensional collapse method to reduce time and memory consumption significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Although the performance of diffusion distance alone is poor in complex environment, AM-RRT* algorithm, as an assisting metric of RRT*, not only guarantees probability completeness, but also achieves good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' It quickly finds collision-free paths, especially when obstacles appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 1 shows the obstacle avoidance performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' When the obstacle appears on the path, AM-RRT* does not regenerate the tree, but use the information in the whole tree for obstacle avoidance action, especially the node information around the obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' As can be viewed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 1, when obstacles appear, with IN PREPARATION FOR SUBMITTING TO IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING 4 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The obstacle avoidance process of AM-RRT*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' (a) The branch information (green circular area) is used for motion planning, and (b) obstacle avoidance when encountering a dynamic obstacle, where red represents the agent, blue represents the goal point, and black represents the obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' the help of diffusion metric, a feasible path can be quickly drawn based on node rewiring by using the branch information of a tree below the original path to ensure the real-time performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' During agent movement, the tree is maintained in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' At the same time, the planned path is also rewired for optimization, and the path lengths are made approximately optimal by successive iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The whole process of tree growth is similar to RRT and its variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In this process, the diffusion map only plays a leading role in guiding the tree to find the goal point quickly and cover the full space faster while maintaining its probabilistic completeness in a complex environment [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Our proposed Bi-AM-RRT* can further shorten the time of feasible path generation and improve the real-time perfor- mance of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' When the path is generated, the reverse tree stops exploring and initializes, and is then maintained sep- arately by the forward tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Besides, a novel rewiring method is proposed in this paper to accelerate the path optimization process and shorten the path length, thereby achieving high- quality motion planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' PROPOSED METHOD In this paper, we propose the Bi-AM-RRT* for real-time optimal motion planning of mobile robots in dynamic environ- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Generally, our method extends AM-RRT*, which uses both forward and reverse trees for searching and accelerates the path optimization by a new rewiring process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 2 illustrates the bidirectional tree growth rewiring process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' First, the two trees grow simultaneously, and when the two trees are close enough to connect into one tree and the complete path to the goal point is generated, the reverse tree stops growing and initializes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The forward tree continues to grow to the full map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In this process, the branch information of the tree is used for obstacle avoidance and path optimization (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Besides, a path optimization strategy is proposed in this paper to further improve the tree optimization efficiency and shorten the path length, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In the following, the Bi-AM-RRT* process and the use of optimization strategy are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Bidirectional tree growth rewiring process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' (a) The forward tree (in blue) and the reverse tree (in green) grow at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' (b) When the two trees are close enough to connect into one tree, the reverse tree stops growing and initializes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The path optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The tree path is further optimized to A-B on the right when there is a less costly proximity point a around point D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Although the A-C path is better when there is a less costly proximity point b around point E, the path is not optimized due to the obstacle (black square) blocking it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Bi-AM-RRT* Algorithm 1 describes the whole process of Bi-AM-RRT*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' First, the forward tree and reverse tree information are initial- ized and the map information is loaded (Lines 1∼2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Then, the goal points are set and the Xfree and Xobs information is continuously updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The root state of the forward tree follows the position state of the agent, and the goal state is provided by someone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The root state of the reverse tree is set to the goal state, while the goal state is set to the initial state of the agent position and does not change as the agent position moves (Lines 3∼9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' To limit the growth time of the tree, texp is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' When two trees are not connected (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=', xgoal ∈ Tr, xgoal /∈ Tf), they grow simultaneously, and each tree generates a path from the point closest to the goal point to the root node (Lines 11∼16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' When the Euler distance from the farthest point to the respective root node is less than σ and there are no obstacles in the two paths, two trees are connected as one tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Here the information from the reverse path is merged into the forward tree (xgoal /∈ Tf), and the reverse tree stops growing and initializes (Lines 22∼28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Whereafter, the entire path is maintained by the forward tree until agent reaches the goal point (Line 29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' As the agent moves along the path, the tree root is also changed with the path (Lines 30∼31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' It is notable that the agent can be moved along the path using any control method (Line 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In the algorithm, the update of the root and the movement of the agent are guaranteed by limiting the growth time of the tree, enabling real-time response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The bidirectional tree can generate the path from the agent to the goal point faster and make the tree grow radially along the L HLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' C b D BIN PREPARATION FOR SUBMITTING TO IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING 5 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Path optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' (a) When a bidirectional tree connection produces a suboptimal path (red zone), (b) it can be optimized due to the continuous growth of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' whole path in full space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' It optimizes the suboptimal paths generated by the bidirectional tree connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 4, the impact of the suboptimal paths is almost eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The tree is grown in a way that maintains the probabilistic completeness of random sampling while using AM for guid- ance, which make the tree growth more aggressive and effi- cient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The growth of the forward tree is presented in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' When the goal point is given (Line 1), the forward tree actively grows toward the goal point under the guidance of the AM, and finds the goal point when successfully connected to the reverse tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The uniform sampling is then continued so that the tree can cover the entire space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Notably, sampling in the reconnected ellipse can significantly increase the efficiency of path optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' When there is an obstacle between the sampling point and the nearest point, a node is found near the sampling point between the two points by using the AM (Lines 2∼5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In order to control the size of the tree, the number of sampling points around xnew is limited in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' When the number of sampling points is less than nmax or the distance between the sampling point and its nearest point is greater than emax, xnew is added to the tree and connected with the nearest neighbor to minimize the cost (Lines 6∼14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' The root rewiring (see Algorithm 3) is performed to optimize the path (Line 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' After finding the goal point in the forward tree, the goal rewiring (see Algorithm 5) is executed to optimize the path (Lines 16∼17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' In particular, the growth of the reverse tree is basically the same as that of the forward tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Since the reverse tree does not need to reach its own goal point, there is no goal rewiring step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Path optimization Algorithm 3 provides the general process of root rewiring, which optimizes the path when the position of the root changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' When the root queue is empty, the information of the offset root is added and the root queue is reset (Lines 1∼2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' As shown in Line 5∼6, the nearest neighbor optimization is maintained when the root queue length is greater than 0 and less than or equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' When the queue length is greater than 2, the optimization is performed according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 3, which is given in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' This method can speed up the path optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Also, in combination with the nearest neighbor Algorithm 1: Bi-AM-RRT* Input: Agent, Goal, Map, Qroot←[], Q rroot←[], Qgoal←[], Sgoal←[] Output: Path 1 Path← φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Tf← φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' Tr← φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 2 load()←Map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 3 while xgoal is not reached do 4 load()←Xfree, Xobs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 5 xagent←Agent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 6 xroot←Agent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 7 xgoal←Goal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 8 xroot r←Agent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 9 xgoal r←xroot(time=0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 10 start←clock();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFKT4oBgHgl3EQfci4v/content/2301.11816v1.pdf'} +page_content=' 11 while clock() - startquan&zed query 𝑥! +reference.fasta +>quantized 𝑡" +![𝑠" − 𝑏": 𝑒" + 𝑏"] +>quantized 𝑡# +![𝑠# − 𝑏#: 𝑒# + 𝑏#] +target: 𝑡! +target: 𝑡" +query: 𝑥 +𝑠! +𝑠" +𝑒" +𝑒! +𝑏! +𝑏! +𝑛𝑓! +𝑏" +𝑏" +𝑛𝑓" +rc_query.fasta +>quantized rc query ̅𝑥! +(b) +Genome +Read 1 +Read 2 +𝑖! +𝑗! +𝑖" +𝑗" +𝑖! +# +𝑖" +# +𝑗! +# +𝑗" +# +Aligned section on Genome +(c) +Genome +𝑖! +" +𝑗! +SV in the truth set +𝑖! +𝑗! +" +SV from alignment method: +minimap2/HQ3 +(d) +Figure 2: (a) An example to demonstrate the ability of HQAlign pipeline to align inverted +sequences where QAlign fails (b) An example of HQAlign pipeline. (c) An example of read- +to-genome alignment. +(d) Comparison of SV in truth set to SV determined by method: +minimap2/HQ3. +The HQAlign strategy consists of two steps: (1) the initial alignment of the standard base- +called query sequence x to a target sequence t using Minimap2. This primitive step identifies the +region of interests on the target sequence where x aligns. (2) the hybrid step is re-aligning the +query x only to the region of interest determined in the first step using a modified pipeline for +QAlign method. QAlign method converts the nucleotide sequences in both query x and target +region t to quantized (three levels) sequence, and then aligns these quantized sequences using +any state-of-the-art aligner. We have modified the minimap2 (v2.24) pipeline for the align- +ment of the quantized sequences in the second step so that it enables detection of the inversion +variants in alignment using the quantized sequences. This is explained in detail in section 2.2. +Further, this strategy is about 2x faster than QAlign standalone as it narrows down the seed +7 + +search domain for lower alphabet size (e.g. three levels) in QAlign. This strategy is explained +in Figure 2b, and mathematically in the following sections. +Table 1: Comparison of computation time for alignment of 500k randomly sampled ONT reads +to CHM13 assembly using 20 threads for each method. +Method +minimizer +(seed) length +Real time (in +seconds) +CPU time (in +seconds) +QAlign (Q3) +18 +10, 312 +177, 516 +HQAlign (HQ3) +18 +5, 033 +76, 375 +2.1 +Initial alignment +The nucleotide query x is aligned to a nucleotide target sequence t using minimap2. +This +is similar to aligning a read to a genome with one chromosome. Here we consider only one +chromosome in target t for simplicity but the method generalizes to multiple chromosomes in t +such as t = (t1, t2, . . . , tm) (this generalization is explained in detail in Supplementary material +section A.2). This step identifies the region of interests on the target t, say, t[si : ei], where +i ∈ {1, 2, 3, . . . } represent one or more alignments on t and si and ei are the corresponding start +and end location of each alignment i on target t, respectively. +2.2 +Hybrid alignment +In this step, the query x is re-aligned to an extended region of interest on the target t[sq +i : eq +i] using +the modified pipeline for QAlign method, where sq +i = si−bi and eq +i = ei+bi, bi = (1−fi+0.25)n +is an appended extension of the region of interest on target, fi = (ei − si)/n is the fraction of +read aligned in initial step, and n is the length of the query x. The nucleotide query x and +the nucleotide extended target t[sq +i : eq +i] are converted to the quantized query xq and quantized +extended target tq[sq +i : eq +i], respectively, using the quantization method demonstrated in QAlign +(refer to Supplementary material section A.1 for more details on quantization process). It is +important to note that we do not use any additional soft information such as raw current signals +from nanopore sequencing in the quantization process, instead, we translate the basecalled nu- +8 + +cleotide reads to current levels using the Q-mer map (in Figure 1b) and then hard threshold the +current levels to finite (e.g. three) levels to get the quantized (HQ3) reads (refer Supplemen- +tary material Figure 15 for more details on quantization process). These quantized sequences +are then aligned using a modified pipeline of minimap2 (v2.24). We have modified minimap2 +pipeline for this hybrid step to accept the quantized reverse complement query ¯xq as an input +which helps in indentifing the inversion SVs in contiguous alignment with quantized sequences +which was not possible with the earlier QAlign method as shown in Figure 2a. QAlign uses the +default minimap2 pipeline for the alignment of quantized sequences which inherently aligns both +the forward and the reverse complement strand of the sequences in nucleotide domain. How- +ever, the quantized reverse complement sequence cannot be computed given only the forward +quantized sequence, therefore, QAlign separately aligns both quantized forward and quantized +reverse complement sequence. This method, however, fails to identify an inverted alignment +as shown in Figure 2a. Therefore, in HQAlign, we have modified the minimap2 pipeline to +enable alignment using both quantized forward and quantized reverse complement sequence, +simultaneously. Note that the quantized alignment employs a different minimizer length k = 18 +in minimap2 for ternary (HQ3) quantization. +We define several metrics that are used for the performance evaluation of HQAlign against +minimap2 (these metrics are used from the earlier QAlign method [13]). +(i) well-aligned: Consider in Figure 2c, Read 1 aligns at location i1 through j1 on the +genome determined using nucleotide alignment. We say that the read is well-aligned, if +at-least 90% of the read is aligned onto the genome (i.e., j1−i1≥0.9(length(Read 1))), and +has high mapping quality (greater than 20). This metric quantifies the reads that are +mapped almost entirely to the reference. +(ii) normalized edit distance: In order to compare the quality of the alignments at fine- +grained level, we further define normalized edit distance. The normalized edit distance +9 + +for nucleotide alignment is defined as +edit distance{r; G[i1 : j1]} +length(r) +(1) +and for quantized alignment is +edit distance{r; G[iq +1 : jq +1]} +length(r) +(2) +where i1, j1 are the start and end location of alignment on genome in nucleotide space and +iq +1, jq +1 are the start and end location of alignment on genome in the quantized space, r is +the entire read and G is the genome as shown in Figure 2c. It is important to note that +for computing the normalized edit distance for alignments in the quantized space, we only +leverage the information of location of the alignment on genome from quantized space, +i.e. iq +1 and jq +1, but the edit distance between read and the aligned section on genome +is computed on the nucleotide sequences. This metric gives a measure of the distance +similarity between two sequences, especially, used for the real data where the truth of +sequence sampling location is not known. +(iii) normalized alignment length: Another metric at the fine-grained level is normalized +alignment length, which is the ratio of the length of the section on genome where a read +aligns to the length of the read. It is +j1 − i1 +len(r) +(3) +for nucleotide alignment, and +jq +1 − iq +1 +len(rQ) +(4) +for quantized alignment. A contiguous alignment tends to have this metric as 1. This +metric gives a measure of the contiguity of the alignment. +10 + +2.3 +SV calling +The alignments from HQAlign and minimap2 in sorted bam format are used to detect structural +variants using Sniffles2. These calls are benchmarked against a truth set using Truvari [16]. We +have used F1 score, precision and recall as the metric to analyze the performance of HQAlign +and compare them with minimap2. Precision (P) is defined as the fraction of SVs detected by +the algorithm in the truth set among the total SVs detected by the algorithm. Recall (R) is the +fraction of SVs detected by the algorithm in the truth set among the total SVs in the truth set. +F1 score is the harmonic mean of precision and recall (= 2P·R +P+R). Further, we have observed that +there are many complementary SV calls made by both minimap2 and HQ3 that are missed by +the other method. Therefore, we have defined a union model which takes a union of the SV +calls from both minimap2 and HQ3. The precision, recall, and F1 score of the union model are +also computed and reported in Table 3. +Further, the quality of the SVs for the common calls in minimap2 and HQAlign is evaluated +by comparing the following metrics w.r.t. the SVs in truth set +(i) breakpoint accuracy: Breakpoint accuracy is measured by taking an average of the +difference in the in the start and end breakpoint of the SV w.r.t. the SV in truth set. For +instance, as shown in Figure 2d, i1 and j1 are the start and the end point on genome of SV +in the truth set, and i +′ +1 and j +′ +1 are the start and the end point of the same SV determined +by any alignment method (minimap2/HQ3), then breakpoint score is calculated as +|i +′ +1 − i1| + |j +′ +1 − j1| +2 +(5) +where | · | is absolute value function. Therefore, lower the score higher is the breakpoint +accuracy of the SV determined by the alignment method. +(ii) SV length similarity: SV length similarity is measured as the ratio of minimum SV +length in truth set and from algorithm to the maximum of two values. Mathematically, +11 + +it is +min(j1 − i1, j +′ +1 − i +′ +1) +max(j1 − i1, j +′ +1 − i +′ +1) +(6) +for the example shown in Figure 2d. +3 +Results +In this section, we demonstrate the results for (1) comparison of alignments from HQ3 and +minimap2 on real as well as simulated data, and (2) comparison of SV calls from HQ3 and +minimap2 alignments using Sniffles2 as the variant caller on real and simulated data. +3.1 +DNA read-to-genome alignment +3.1.1 +Datasets +We have used the publicly available R9.4.1 ONT PromethION reads dataset from HG002 sample +[17]. These reads are aligned to the recent telomere-to-telomere assembly CHM13 and to the +human reference genome GRCh37. GRCh37 is used as the reference build to map the real +data so that the curated variants can be used for accuracy analysis [18]. Further, we have also +benchmarked the performance of HQAlign and minimap2 on simulated data for both alignments +and SV calling. +3.1.2 +Alignment results +The alignment of DNA reads to the genome is a primitive step in structural variant calling +pipelines [19]. +HQ3 alignments shows an improvement over minimap2 alignments in terms +of contiguity measured by normalized alignment length and alignment quality measured by +normalized edit distance. +The results are illustrated in the Figures 3, 4, 5, and Table 2. +At a coarse level, the +performance is measured by the fraction of the reads that are well-aligned by the algorithm. A +read is well-aligned if at-least 90% of the read is aligned to genome and has a high mapping +12 + +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Comparison of normalized edit distance for HG002 ONT reads alignment +to T2T CHM13 reference genome +4.0% reads well aligned in +HQ3 only +85.3% reads well alignedin +both HQ3 and minimap2 +0.3% reads well aligned in +minimap2 only +Regression line: +slope: 0.79 +intercept: 0.02 +10.3% reads not well +aligned in both HQ3 and +minimap2 +(a) +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Comparison of normalized alignment length for HG002 ONT reads alignment +to T2T CHM13 reference genome +85.3% reads well aligned in +both HQ3 and minimap2 +10.3% reads not well aligned +in both HQ3 and minimap2 +0.3% reads well aligned in +minimap2 only +4.0% reads well aligned in +HQ3 only +Regression line: +slope: 0.61832 +intercept: 1 +(b) +Figure 3: HG002 nanopore long DNA reads alignment onto T2T CHM13 genome. +(a) Comparison of normalized edit distance for HG002 R9.4.1 PromethION reads data. Smaller +values for normalized edit distance is desirable as it represents better alignment. The slope +of the regression line is 0.79 < 1, therefore, representing better alignments with HQ3 than +minimap2 alignments for same reads on average. (b) Comparison of normalized alignment length +for HG002 R9.4.1 PromethION reads data. Normalized alignment length of 1 is desirable as it +represents that entire read is aligned. The majority of the reads are above y = x line representing +longer alignment length in HQ3 than minimap2 alignment. +Table 2: Comparison for the percentage of well-aligned reads onto genome, and slope of the +regression line (for normalized edit distance comparison plot of HQ3 vs minimap2 alignments) +with randomly sampled reads for each datasets. The slope of the regression line shows the +average gain in the normalized edit distance. +Dataset (No. +of sampled +reads) +Method of +alignment +Percentage +well- +aligned +reads +Slope +of +regression +line +Intercept +HG002 R9.4.1 reads to +CHM13 (50k) +minimap2 +85.64 +0.7940 +0.0206 +HQ3 +89.35 +HG002 R9.4.1 reads to +GRCh37 (50k) +minimap2 +83.48 +0.8301 +0.0181 +HQ3 +86.65 +Simulated reads from chr 8 & +X of CHM13 assembly (50k) +minimap2 +81.01 +0.9860 +0.0028 +HQ3 +81.57 +quality (see Methods). HQAlign improves the fraction of well-aligned reads than minimap2 +- in particular, in the HG002 R9.4.1 reads alignment to T2T CHM13 reference, this metric +13 + +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Comparison of normalized edit distance for HG002 ONT reads alignment +to GRCh37 reference genome +3.7% reads well aligned in +HQ3 only +83.0% reads well alignedin +both HQ3 and minimap2 +0.5% reads well aligned in +minimap2 only +Regression line: +slope: 0.82 +intercept: 0.02 +12.8% reads not well +aligned in both HQ3 and +minimap2 +(a) +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Comparison of normalized alignment length for HG002 ONT reads alignment to +GRCh37 reference genome +83.0% reads well alignedin +both HQ3 and minimap2 +Regression line: +slope: 0.66 +intercept: 0.33 +0.5% reads well aligned in +minimap2 only +12.8% reads not well aligned +in both HQ3 and minimap2 +3.7% reads well aligned in +HQ3 only +(b) +Figure 4: HG002 nanopore long DNA reads alignment onto GRCh37 genome. (a) +Comparison of normalized edit distance for HG002 R9.4.1 PromethION reads data. Smaller +values for normalized edit distance is desirable as it represents better alignment. The slope +of the regression line is 0.82 < 1, therefore, representing better alignments with HQ3 than +minimap2 alignments for same reads on average. (b) Comparison of normalized alignment length +for HG002 R9.4.1 PromethION reads data. Normalized alignment length of 1 is desirable as it +represents that entire read is aligned. The majority of the reads are above y = x line representing +longer alignment length in HQ3 than minimap2 alignment. +14 + +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Comparison of normalized edit distance for simulated reads alignment +to T2T CHM13 reference genome +80.4% reads well alignedin +both HQ3 and minimap2 +0.6% reads well aligned in +minimap2 only +Regression line: +slope: 0.99 +intercept: 0.00 +17.8% reads not well +aligned in both HQ3 and +minimap2 +1.2% reads well aligned in +HQ3 only +(a) +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Comparison of normalized alignment length for simulated reads alignment +to T2T CHM13 reference genome +17.8% reads not well aligned +in both HQ3 and minimap2 +1.2% reads well aligned in +HQ3 only +0.6% reads well aligned in +minimap2 only +Regression line: +slope: 0.99 +intercept: 0.00 +80.4% reads well aligned in +both HQ3 and minimap2 +(b) +Figure 5: Simulated nanopore reads alignment onto T2T CHM13 genome. (a) Com- +parison of normalized edit distance for simulated nanopore reads data. Smaller values for nor- +malized edit distance is desirable as it represents better alignment. The slope of the regression +line is 0.99 < 1, therefore, representing marginally better alignments with HQ3 than minimap2 +alignments for same reads on average. (b) Comparison of normalized alignment length for sim- +ulated nanopore reads data. Normalized alignment length of 1 is desirable as it represents a +contiguous alignment of the entire read. +improves to 89.35% from 85.64%, and for the alignments to GRCh37 reference, this metric +improves to 86.65% from 83.48%. +Furthermore, there are 310, 036 reads with at-least 1kb +additional bases aligned using HQAlign compared to minimap2 alignments for T2T CHM13 +reference, and there are 299, 896 reads with at-least 1kb additional bases aligned using HQAlign +compared to minimap2 for GRCh37 reference. +The results in Figure 3 and 4 compares the quality of the alignments using minimap2 and +HQAlign at a fine-grained level for HG002 ONT reads alignment to T2T CHM13 genome and +GRCh37 genome, respectively. Figure 3a and 4a compares the normalized edit distance for +HQAlign and minimap2. The normalized edit distance is the edit distance between the entire +read and the aligned section on the genome normalized by the length of the read, in nucleotide +domain for both minimap2 alignment and quantized alignment (HQ3). In case of HQ3, the +information of the location of the alignment on the genome is leveraged from the quantized +read and the quantized genome alignment, and the edit distance is computed between the +15 + +corresponding nucleotide read and the aligned region on the nucleotide genome (see Methods +for details). Intuitively, normalized edit distance gives a measure of how close the two sequences +are. Therefore, the smaller the normalized edit distance, better is the alignment. +Figure 3a shows that for alignments of the reads to T2T CHM13 reference, the normalized +edit distance is on average smaller for HQ3 alignments than minimap2 alignments. The better +alignment in HQ3 is also evident from the slope of the regression line in Figure 3a. It shows that +on average HQ3 alignments has 21% improvement in terms of the normalized edit distance than +the minimap2 alignments. Well aligned reads in both HQ3 and minimap2 are represented by +blue circles in Figure 3, well aligned reads in HQ3 only are represented in black asterisks, well +aligned in minimap2 only are represented in green diamonds and reads that are not well aligned +in both are represented in grey squares. Further, it is important to note that for normalized +edit distance less than 0.1, the alignments are marginally better in the DNA space, but for +normalized edit distance higher than 0.1, the alignments are significantly better in HQ3 space, +especially, the 4% reads that are well aligned in HQ3 and not well aligned in minimap2. This +is because of higher contiguity of alignments in HQ3 space and signifies the improvement by +HQ3 when the error rates is higher. For alignments to GRCh37 reference, HQ3 has an average +improvement of 17%, as shown in Figure 4a. +The results for another fine-grained metric are shown in Figure 3b and 4b, which compares +the normalized alignment length in HQ3 to the normalized alignment length in minimap2 +alignments. The normalized alignment length is the ratio of the length of the section on genome +where a read aligns to the length of the read. In Figure 3b, there are 4% reads that are well- +aligned in HQ3 only, and the normalized alignment length is close to 1 in HQ3 but it is much +less than 1 in minimap2, therefore representing several non-contiguous alignments in nucleotide +domain that are captured as contiguous alignment in HQ3. In Figure 4b, there are 3.7% that +are well-aligned in HQ3 only. +We have also benchmarked the performance of HQAlign with the simulated reads data and +compared its alignment performance with minimap2 in Figure 5. The ONT reads are simulated +from chromosome 8 and X of CHM13 T2T assembly using nanosim [20] with a coverage of +16 + +40x, median and mean read length 4.5 kb and 14 kb, respectively. The results shows that the +alignment performance of both HQAlign and minimap2 are at par with each other. +3.2 +SV calling +3.2.1 +Dataset +Long read sequencing plays an important role in detecting structural variations. We evaluated +SV detection using minimap2 and HQAlign with Sniffles2 as the variant calling algorithm on +both real and simulated data. We simulated 2000 INDELS and 200 Inversion SVs on chromo- +some 8 and X of T2T CHM13 reference genome using SURVIVOR [21] with SV length uniformly +distributed between 50 and 10000, and the ONT reads are simulated using nanosim with av- +erage length of 14k, median length of 4.5k and maximum length 2.5Mbp at a coverage of 40x. +We have used Truvari to benchmark the calls against the truth set. For real data alignment +with GRCh37 as the reference genome, the SV calls are compared against the ground truth sets +from (1) Genome In A Bottle (GIAB) Tier 1 calls [18] and (2) another truth set is constructed +by comparing the haplotype-resolved assembly of HG002 against GRCh37 reference genome +using dipcall [22]. For T2T CHM13 reference genome, since the ground truth for SVs is not +available, we have constructed the truth set by comparing the haplotype-resolved assembly of +HG002 against CHM13 reference using dipcall. However, it is hard to establish ground truth +for the SV calls that are made in the centromere regions, even though the assembly is likely to +be correct. Therefore, we have provided both the analysis including the SV calls in centromere +regions (in Figures 6 and 11) and the analysis for SV calls excluding the centromere regions (in +Figures 10 and 14). +3.2.2 +SV calling results +The standalone performance of both HQ3 and minimap2 is at par with each other across +different references and truth set used in this study for real data as well as for the simulated +data in terms of the F1 score. However, both HQ3 and minimap2 detects complementary SV +17 + +Minimap2 +HQ3 +True positives +890 +(5.8%) +1039 +(6.7%) +14,506 +15,396 +(57%) +15,545 +(58%) +Out of 890 calls in minimap2: +461: calls are captured by HQ3 at low SV length similarity +429: unique region calls +Out of 1039 calls in HQ3: +358: calls are captured by minimap2 at low SV length similarity +681: unique region calls +(a) +Minimap2 +HQ3 +False positives +1595 +2044 +3275 +4870 +5319 +(b) +Figure 6: Comparison of SV calls from HQ3 and minimap2 with HG002-to-CHM13 +dipcall as truth set. (a) Comparison of true positive calls. (b) Comparison of false positive +calls. +Minimap2 +HQ3 +True positives +105 +(1.2%) +103 +(1.2%) +8,845 +8,950 +(94%) +8,948 +(94%) +Out of 105 calls in minimap2: +51: calls are captured by HQ3 at low SV length similarity +54: unique region calls +Out of 1039 calls in HQ3: +41: calls are captured by minimap2 at low SV length similarity +62: unique region calls +(a) +Minimap2 +HQ3 +False positives +234 +216 +366 +600 +582 +(b) +Figure 7: Comparison of SV calls from HQ3 and minimap2 with Genome in a Bottle +(GIAB) Tier 1 truth set for GRCh37 build. (a) Comparison of true positive calls. (b) +Comparison of false positive calls. +Minimap2 +HQ3 +True positives +861 +(5.2%) +703 +(4.3%) +15,759 +16,620 +(76%) +16,462 +(74%) +Out of 861 calls in minimap2: +524: calls are captured by HQ3 at low SV length similarity +337: unique region calls +Out of 1039 calls in HQ3: +376: calls are captured by minimap2 at low SV length similarity +327: unique region calls +(a) +Minimap2 +HQ3 +False positives +1679 +1472 +3077 +4756 +4549 +(b) +Figure 8: Comparison of SV calls from HQ3 and minimap2 with HG002-to-GRCh37 +dipcall as truth set. (a) Comparison of true positive calls. (b) Comparison of false positive +calls. +18 + +Minimap2 +HQ3 +True positives +17 +6 +2125 +2142 +(97%) +2131 +(97%) +(a) +Minimap2 +HQ3 +False positives +2 +10 +6 +8 +16 +(b) +Figure 9: Comparison of SV calls from HQ3 and minimap2 with simulated data. (a) +Comparison of true positive calls. (b) Comparison of false positive calls. +Minimap2 +HQ3 +True positives +763 +(5.8%) +641 +(4.9%) +12,495 +13,258 +(75%) +13,136 +(74%) +Out of 763 calls in minimap2: +83: calls are captured by HQ3 at low SV length similarity +680: unique region calls +Out of 641 calls in HQ3: +77: calls are captured by minimap2 at low SV length similarity +564: unique region calls +(a) +Minimap2 +HQ3 +False positives +995 +974 +1991 +2986 +2965 +(b) +Figure 10: Comparison of SV calls from HQ3 and minimap2 with HG002-to-CHM13 +dipcall as truth set and excluding the calls from the centromere regions. (a) Com- +parison of true positive calls. (b) Comparison of false positive calls. +calls most likely in the repeat regions where accurate alignment is difficult and therefore, leads +to many broken calls. +The analysis with comparison of SV calls from HQ3 and minimap2 with GIAB Tier 1 truth +set gives a precision, recall and F1 score of 0.94, 0.94, and 0.94, respectively for both minimap2 +and HQ3. A union model of minimap2 and HQ3 can improve the recall rate at the same +F1 score, and the union model has a precision, recall, and F1 score of 0.93, 0.95, and 0.94, +respectively. Moreover, out of 103 SV calls that are made by HQ3 only (Figure 7), 41 calls are +made by minimap2 alignments at a lower SV length similarity and 62 calls are unique region +calls. Out of 105 SV calls made by minimap2 only, 51 are captured by HQ3 at a lower SV length +similarity and 54 are unique region calls. HQ3 improves the breakpoint accuracy for 14.11% +19 + +calls that have higher than 50 difference in breakpoints and it improves the length similarity of +19.97% calls that have SV length similarity lower than 0.95 (Figure 12). +We have compared the SV calls made by HG002 reads against T2T CHM13 reference genome +using both minimap2 and HQ3 and benchmarking them against truth set generated by compar- +ing HG002 haplotype-resolved assembly to T2T CHM13 assembly. The standalone performance +have precision, recall and F1 score of 0.77, 0.57 and 0.66, respectively for minimap2 and 0.75, +0.58 and 0.65, respectively for HQ3. However, because of high number of complementary true +positive calls in minimap2 and HQ3, the union model has a significant improved recall at the +same F1 score with precision, recall and F1 score of 0.71, 0.61 and 0.66, respectively. Out of +1039 (6.7%) calls that are made in HQ3 only, 358 are captured by minimap2 at a lower SV +length similarity threshold and 681 are unique calls, whereas out of 890 (5.8%) calls that are +made by minimap2 only, 461 are captured by HQ3 at a lower SV length similarity threshold +and 429 are unique (as shown in Figure 6a). Further, for the common true positive calls in +both minimap2 and HQ3, we observe a similar pattern as the other datasets in improvement +of breakpoint accuracy with HQ3 for 18.66% calls that have difference in breakpoint greater +than 50, and improvement in SV length similarity for 19.76% calls with similarity less than 0.95 +(Figure 11a-b). +For SV calls from HG002 reads alignment to GRCh37 and benchmarking them against truth +set generated by comparing HG002 haplotype-resolved assembly to GRCh37 build, minimap2 +has precision 0.78, recall 0.76, and F1 score 0.77 while HQ3 has precision 0.79, recall 0.75, +and F1 score 0.77. Out of 16462 true positive calls in HQ3, 703 (4.27%) are made only in +HQ3 with SV length similarity to the truth set greater than 0.7 (default parameter in Truvari). +However, 376/703 calls that are captured by minimap2 with SV length similarity less than +0.7 and 327/703 calls that are uniquely made by HQ3. Likewise, out of 16620 true positive +calls in minimap2, 861 (5.18%) are made only in minimap2 with SV length similarity greater +than 0.7. However, 524/861 are captured by HQ3 with SV length similarity less than 0.7 and +337/861 are uniquely made by minimap2. A fine-grain analysis of the common true positive +calls by minimap2 and HQ3 in Figure 13a, shows that a major density of SV calls (81.85%) have +20 + +Table 3: Comparison for precision, recall and and F1 score for SV calls made by HQ3, minimap2, +and the Union model. +Dataset +Truth set +Method +of +align- +ment +Precision Recall +F1 score +HG002 reads +to GRCh37 +GIAB Tier 1 calls +minimap2 +0.94 +0.94 +0.94 +HQ3 +0.94 +0.94 +0.94 +Union +0.93 +0.95 +0.94 +HG002 reads +to CHM13 +comparing HG002 assembly +to CHM13 (including +centromere calls) +minimap2 +0.77 +0.57 +0.66 +HQ3 +0.75 +0.58 +0.65 +Union +0.71 +0.61 +0.66 +HG002 reads +to CHM13 +comparing HG002 assembly +to CHM13 (excluding +centromere calls) +minimap2 +0.82 +0.75 +0.78 +HQ3 +0.82 +0.74 +0.78 +Union +0.78 +0.79 +0.78 +HG002 reads +to GRCh37 +comparing HG002 +assembly to GRCh37 +minimap2 +0.78 +0.76 +0.77 +HQ3 +0.79 +0.74 +0.77 +Union +0.74 +0.79 +0.77 +Simulated +reads to +CHM13 +Simulated SVs on chr 8 +and X +minimap2 +0.99 +0.97 +0.98 +HQ3 +0.99 +0.97 +0.98 +Union +0.99 +0.98 +0.98 +difference in breakpoint below 50 in both minimap2 and HQ3, and minimap2 has marginally +better performance in terms of lower difference in breakpoint of SVs for difference in breakpoint +below 50. +Whereas, for a large difference in the SV breakpoint (greater than 50), HQ3 is +better in terms of the breakpoint accuracy of the SV calls (on average across all SV calls). +Therefore, HQ3 improves the SV breakpoint for the rest 18.15% calls that have high difference +in breakpoints. Further, Figure 13b demonstrate that HQ3 has better SV length similarity +when the length similarity is below 0.95 which corresponds to 21.82% calls. +4 +Discussion +HQAlign method is an alignment method designed for the detection of structural variants for +nanopore sequencing reads. HQAlign provides alignment that outperforms the recent minimap2 +aligner in terms of the accuracy and quality of the alignments. The SV calling from HQAlign is +also at par with minimap2 in terms of F1 score and it outperforms minimap2 SV calls in terms +21 + +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000 +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000 +Comparison of average breakpoint difference for T2T CHM13 truth set +(including the centromere regions) +2000 +4000 +6000 +8000 +10000 +12000 +Regression line: +slope: 0.58 +intercept: 19.66 +(a) +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +Comparison of SV length similarity for T2T CHM13 truth set +(including the centromere regions) +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +4500 +Regression line: +slope: 0.76 +intercept: 0.23 +(b) +Figure 11: SV quality comparison for common true positive calls in HQ3 and min- +imap2 against HG002-to-CHM13 dipcall truth set. (a) Comparison of SV breakpoint +accuracy in HQ3 and minimap2 for common true positive calls. The difference of SV breakpoint +is compared to the truth set generated from comparing HG002 haplotype-resolved assembly to +T2T CHM13 build. A smaller difference represents better breakpoint accuracy. Therefore, slope +of the regression line 0.58 < 1 represents better accuracy of HQ3 than minimap2 on average. +(b) Comparison of SV length similarity in HQ3 and minimap2 for common true positive calls. +The slope of the regression line 0.76 < 1 represents better SV length in minimap2 than HQ3 +on average, but the intercept is high (0.23). However, this is due to a large density of SVs with +length similarity ≥ 0.95 in both minimap2 and HQ3. For length similarity less than 0.95, HQ3 +has better performance than minimap2. +of the quality of SVs measured in breakpoint accuracy and SV length similarity. Moreover, there +are many complementary SVs captured by HQAlign that are missed by minimap2 alignments. +The reason for this improvement in the performance of alignment and SV calling with +HQAlign is that it takes into account the underlying physics of nanopore sequencer through +the Q-mer map, which could be one of the major causes of the high error rates in nanopore +sequencing, and also it focuses on a narrow region of the genome (where the read aligns in +nucleotide domain) for alignment with quantized sequences. Further, this pipeline is adapted +specifically for the detection of SVs. We demonstrated how HQAlign utilizes the bias of Q- +mer map without accessing the raw current signal of nanopore sequencer by translating the +basecalled nucleotide sequences to quantized current level (of finite alphabet size) sequences. +This improvement help in detecting several SVs that are missed by minimap2 due to high error +22 + +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000 +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000Comparison of average breakpoint difference for GIAB Tier 1 truth set +1000 +2000 +3000 +4000 +5000 +6000 +Regression line: +slope: 0.89 +intercept: 7.92 +(a) +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +Comparison of SV length similarity for GIAB Tier 1 truth set +500 +1000 +1500 +2000 +2500 +Regression line: +slope: 0.84 +intercept: 0.16 +(b) +Figure 12: SV quality comparison for common true positive calls in HQ3 and min- +imap2 against GIAB Tier 1 truth set. +(a) Comparison of SV breakpoint accuracy in +HQ3 and minimap2 for common true positive calls. The difference of SV breakpoint is com- +pared to the GIAB Tier 1 truth set. A smaller difference represents better breakpoint accuracy. +Therefore, slope of the regression line < 1 represents better accuracy of HQ3 than minimap2 +on average. (b) Comparison of SV length similarity in HQ3 and minimap2 for common true +positive calls. The slope of the regression line < 1 represents better SV length in minimap2 than +HQ3 on average. However, this is due to a large density of SVs with length similarity ≥ 0.95 +in both minimap2 and HQ3. For length similarity less than 0.95, HQ3 has better performance +than minimap2. +rates in the nanopore reads. +Further, the recall rate for SV detection can be improved by +combining the complementary calls from both HQ3 and minimap2 in the union model at the +same F1 score. +Competing interests +The authors declare that they have no competing interests. +Author’s contributions +DJ, SK, and SD conceived the original idea and developed the project. DJ led the development +of the software tool and its open-source development. MC helped with SV metrics, datasets, and +23 + +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000 +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000 Comparison of average breakpoint difference for GRCh37 truth set +2000 +4000 +6000 +8000 +10000 +12000 +Regression line: +slope: 0.54 +intercept: 17.86 +(a) +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +Comparison of SV length similarity for GRCh37 truth set +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +Regression line: +slope: 0.77 +intercept: 0.22 +(b) +Figure 13: SV quality comparison for common true positive calls in HQ3 and min- +imap2 against HG002-to-GRCh37 dipcall truth set. (a) Comparison of SV breakpoint +accuracy in HQ3 and minimap2 for common true positive calls. The difference of SV breakpoint +is compared to the truth set generated from comparing HG002 haplotype-resolved assembly to +GRCh37 build. A smaller difference represents better breakpoint accuracy. Therefore, slope +of the regression line < 1 represents better accuracy of HQ3 than minimap2 on average. (b) +Comparison of SV length similarity in HQ3 and minimap2 for common true positive calls. The +slope of the regression line < 1 represents better SV length in minimap2 than HQ3 on average. +However, this is due to a large density of SVs with length similarity ≥ 0.95 in both minimap2 +and HQ3. For length similarity less than 0.95, HQ3 has better performance than minimap2. +SV comparison analysis between methods. DJ performed the analysis on the various datasets +for both alignment and SV calling. All the authors wrote the manuscript. +Acknowledgements +SD and DJ were supported in part by National Science Foundation grant 1705077. MC was +supported by National Institutes of Health grant R01HG011649. SK was supported in part +by National Institutes of Health grant 1R01HG008164 and National Science Foundation grants +1651236 and 1703403. +24 + +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000 +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000 +Comparison of average breakpoint difference for T2T CHM13 truth set +(excluding calls from the centromere regions) +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +9000 +Regression line: +slope: 0.52 +intercept: 20.58 +(a) +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +Comparison of SV length similarity for T2T CHM13 truth set +(excluding the centromere regions) +500 +1000 +1500 +2000 +2500 +3000 +3500 +Regression line: +slope: 0.75 +intercept: 0.24 +(b) +Figure 14: SV quality comparison for common true positive calls in HQ3 and min- +imap2 against HG002-to-CHM13 dipcall truth set (excluding the centromere re- +gion). (a) Comparison of SV breakpoint accuracy in HQ3 and minimap2 for common true +positive calls. The difference of SV breakpoint is compared to the truth set generated from +comparing HG002 haplotype-resolved assembly to T2T CHM13 build and the SV calls in the +centromere region are excluded in this truth set. A smaller difference represents better break- +point accuracy. Therefore, slope of the regression line < 1 represents better accuracy of HQ3 +than minimap2 on average. (b) Comparison of SV length similarity in HQ3 and minimap2 for +common true positive calls. The slope of the regression line < 1 represents better SV length +in minimap2 than HQ3 on average. However, this is due to a large density of SVs with length +similarity ≥ 0.95 in both minimap2 and HQ3. For length similarity less than 0.95, HQ3 has +better performance than minimap2. +References +[1] Alkan, C., Coe, B. P. & Eichler, E. E. Genome structural variation discovery and genotyp- +ing. Nature reviews genetics 12, 363–376 (2011). +[2] Pan-cancer analysis of whole genomes. 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Nature +methods 15, 595–597 (2018). +27 + +A +Supplementary material +A.1 +Quantization method from QAlign [13] +The nucleotide sequences are inferred from the nanopore current signals by basecallers, there- +fore, using a Q-mer map to translate the basecalled sequences to the current levels implicitly +maintains all of the “equivalent” basecalled sequences that could be inferred from the observed +current levels. These current levels can be quantized to an alphabet of finite size. +Mathematically, the quantization process is as follows. Let Σ = {A, C, G, T} be the alphabet +of nucleotide sequences. For a symbol s ∈ Σ, let ¯s be the Watson-Crick complement of s. A +string x = s1s2 . . . sn over Σ is called a nucleotide sequence, where |x| = n is the string length +and the reverse complement of x is x = s1s2 . . . sn = snsn−1 . . . s1. Let p(x) be a list of all Q-mers +(e.g. Q=6) in the string x, sorted by their occurrences. For example, p(x) = k1k2 . . . kn−Q+1 +and each Q-mer ki = sisi+1 . . . si+Q−1 for i = 1, 2, . . . , n − Q + 1. Now, we define f : ΣQ → R +as the Q-mer map 1, which is a deterministic function that translates each Q-mer (ki) to the +(median) current level (Figure 1b). Now, let C(x) = c1c2 . . . cn−Q+1 be the sequence of the +current levels, so that ci = f(ki) for i = 1, 2, . . . , n − Q + 1. The current sequence C(x) can be +further quantized into w(x) = q1q2 . . . qn−Q+1 by applying hard thresholding function qi = g(ci). +The thresholding can be ternary (qi ∈ {0, 1, 2}) for HQ3 (Figure 1c and Supplemental Figure +15). We define w(x) as the quantized reverse complementary of sequence x, so w(x) = w(x). +Supplemental Figure 15 explains this process using a toy example. +A.2 +Generalization of HQAlign method +A.2.1 +Initial alignment +The nucleotide query x is aligned to a set of nucleotide target sequences t = (t1, t2, . . . , tm) +using Minimap2. This is similar to aligning a read to a genome which has several chromosome +1Q-mer map is determined by the chemistry of the nanopore flow cell, and is therefore dataset dependent, +i.e., the Q-mer map for sequencing using R9 flow cell is different from Q-mer map for sequencing using R9.4.1 +flow cell. The Q-mer maps used in this work are generated by Nanopolish (https://github.com/jts/nanopolish). +28 + +sequences. This step identifies the region of interests on the target t, say, tj[si : ei], where +tj, j ∈ {1, 2, . . . , m} represent alignment to one or more target chromosomes that x aligns to, +i ∈ {1, 2, 3, . . . } represent represent one or more alignments to chromosome j, si and ei are the +corresponding start and end location of each alignment i on the target tj, respectively. +A.2.2 +Hybrid alignment +In this step, the query x is re-aligned to an extended region of interest on the target tj[sq +i : eq +i] +using the QAlign method, where sq +i = si − bi and eq +i = ei + bi, bi = (1 − fi + 0.25)n is +an appended extension of the region of interest on target, fi = (ei − si)/n is the fraction of +read aligned in initial step, and n is the length of the query x. The nucleotide query x and +the nucleotide extended target tj[sq +i : eq +i] are converted to the quantized query xq, quantized +reverse complement query xq and quantized extended target tq +j[sq +i : eq +i], respectively, using the +quantization method demonstrated in QAlign. +These quantized sequences are then aligned +using modified minimap2 pipeline. +29 + +ACGTAACGTATTG +[ ACGTAA , CGTAAC , GTAACG , TAACGT , +AACGTA , ACGTAT , CGTATT , GTATTG ] +[ 82.13 , 104.67 , 81.75 , 88.27 , +97.85 , 80.16 , 104.75 , 77.26 ] +12011020 +Nucleotide seq +List of 6-mers +𝑄-mer map +𝑄-mer map translates 6-mers to median +current value +List of current levels +Q3 seq +Hard thresholding: +(current level) 58 – 82 à 0 (quantize level) +(current level) 82 – 98 à 1 (quantize level) +(current level) 98 – 120 à 2 (quantize level) +Figure 15: An example for the Quantization method for QAlign. The nucleotide se- +quences are first translated to current level sequences using the Q-mer map, and then the +(continuous) current level sequences are quantized to finite levels (e.g. three levels for HQ3) by +hard thresholding the current levels. +A.3 +Accessing HQAlign on github +HQAlign requires python 3, and the installation guideline can be found on github. +The software is available at: https://github.com/joshidhaivat/HQAlign.git +usage: +python hqalign.py [-h] -r REF -i READS -o OUTPUT [-t THREADS] [-k +KMER] +arguments: +-h, --help +show this help message and exit +-r REF, --ref REF +reference genome filename in fasta format +-i READS, --reads READS +directory location of read files in fasta format (with file +extension .fasta) +-o OUTPUT, --output OUTPUT +location of directory of output files +-t THREADS, --threads THREADS +maximum number of parallel threads (default=4) +-k KMER, --kmer KMER +minimizer length for hybrid step (default=18) +30 + +Minimap2 (v2.24) +Reads data (Nucleotide) +Reference Genome (Nucleotide) +Sort & merge alignments using samtools +Sniffles2 +Miniamp2 SV callsets in vcf format +Minimap2 alignments +(in sam & paf format) +Alignments in bam format +Quantization to three +levels +HQAlign with modified +Minimap2 (v2.24) +HQ3 SV callsets in vcf format +Q3 reads +Q3 reference +HQAlign pipeline +Figure 16: Complete pipeline for SV calling using minimap2 and HQAlign. +31 + +0 +1 +2 +3 +4 +5 +6 +7 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Comparison of minimap2 SV calls to same calls in HQ3 +Unique calls in minimap2 +Common calls in HQ3 and minimap2 +412 minimap2 calls that are +captured in HQ3 at low SV +length similarity +397 minimap2 calls +in unique region +(a) +0 +1 +2 +3 +4 +5 +6 +7 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Comparison of HQ3 SV calls to same calls in minimap2 +Unique calls in HQ3 +Common calls in HQ3 and minimap2 +329 HQ3 calls that are captured in +minimap2 at low SV length +similarity +539 HQ3 calls in +unique region +(b) +Figure 17: Comparison of SV calls made by minimap2 and HQ3 to other method. +(a) For the complementary calls (in blue) and common calls (in red) made by minimap2, this +figure compares SV length similarity and distance to nearest SV in HQ3 of the same type. 397 +complementary calls made by minimap2 are in unique region, whereas 412 complementary calls +in minimap2 are captured in neighboring region (within 1000 bp) in HQ3 but with a low SV +length similarity. (b) For the complementary calls (in blue) and common calls (in red) made +by HQ3, this figure compares SV length similarity and distance to nearest SV in minimap2 +of the same type. 539 complementary calls made by HQ3 are in unique region, whereas 329 +complementary calls in HQ3 are captured in neighboring region (within 1000 bp) in minimap2 +but with a low SV length similarity. +32 + +0 +1 +2 +3 +4 +5 +6 +7 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Comparison of minimap2 SV calls to same calls in HQ3 +Unique calls in minimap2 +Common calls in HQ3 and minimap2 +445 minimap2 calls that are +captured in HQ3 at low SV +length similarity +290 minimap2 calls +in unique region +(a) +0 +1 +2 +3 +4 +5 +6 +7 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Comparison of HQ3 SV calls to same calls in minimap2 +Unique calls in HQ3 +Common calls in HQ3 and minimap2 +347 HQ3 calls that are captured in +minimap2 at low SV length +similarity +179 HQ3 calls in +unique region +(b) +Figure 18: Comparison of SV calls to HG002 truth set. (a) For the complementary +calls (in blue) and common calls (in red) made by minimap2, this figure compares SV length +similarity and distance to nearest SV in HQ3 of the same type. 290 complementary calls made +by minimap2 are in unique region, whereas 445 complementary calls in minimap2 are captured +in neighboring region (within 1000 bp) in HQ3 but with a low SV length similarity. (b) For the +complementary calls (in blue) and common calls (in red) made by HQ3, this figure compares SV +length similarity and distance to nearest SV in minimap2 of the same type. 179 complementary +calls made by HQ3 are in unique region, whereas 347 complementary calls in HQ3 are captured +in neighboring region (within 1000 bp) in minimap2 but with a low SV length similarity. +33 + diff --git a/SNE2T4oBgHgl3EQfWgd-/content/tmp_files/load_file.txt b/SNE2T4oBgHgl3EQfWgd-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..87b3b4e4b5c30aeb972f96ddcc77ec3cc90de87f --- /dev/null +++ b/SNE2T4oBgHgl3EQfWgd-/content/tmp_files/load_file.txt @@ -0,0 +1,1059 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf,len=1058 +page_content='HQAlign: Aligning nanopore reads for SV detection using current-level modeling Dhaivat Joshi, Suhas Diggavi∗, Mark J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Chaisson, and Sreeram Kannan † Abstract Motivation: Detection of structural variants (SV) from the alignment of sample DNA reads to the reference genome is an important problem in understanding human diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Long reads that can span repeat regions, along with an accurate alignment of these long reads play an important role in identifying novel SVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Long read sequencers such as nanopore sequencing can address this problem by providing very long reads but with high error rates, making accurate alignment challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Many errors induced by nanopore sequencing have a bias because of the physics of the sequencing process and proper utiliza- tion of these error characteristics can play an important role in designing a robust aligner for SV detection problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' In this paper, we design and evaluate HQAlign, an aligner for SV detection using nanopore sequenced reads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The key ideas of HQAlign include (i) using basecalled nanopore reads along with the nanopore physics to improve alignments for SVs (ii) incorporating SV specific changes to the alignment pipeline (iii) adapting these into existing state-of-the-art long read aligner pipeline, minimap2 (v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='24), for efficient align- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Results: We show that HQAlign captures about 4 − 6% complementary SVs across dif- ferent datasets which are missed by minimap2 alignments while having a standalone per- formance at par with minimap2 for real nanopore reads data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' For the common SV calls ∗SD is corresponding author: suhas@ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='edu †DJ and SD are at the University of California, Los Angeles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' MC is at the Department of Quantitative and Computational Biology, University of Southern California, Los Angeles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' SK is at the University of Washington, Seattle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='03834v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='GN] 10 Jan 2023 between HQAlign and minimap2, HQAlign improves the start and the end breakpoint ac- curacy for about 10 − 50% of SVs across different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Moreover, HQAlign improves the alignment rate to 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='35% from minimap2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='64% for nanopore reads alignment to recent telomere-to-telomere CHM13 assembly, and it improves to 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='65% from 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='48% for nanopore reads alignment to GRCh37 human genome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Availability: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='com/joshidhaivat/HQAlign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='git 1 Introduction Structural variations (SVs) are genomic alterations of size at least 50 bp long, including in- sertions, deletions, inversions, duplications, translocations or a combination of these types [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The study of these genetic variations has an important role in understanding human diseases, including cancer [2], and begins with sequence alignment from the sample back to the reference genome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Accurate alignment of short reads from high throughput sequencing poses a challenge, especially, in the repetitive regions of the genome which are also the hotspots of nearly 70% of the observed structural variations [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Long read sequencing technologies have addressed this problem by producing reads that are longer than the repeat regions, therefore, enabling the detection of variants in the repeat regions at the cost of higher error rates than short read sequencing technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This high error rates in the long reads lead to non-contiguous alignment which poses a challenge in variant detection problem, especially, in the repeat regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Nanopore sequencing [4, 5] is a long read sequencing technology that provides reads (with average read length 10-kb and the longest read sequenced more than 2-Mb long) that can span these repetitive regions but it has a high error rate of (average) 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This high error rates result in low accuracy alignments [6] using state-of-the-art methods including minimap2 (v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='24) [7] which is a fast method designed for the computationally challenging task of long sequence alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This problem is further amplified in the repetitive regions such as variable- number tandem repeats (VNTR) region that accounts for a significant fraction of SVs [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 2 DNA sequence Nanopore channel Current signals Basecaller Basecalled sequence (Reads) G C A T G A C A G G C G G C A A C C G A Sequence Aligner Quantizer Edit distance = 6 Q-mer Map Current levels: 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='32, 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='86, 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='96, 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='53, 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='59 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='50, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='92, 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='91, 85.' 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+page_content='00000000000000000000000000000000000000000000000000000000000000000000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='00000000000000000000000000000000000000000000000000000000000000000000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='00000000000000000000000000000000000000000000000000000000000000000000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='00000000000000000000000000000000000000000000000000000002100000000000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='0000000000000000000000000000000000000000000000002100000000000000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='Edit distance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='= 66 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='Edit distance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='= 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='Figure 1: (a) An example to illustrate the error biases in nanopore basecalled reads which can ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='be resolved through the Q-mer map ability of HQAlign to perform accurate alignment despite ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='of the errors (the edit distance used here is domain specific and is used to demonstrate accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='of the alignment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (b) Q-mer map for Nanopore R9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='4 1D flow cell (for Q = 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' It represents the physics of nanopore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The median current value along with the standard deviation (as error bars) are plotted for all 6-mers in the Q-mer map for R9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='4 1D nanopore flow cell (the Q-mers are sorted in increasing median current levels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Note that the difference between the median current levels of any two consecutive Q-mers is very small, therefore, resulting in large overlaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (c) An example from PromethION R9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 ONT data in the neighborhood of a SV in repeat region showing the two different nucleotide sequences have similar current levels and therefore, the edit distance as observed through the lens of quantized sequence is significantly lower in HQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' However, these errors in nanopore sequencing have a bias induced from nanopore physics which is missed by many long read aligners since they consider the errors as independent insertions, 3 Current Signal 115 Current signal Current level 110 105 100 95 90 85 80 75 0 200 400 600 800 1000 1200+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='777e7 Read[33l=SRR13062494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='345919 Minimap2 HQ3 9000 8500 8000 Reference 7500 7000 6500 6000 5500 0 1000 2000 3000 4000 5000 Readdeletions, and substitutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' In nanopore sequencing, a DNA strand migrates through the nanopore, and an ionic current according to the nucleotide sequence in or near the nanopore is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' However, because of the physics and non-idealities of the nanopore sequencing, each current level recorded depends on a Q-mer (a set of Q consecutive nucleotide bases which influence the measurement in the nanopore) [10, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' These current readings are translated back to nucleotide sequences by basecalling algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Therefore, the error biases could be introduced in basecalling, especially, between different Q-mers that have similar current levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This similarity in the median current levels for different Q-mers is captured by the Q-mer map as shown in Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' A Q-mer map represents the median current level for different Q-mers (Q = 6) for nanopore flow cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' It is evident from this figure that there is a significant overlap between the current levels observed for different Q-mers migrating the nanopore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' We propose a new alignment method, HQAlign (which is based on QAlign [13]), which is designed specifically for detecting SVs while incorporating the error biases inherent in the nanopore sequencing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' HQAlign pipeline is modified to enable detection of inversion variants which was not feasible with the earlier QAlign pipeline (refer to methods section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' HQAlign takes the dependence of Q-mer map into account to perform accurate alignment with modifications specifically for discovery of SVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Figure 1a gives an example where a DNA sequence (GCATGACAGG) is sequenced incorrectly as (CGGCAACCGA) due to the error bias in nanopore sequencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Therefore, the sequences are different in the nucleotide space but they are identical in the Q-mer map space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' It is important to note that no additional soft information is used to establish this identity such as raw nanopore current values for the nanopore reads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Instead, the nucleotide sequences that have indistinguishable current levels from the lens of the Q-mer map are mapped to a common quantized sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' A nucleotide sequence is converted to a quantized sequence by first converting the nucleotide sequence to a sequence of current levels using the Q-mer map and then converting the sequence of (continuous) current levels to a (finite level) quantized sequence by hard thresholding the current levels (refer to Supplementary material section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Therefore, the additional information about the raw current signals is not used in the quantization process but only the Q-mer map is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This 4 process is explained in detail in Supplementary material Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Further, the quantization of continuous current levels to finite discrete levels enables the use of existing software pipelines of state-of-the-art long read aligners such as minimap2 as the core seed and extend algorithm for the alignment of quantized sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' In HQAlign, we first perform the alignment of reads onto genome using minimap2 to deter- mine the region of interest where a read can possibly align to, and then re-align the quantized read to the quantized genome region from the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This helps in performing an accu- rate alignment of the read to the region of genome without dropping the frequently occurring seed matches from the chain in minimap2 algorithm while taking the error biases of nanopore sequencing into account through quantized sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Moreover, HQAlign pipeline enables detection of inversion variants unlike QAlign pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' In QAlign, the quantized reverse com- plement of read is aligned separately to the quantized genome, therefore, the alignment of inverted sequence is not observed in QAlign (as shown in Figure 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' However, in HQAlign, we have modified the minimap2 pipeline to align the reverse complement of quantized read along with the aligning the forward quantized read sequence to the quantized genome, simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This is necessary for detection of inverted alignment using quantized sequences because unlike nucleotide sequences where minimap2 can inherently produce the reverse complement of the input nucleotide query, the reverse complement of the quantized sequence is to be computed separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Further, HQAlign is about 2x faster than QAlign as the seed search domain is reduced to a region of interest determined in the first step of the pipeline for the quantized sequences (as shown in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Figure 1c demonstrate an example from real ONT reads data in a repeat region (note that a pattern of a few consecutive nucleotide bases is repeated in the example) that is flanking around an insertion structural variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Minimap2 alignment of nucleotide reference and read (both of length 356 from the region highlighted with a box) have an edit distance of 66 whereas the HQ3 alignment (HQ3 is an alignment from HQAlign pipeline where the nucleotide sequences are translated to three level quantized sequences, refer to section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2 for details) of quantized reference and read sequences from the same region have a significantly smaller edit distance 5 of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This is because the current level sequence (by converting the nucleotide sequences using the Q-mer map in Figure 1b) for the reference and the read are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Therefore, the sequences that are far apart in nucleotide space are inherently very similar in the HQ3 space in terms of the edit distance in the transformed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' We show that HQAlign gives significant performance improvements in quality of read align- ment across real and simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The well-aligned reads (a read is defined as well aligned if at least 90% of the read is aligned on the genome with a mapping quality more than 20) improves to 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='65% with HQ3 from 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='48% with minimap2 (v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='24) for the alignment of ONT reads from HG002 sample to GRCh37 human genome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The metric improves to 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='35% from 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='64% for HG002 reads alignment to T2T CHM13 assembly [14], and improves to 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='57% from 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='01% for the simulated reads data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' These results are presented in the results section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' In terms of SV detection, HQAlign has F1 score at par with minimap2 (v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='24) with Sniffles2 [15] as the variant calling algorithm across both real and simulated dataset (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' However, both HQAlign and minimap2 captures many complementary calls ( 4 − 6%) which are missed by the other method (as shown in Figure 6, 7, 8, 9, 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' For instance, the complementary HQAlign calls are SVs that are uniquely called by HQAlign or labeled missed in minimap2 due to breaking in the SV and vice-versa for the complementary calls in minimap2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Further,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' the analysis of common true positive SV calls in HQAlign and minimap2 against the truth set shows that HQAlign has on average a significant improvement (10 − 50%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' from the slope of the regression line in Figures 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' and weighted average across all datasets for 39% SVs) in the breakpoint accuracy than minimap2 for the calls with difference in breakpoint greater than 50 bp (breakpoint accuracy is determined from the difference in the start and end breakpoints of a SV with respect to the match SV in truth set,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' lower the difference higher is the breakpoint accuracy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' refer to section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='3 for precise definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Moreover, for the common true positive calls, HQAlign has (on average) better SV length similarity than minimap2 (when SV length similarity is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95, SV length similarity is a measure of how similar is the length of SV from an alignment method relative to the match SV in truth set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' refer to section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='3 for 6 a precise definition) as shown in Figures 11, 12, 13, and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 2 Methods 0 2000 4000 6000 8000 10000 12000 Read 0 2000 4000 6000 8000 10000 12000 Reference Detection of Inversion alignment in HQAlgin vs QAlign QAlign HQAlign (a) Step 1: Initial alignment using minimap2 on Nucleotide sequences Step 2: Hybrid alignment using modified minimap2 on quantized sequences query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='fasta >quan&zed query 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='fasta >quantized 𝑡" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' [𝑠" − 𝑏": 𝑒" + 𝑏"] >quantized 𝑡# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' [𝑠# − 𝑏#: 𝑒# + 𝑏#] target: 𝑡!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' target: 𝑡" query: 𝑥 𝑠!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 𝑠" 𝑒" 𝑒!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 𝑏!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 𝑏!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 𝑛𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 𝑏" 𝑏" 𝑛𝑓" rc_query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='fasta >quantized rc query ̅𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (b) Genome Read 1 Read 2 𝑖!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 𝑗!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 𝑖" 𝑗" 𝑖!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' # 𝑖" # 𝑗!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' # 𝑗" # Aligned section on Genome (c) Genome 𝑖!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' " 𝑗!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' SV in the truth set 𝑖!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 𝑗!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' " SV from alignment method: minimap2/HQ3 (d) Figure 2: (a) An example to demonstrate the ability of HQAlign pipeline to align inverted sequences where QAlign fails (b) An example of HQAlign pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (c) An example of read- to-genome alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (d) Comparison of SV in truth set to SV determined by method: minimap2/HQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The HQAlign strategy consists of two steps: (1) the initial alignment of the standard base- called query sequence x to a target sequence t using Minimap2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This primitive step identifies the region of interests on the target sequence where x aligns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (2) the hybrid step is re-aligning the query x only to the region of interest determined in the first step using a modified pipeline for QAlign method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' QAlign method converts the nucleotide sequences in both query x and target region t to quantized (three levels) sequence, and then aligns these quantized sequences using any state-of-the-art aligner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' We have modified the minimap2 (v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='24) pipeline for the align- ment of the quantized sequences in the second step so that it enables detection of the inversion variants in alignment using the quantized sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This is explained in detail in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Further, this strategy is about 2x faster than QAlign standalone as it narrows down the seed 7 search domain for lower alphabet size (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' three levels) in QAlign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This strategy is explained in Figure 2b, and mathematically in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Table 1: Comparison of computation time for alignment of 500k randomly sampled ONT reads to CHM13 assembly using 20 threads for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Method minimizer (seed) length Real time (in seconds) CPU time (in seconds) QAlign (Q3) 18 10, 312 177, 516 HQAlign (HQ3) 18 5, 033 76, 375 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 Initial alignment The nucleotide query x is aligned to a nucleotide target sequence t using minimap2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This is similar to aligning a read to a genome with one chromosome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Here we consider only one chromosome in target t for simplicity but the method generalizes to multiple chromosomes in t such as t = (t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' , tm) (this generalization is explained in detail in Supplementary material section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This step identifies the region of interests on the target t, say, t[si : ei], where i ∈ {1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' } represent one or more alignments on t and si and ei are the corresponding start and end location of each alignment i on target t, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2 Hybrid alignment In this step, the query x is re-aligned to an extended region of interest on the target t[sq i : eq i] using the modified pipeline for QAlign method, where sq i = si−bi and eq i = ei+bi, bi = (1−fi+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='25)n is an appended extension of the region of interest on target, fi = (ei − si)/n is the fraction of read aligned in initial step, and n is the length of the query x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The nucleotide query x and the nucleotide extended target t[sq i : eq i] are converted to the quantized query xq and quantized extended target tq[sq i : eq i], respectively, using the quantization method demonstrated in QAlign (refer to Supplementary material section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 for more details on quantization process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' It is important to note that we do not use any additional soft information such as raw current signals from nanopore sequencing in the quantization process, instead, we translate the basecalled nu- 8 cleotide reads to current levels using the Q-mer map (in Figure 1b) and then hard threshold the current levels to finite (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' three) levels to get the quantized (HQ3) reads (refer Supplemen- tary material Figure 15 for more details on quantization process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' These quantized sequences are then aligned using a modified pipeline of minimap2 (v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' We have modified minimap2 pipeline for this hybrid step to accept the quantized reverse complement query ¯xq as an input which helps in indentifing the inversion SVs in contiguous alignment with quantized sequences which was not possible with the earlier QAlign method as shown in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' QAlign uses the default minimap2 pipeline for the alignment of quantized sequences which inherently aligns both the forward and the reverse complement strand of the sequences in nucleotide domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' How- ever, the quantized reverse complement sequence cannot be computed given only the forward quantized sequence, therefore, QAlign separately aligns both quantized forward and quantized reverse complement sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This method, however, fails to identify an inverted alignment as shown in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Therefore, in HQAlign, we have modified the minimap2 pipeline to enable alignment using both quantized forward and quantized reverse complement sequence, simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Note that the quantized alignment employs a different minimizer length k = 18 in minimap2 for ternary (HQ3) quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' We define several metrics that are used for the performance evaluation of HQAlign against minimap2 (these metrics are used from the earlier QAlign method [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (i) well-aligned: Consider in Figure 2c, Read 1 aligns at location i1 through j1 on the genome determined using nucleotide alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' We say that the read is well-aligned, if at-least 90% of the read is aligned onto the genome (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=', j1−i1≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='9(length(Read 1))), and has high mapping quality (greater than 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This metric quantifies the reads that are mapped almost entirely to the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (ii) normalized edit distance: In order to compare the quality of the alignments at fine- grained level, we further define normalized edit distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The normalized edit distance 9 for nucleotide alignment is defined as edit distance{r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' G[i1 : j1]} length(r) (1) and for quantized alignment is edit distance{r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' G[iq 1 : jq 1]} length(r) (2) where i1, j1 are the start and end location of alignment on genome in nucleotide space and iq 1, jq 1 are the start and end location of alignment on genome in the quantized space, r is the entire read and G is the genome as shown in Figure 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' It is important to note that for computing the normalized edit distance for alignments in the quantized space, we only leverage the information of location of the alignment on genome from quantized space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' iq 1 and jq 1, but the edit distance between read and the aligned section on genome is computed on the nucleotide sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This metric gives a measure of the distance similarity between two sequences, especially, used for the real data where the truth of sequence sampling location is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (iii) normalized alignment length: Another metric at the fine-grained level is normalized alignment length, which is the ratio of the length of the section on genome where a read aligns to the length of the read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' It is j1 − i1 len(r) (3) for nucleotide alignment, and jq 1 − iq 1 len(rQ) (4) for quantized alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' A contiguous alignment tends to have this metric as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This metric gives a measure of the contiguity of the alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='3 SV calling The alignments from HQAlign and minimap2 in sorted bam format are used to detect structural variants using Sniffles2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' These calls are benchmarked against a truth set using Truvari [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' We have used F1 score, precision and recall as the metric to analyze the performance of HQAlign and compare them with minimap2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Precision (P) is defined as the fraction of SVs detected by the algorithm in the truth set among the total SVs detected by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Recall (R) is the fraction of SVs detected by the algorithm in the truth set among the total SVs in the truth set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' F1 score is the harmonic mean of precision and recall (= 2P·R P+R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Further, we have observed that there are many complementary SV calls made by both minimap2 and HQ3 that are missed by the other method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Therefore, we have defined a union model which takes a union of the SV calls from both minimap2 and HQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The precision, recall, and F1 score of the union model are also computed and reported in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Further, the quality of the SVs for the common calls in minimap2 and HQAlign is evaluated by comparing the following metrics w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' the SVs in truth set (i) breakpoint accuracy: Breakpoint accuracy is measured by taking an average of the difference in the in the start and end breakpoint of the SV w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' the SV in truth set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' For instance, as shown in Figure 2d, i1 and j1 are the start and the end point on genome of SV in the truth set, and i ′ 1 and j ′ 1 are the start and the end point of the same SV determined by any alignment method (minimap2/HQ3), then breakpoint score is calculated as |i ′ 1 − i1| + |j ′ 1 − j1| 2 (5) where | · | is absolute value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Therefore, lower the score higher is the breakpoint accuracy of the SV determined by the alignment method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (ii) SV length similarity: SV length similarity is measured as the ratio of minimum SV length in truth set and from algorithm to the maximum of two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Mathematically, 11 it is min(j1 − i1, j ′ 1 − i ′ 1) max(j1 − i1, j ′ 1 − i ′ 1) (6) for the example shown in Figure 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 3 Results In this section, we demonstrate the results for (1) comparison of alignments from HQ3 and minimap2 on real as well as simulated data, and (2) comparison of SV calls from HQ3 and minimap2 alignments using Sniffles2 as the variant caller on real and simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 DNA read-to-genome alignment 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 Datasets We have used the publicly available R9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 ONT PromethION reads dataset from HG002 sample [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' These reads are aligned to the recent telomere-to-telomere assembly CHM13 and to the human reference genome GRCh37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' GRCh37 is used as the reference build to map the real data so that the curated variants can be used for accuracy analysis [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Further, we have also benchmarked the performance of HQAlign and minimap2 on simulated data for both alignments and SV calling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2 Alignment results The alignment of DNA reads to the genome is a primitive step in structural variant calling pipelines [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' HQ3 alignments shows an improvement over minimap2 alignments in terms of contiguity measured by normalized alignment length and alignment quality measured by normalized edit distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The results are illustrated in the Figures 3, 4, 5, and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' At a coarse level, the performance is measured by the fraction of the reads that are well-aligned by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' A read is well-aligned if at-least 90% of the read is aligned to genome and has a high mapping 12 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='9 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='9 1 Comparison of normalized edit distance for HG002 ONT reads alignment to T2T CHM13 reference genome 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='0% reads well aligned in HQ3 only 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='3% reads well alignedin both HQ3 and minimap2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='3% reads well aligned in minimap2 only Regression line: slope: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='79 intercept: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='02 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='3% reads not well aligned in both HQ3 and minimap2 (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='9 1 Comparison of normalized alignment length for HG002 ONT reads alignment to T2T CHM13 reference genome 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='3% reads well aligned in both HQ3 and minimap2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='3% reads not well aligned in both HQ3 and minimap2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='3% reads well aligned in minimap2 only 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='0% reads well aligned in HQ3 only Regression line: slope: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='61832 intercept: 1 (b) Figure 3: HG002 nanopore long DNA reads alignment onto T2T CHM13 genome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (a) Comparison of normalized edit distance for HG002 R9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 PromethION reads data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Smaller values for normalized edit distance is desirable as it represents better alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The slope of the regression line is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='79 < 1, therefore, representing better alignments with HQ3 than minimap2 alignments for same reads on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (b) Comparison of normalized alignment length for HG002 R9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 PromethION reads data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Normalized alignment length of 1 is desirable as it represents that entire read is aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The majority of the reads are above y = x line representing longer alignment length in HQ3 than minimap2 alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Table 2: Comparison for the percentage of well-aligned reads onto genome, and slope of the regression line (for normalized edit distance comparison plot of HQ3 vs minimap2 alignments) with randomly sampled reads for each datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The slope of the regression line shows the average gain in the normalized edit distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Dataset (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' of sampled reads) Method of alignment Percentage well- aligned reads Slope of regression line Intercept HG002 R9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 reads to CHM13 (50k) minimap2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7940 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='0206 HQ3 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='35 HG002 R9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 reads to GRCh37 (50k) minimap2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='8301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='0181 HQ3 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='65 Simulated reads from chr 8 & X of CHM13 assembly (50k) minimap2 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='9860 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='0028 HQ3 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='57 quality (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' HQAlign improves the fraction of well-aligned reads than minimap2 in particular, in the HG002 R9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 reads alignment to T2T CHM13 reference, this metric 13 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='9 1 Comparison of normalized edit distance for HG002 ONT reads alignment to GRCh37 reference genome 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7% reads well aligned in HQ3 only 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='0% reads well alignedin both HQ3 and minimap2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='5% reads well aligned in minimap2 only Regression line: slope: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='82 intercept: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='02 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='8% reads not well aligned in both HQ3 and minimap2 (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='9 1 Comparison of normalized alignment length for HG002 ONT reads alignment to GRCh37 reference genome 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='0% reads well alignedin both HQ3 and minimap2 Regression line: slope: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='66 intercept: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='5% reads well aligned in minimap2 only 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='8% reads not well aligned in both HQ3 and minimap2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7% reads well aligned in HQ3 only (b) Figure 4: HG002 nanopore long DNA reads alignment onto GRCh37 genome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (a) Comparison of normalized edit distance for HG002 R9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 PromethION reads data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Smaller values for normalized edit distance is desirable as it represents better alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The slope of the regression line is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='82 < 1, therefore, representing better alignments with HQ3 than minimap2 alignments for same reads on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (b) Comparison of normalized alignment length for HG002 R9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 PromethION reads data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Normalized alignment length of 1 is desirable as it represents that entire read is aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The majority of the reads are above y = x line representing longer alignment length in HQ3 than minimap2 alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 14 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} 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reads alignment to T2T CHM13 reference genome 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='8% reads not well aligned in both HQ3 and minimap2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2% reads well aligned in HQ3 only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='6% reads well aligned in minimap2 only Regression line: slope: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='99 intercept: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='00 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='4% reads well aligned in both HQ3 and minimap2 (b) Figure 5: Simulated nanopore reads alignment onto T2T CHM13 genome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (a) Com- parison of normalized edit distance for simulated nanopore reads data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Smaller values for nor- malized edit distance is desirable as it represents better alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The slope of the regression line is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='99 < 1, therefore, representing marginally better alignments with HQ3 than minimap2 alignments for same reads on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (b) Comparison of normalized alignment length for sim- ulated nanopore reads data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Normalized alignment length of 1 is desirable as it represents a contiguous alignment of the entire read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' improves to 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='35% from 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='64%, and for the alignments to GRCh37 reference, this metric improves to 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='65% from 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='48%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Furthermore, there are 310, 036 reads with at-least 1kb additional bases aligned using HQAlign compared to minimap2 alignments for T2T CHM13 reference, and there are 299, 896 reads with at-least 1kb additional bases aligned using HQAlign compared to minimap2 for GRCh37 reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The results in Figure 3 and 4 compares the quality of the alignments using minimap2 and HQAlign at a fine-grained level for HG002 ONT reads alignment to T2T CHM13 genome and GRCh37 genome, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Figure 3a and 4a compares the normalized edit distance for HQAlign and minimap2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The normalized edit distance is the edit distance between the entire read and the aligned section on the genome normalized by the length of the read, in nucleotide domain for both minimap2 alignment and quantized alignment (HQ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' In case of HQ3, the information of the location of the alignment on the genome is leveraged from the quantized read and the quantized genome alignment, and the edit distance is computed between the 15 corresponding nucleotide read and the aligned region on the nucleotide genome (see Methods for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Intuitively, normalized edit distance gives a measure of how close the two sequences are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Therefore, the smaller the normalized edit distance, better is the alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Figure 3a shows that for alignments of the reads to T2T CHM13 reference, the normalized edit distance is on average smaller for HQ3 alignments than minimap2 alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The better alignment in HQ3 is also evident from the slope of the regression line in Figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' It shows that on average HQ3 alignments has 21% improvement in terms of the normalized edit distance than the minimap2 alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Well aligned reads in both HQ3 and minimap2 are represented by blue circles in Figure 3, well aligned reads in HQ3 only are represented in black asterisks, well aligned in minimap2 only are represented in green diamonds and reads that are not well aligned in both are represented in grey squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Further, it is important to note that for normalized edit distance less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1, the alignments are marginally better in the DNA space, but for normalized edit distance higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1, the alignments are significantly better in HQ3 space, especially, the 4% reads that are well aligned in HQ3 and not well aligned in minimap2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This is because of higher contiguity of alignments in HQ3 space and signifies the improvement by HQ3 when the error rates is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' For alignments to GRCh37 reference, HQ3 has an average improvement of 17%, as shown in Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The results for another fine-grained metric are shown in Figure 3b and 4b, which compares the normalized alignment length in HQ3 to the normalized alignment length in minimap2 alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The normalized alignment length is the ratio of the length of the section on genome where a read aligns to the length of the read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' In Figure 3b, there are 4% reads that are well- aligned in HQ3 only, and the normalized alignment length is close to 1 in HQ3 but it is much less than 1 in minimap2, therefore representing several non-contiguous alignments in nucleotide domain that are captured as contiguous alignment in HQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' In Figure 4b, there are 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7% that are well-aligned in HQ3 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' We have also benchmarked the performance of HQAlign with the simulated reads data and compared its alignment performance with minimap2 in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The ONT reads are simulated from chromosome 8 and X of CHM13 T2T assembly using nanosim [20] with a coverage of 16 40x, median and mean read length 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='5 kb and 14 kb, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The results shows that the alignment performance of both HQAlign and minimap2 are at par with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2 SV calling 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 Dataset Long read sequencing plays an important role in detecting structural variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' We evaluated SV detection using minimap2 and HQAlign with Sniffles2 as the variant calling algorithm on both real and simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' We simulated 2000 INDELS and 200 Inversion SVs on chromo- some 8 and X of T2T CHM13 reference genome using SURVIVOR [21] with SV length uniformly distributed between 50 and 10000, and the ONT reads are simulated using nanosim with av- erage length of 14k, median length of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='5k and maximum length 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='5Mbp at a coverage of 40x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' We have used Truvari to benchmark the calls against the truth set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' For real data alignment with GRCh37 as the reference genome, the SV calls are compared against the ground truth sets from (1) Genome In A Bottle (GIAB) Tier 1 calls [18] and (2) another truth set is constructed by comparing the haplotype-resolved assembly of HG002 against GRCh37 reference genome using dipcall [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' For T2T CHM13 reference genome, since the ground truth for SVs is not available, we have constructed the truth set by comparing the haplotype-resolved assembly of HG002 against CHM13 reference using dipcall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' However, it is hard to establish ground truth for the SV calls that are made in the centromere regions, even though the assembly is likely to be correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Therefore, we have provided both the analysis including the SV calls in centromere regions (in Figures 6 and 11) and the analysis for SV calls excluding the centromere regions (in Figures 10 and 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2 SV calling results The standalone performance of both HQ3 and minimap2 is at par with each other across different references and truth set used in this study for real data as well as for the simulated data in terms of the F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' However, both HQ3 and minimap2 detects complementary SV 17 Minimap2 HQ3 True positives 890 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='8%) 1039 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7%) 14,506 15,396 (57%) 15,545 (58%) Out of 890 calls in minimap2: 461: calls are captured by HQ3 at low SV length similarity 429: unique region calls Out of 1039 calls in HQ3: 358: calls are captured by minimap2 at low SV length similarity 681: unique region calls (a) Minimap2 HQ3 False positives 1595 2044 3275 4870 5319 (b) Figure 6: Comparison of SV calls from HQ3 and minimap2 with HG002-to-CHM13 dipcall as truth set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (a) Comparison of true positive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (b) Comparison of false positive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Minimap2 HQ3 True positives 105 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2%) 103 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2%) 8,845 8,950 (94%) 8,948 (94%) Out of 105 calls in minimap2: 51: calls are captured by HQ3 at low SV length similarity 54: unique region calls Out of 1039 calls in HQ3: 41: calls are captured by minimap2 at low SV length similarity 62: unique region calls (a) Minimap2 HQ3 False positives 234 216 366 600 582 (b) Figure 7: Comparison of SV calls from HQ3 and minimap2 with Genome in a Bottle (GIAB) Tier 1 truth set for GRCh37 build.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (a) Comparison of true positive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (b) Comparison of false positive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Minimap2 HQ3 True positives 861 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2%) 703 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='3%) 15,759 16,620 (76%) 16,462 (74%) Out of 861 calls in minimap2: 524: calls are captured by HQ3 at low SV length similarity 337: unique region calls Out of 1039 calls in HQ3: 376: calls are captured by minimap2 at low SV length similarity 327: unique region calls (a) Minimap2 HQ3 False positives 1679 1472 3077 4756 4549 (b) Figure 8: Comparison of SV calls from HQ3 and minimap2 with HG002-to-GRCh37 dipcall as truth set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (a) Comparison of true positive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (b) Comparison of false positive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 18 Minimap2 HQ3 True positives 17 6 2125 2142 (97%) 2131 (97%) (a) Minimap2 HQ3 False positives 2 10 6 8 16 (b) Figure 9: Comparison of SV calls from HQ3 and minimap2 with simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (a) Comparison of true positive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (b) Comparison of false positive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Minimap2 HQ3 True positives 763 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='8%) 641 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='9%) 12,495 13,258 (75%) 13,136 (74%) Out of 763 calls in minimap2: 83: calls are captured by HQ3 at low SV length similarity 680: unique region calls Out of 641 calls in HQ3: 77: calls are captured by minimap2 at low SV length similarity 564: unique region calls (a) Minimap2 HQ3 False positives 995 974 1991 2986 2965 (b) Figure 10: Comparison of SV calls from HQ3 and minimap2 with HG002-to-CHM13 dipcall as truth set and excluding the calls from the centromere regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (a) Com- parison of true positive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (b) Comparison of false positive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' calls most likely in the repeat regions where accurate alignment is difficult and therefore, leads to many broken calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The analysis with comparison of SV calls from HQ3 and minimap2 with GIAB Tier 1 truth set gives a precision, recall and F1 score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='94, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='94, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='94, respectively for both minimap2 and HQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' A union model of minimap2 and HQ3 can improve the recall rate at the same F1 score, and the union model has a precision, recall, and F1 score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='93, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='94, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Moreover, out of 103 SV calls that are made by HQ3 only (Figure 7), 41 calls are made by minimap2 alignments at a lower SV length similarity and 62 calls are unique region calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Out of 105 SV calls made by minimap2 only, 51 are captured by HQ3 at a lower SV length similarity and 54 are unique region calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' HQ3 improves the breakpoint accuracy for 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='11% 19 calls that have higher than 50 difference in breakpoints and it improves the length similarity of 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='97% calls that have SV length similarity lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95 (Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' We have compared the SV calls made by HG002 reads against T2T CHM13 reference genome using both minimap2 and HQ3 and benchmarking them against truth set generated by compar- ing HG002 haplotype-resolved assembly to T2T CHM13 assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The standalone performance have precision, recall and F1 score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='77, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='57 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='66, respectively for minimap2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='58 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='65, respectively for HQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' However, because of high number of complementary true positive calls in minimap2 and HQ3, the union model has a significant improved recall at the same F1 score with precision, recall and F1 score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='71, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='61 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='66, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Out of 1039 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7%) calls that are made in HQ3 only, 358 are captured by minimap2 at a lower SV length similarity threshold and 681 are unique calls, whereas out of 890 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='8%) calls that are made by minimap2 only, 461 are captured by HQ3 at a lower SV length similarity threshold and 429 are unique (as shown in Figure 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Further, for the common true positive calls in both minimap2 and HQ3, we observe a similar pattern as the other datasets in improvement of breakpoint accuracy with HQ3 for 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='66% calls that have difference in breakpoint greater than 50, and improvement in SV length similarity for 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='76% calls with similarity less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95 (Figure 11a-b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' For SV calls from HG002 reads alignment to GRCh37 and benchmarking them against truth set generated by comparing HG002 haplotype-resolved assembly to GRCh37 build, minimap2 has precision 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='78, recall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='76, and F1 score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='77 while HQ3 has precision 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='79, recall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='75, and F1 score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Out of 16462 true positive calls in HQ3, 703 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='27%) are made only in HQ3 with SV length similarity to the truth set greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7 (default parameter in Truvari).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' However, 376/703 calls that are captured by minimap2 with SV length similarity less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7 and 327/703 calls that are uniquely made by HQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Likewise, out of 16620 true positive calls in minimap2, 861 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='18%) are made only in minimap2 with SV length similarity greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' However, 524/861 are captured by HQ3 with SV length similarity less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7 and 337/861 are uniquely made by minimap2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' A fine-grain analysis of the common true positive calls by minimap2 and HQ3 in Figure 13a, shows that a major density of SV calls (81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='85%) have 20 Table 3: Comparison for precision, recall and and F1 score for SV calls made by HQ3, minimap2, and the Union model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Dataset Truth set Method of align- ment Precision Recall F1 score HG002 reads to GRCh37 GIAB Tier 1 calls minimap2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='94 HQ3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='94 Union 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='94 HG002 reads to CHM13 comparing HG002 assembly to CHM13 (including centromere calls) minimap2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='66 HQ3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='65 Union 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='66 HG002 reads to CHM13 comparing HG002 assembly to CHM13 (excluding centromere calls) minimap2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='78 HQ3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='78 Union 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='78 HG002 reads to GRCh37 comparing HG002 assembly to GRCh37 minimap2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='77 HQ3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='77 Union 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='77 Simulated reads to CHM13 Simulated SVs on chr 8 and X minimap2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='98 HQ3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='98 Union 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='98 difference in breakpoint below 50 in both minimap2 and HQ3, and minimap2 has marginally better performance in terms of lower difference in breakpoint of SVs for difference in breakpoint below 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Whereas, for a large difference in the SV breakpoint (greater than 50), HQ3 is better in terms of the breakpoint accuracy of the SV calls (on average across all SV calls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Therefore, HQ3 improves the SV breakpoint for the rest 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='15% calls that have high difference in breakpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Further, Figure 13b demonstrate that HQ3 has better SV length similarity when the length similarity is below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95 which corresponds to 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='82% calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 4 Discussion HQAlign method is an alignment method designed for the detection of structural variants for nanopore sequencing reads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' HQAlign provides alignment that outperforms the recent minimap2 aligner in terms of the accuracy and quality of the alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The SV calling from HQAlign is also at par with minimap2 in terms of F1 score and it outperforms minimap2 SV calls in terms 21 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 Comparison of average breakpoint difference for T2T CHM13 truth set (including the centromere regions) 2000 4000 6000 8000 10000 12000 Regression line: slope: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='58 intercept: 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='66 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95 1 Comparison of SV length similarity for T2T CHM13 truth set (including the centromere regions) 500 1000 1500 2000 2500 3000 3500 4000 4500 Regression line: slope: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='76 intercept: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='23 (b) Figure 11: SV quality comparison for common true positive calls in HQ3 and min- imap2 against HG002-to-CHM13 dipcall truth set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (a) Comparison of SV breakpoint accuracy in HQ3 and minimap2 for common true positive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The difference of SV breakpoint is compared to the truth set generated from comparing HG002 haplotype-resolved assembly to T2T CHM13 build.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' A smaller difference represents better breakpoint accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Therefore, slope of the regression line 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='58 < 1 represents better accuracy of HQ3 than minimap2 on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (b) Comparison of SV length similarity in HQ3 and minimap2 for common true positive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The slope of the regression line 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='76 < 1 represents better SV length in minimap2 than HQ3 on average, but the intercept is high (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' However, this is due to a large density of SVs with length similarity ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95 in both minimap2 and HQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' For length similarity less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95, HQ3 has better performance than minimap2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' of the quality of SVs measured in breakpoint accuracy and SV length similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Moreover, there are many complementary SVs captured by HQAlign that are missed by minimap2 alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The reason for this improvement in the performance of alignment and SV calling with HQAlign is that it takes into account the underlying physics of nanopore sequencer through the Q-mer map, which could be one of the major causes of the high error rates in nanopore sequencing, and also it focuses on a narrow region of the genome (where the read aligns in nucleotide domain) for alignment with quantized sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Further, this pipeline is adapted specifically for the detection of SVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' We demonstrated how HQAlign utilizes the bias of Q- mer map without accessing the raw current signal of nanopore sequencer by translating the basecalled nucleotide sequences to quantized current level (of finite alphabet size) sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This improvement help in detecting several SVs that are missed by minimap2 due to high error 22 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000Comparison of average breakpoint difference for GIAB Tier 1 truth set 1000 2000 3000 4000 5000 6000 Regression line: slope: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='89 intercept: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='92 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95 1 Comparison of SV length similarity for GIAB Tier 1 truth set 500 1000 1500 2000 2500 Regression line: slope: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='84 intercept: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='16 (b) Figure 12: SV quality comparison for common true positive calls in HQ3 and min- imap2 against GIAB Tier 1 truth set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (a) Comparison of SV breakpoint accuracy in HQ3 and minimap2 for common true positive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The difference of SV breakpoint is com- pared to the GIAB Tier 1 truth set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' A smaller difference represents better breakpoint accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Therefore, slope of the regression line < 1 represents better accuracy of HQ3 than minimap2 on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (b) Comparison of SV length similarity in HQ3 and minimap2 for common true positive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The slope of the regression line < 1 represents better SV length in minimap2 than HQ3 on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' However, this is due to a large density of SVs with length similarity ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95 in both minimap2 and HQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' For length similarity less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95, HQ3 has better performance than minimap2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' rates in the nanopore reads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Further, the recall rate for SV detection can be improved by combining the complementary calls from both HQ3 and minimap2 in the union model at the same F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Competing interests The authors declare that they have no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Author’s contributions DJ, SK, and SD conceived the original idea and developed the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' DJ led the development of the software tool and its open-source development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' MC helped with SV metrics, datasets, and 23 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 Comparison of average breakpoint difference for GRCh37 truth set 2000 4000 6000 8000 10000 12000 Regression line: slope: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='54 intercept: 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='86 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95 1 Comparison of SV length similarity for GRCh37 truth set 500 1000 1500 2000 2500 3000 3500 4000 Regression line: slope: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='77 intercept: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='22 (b) Figure 13: SV quality comparison for common true positive calls in HQ3 and min- imap2 against HG002-to-GRCh37 dipcall truth set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (a) Comparison of SV breakpoint accuracy in HQ3 and minimap2 for common true positive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The difference of SV breakpoint is compared to the truth set generated from comparing HG002 haplotype-resolved assembly to GRCh37 build.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' A smaller difference represents better breakpoint accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Therefore, slope of the regression line < 1 represents better accuracy of HQ3 than minimap2 on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (b) Comparison of SV length similarity in HQ3 and minimap2 for common true positive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The slope of the regression line < 1 represents better SV length in minimap2 than HQ3 on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' However, this is due to a large density of SVs with length similarity ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95 in both minimap2 and HQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' For length similarity less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95, HQ3 has better performance than minimap2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' SV comparison analysis between methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' DJ performed the analysis on the various datasets for both alignment and SV calling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' All the authors wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Acknowledgements SD and DJ were supported in part by National Science Foundation grant 1705077.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' MC was supported by National Institutes of Health grant R01HG011649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' SK was supported in part by National Institutes of Health grant 1R01HG008164 and National Science Foundation grants 1651236 and 1703403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 24 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 Comparison of average breakpoint difference for T2T CHM13 truth set (excluding calls from the centromere regions) 1000 2000 3000 4000 5000 6000 7000 8000 9000 Regression line: slope: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='52 intercept: 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='58 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='75 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95 1 Comparison of SV length similarity for T2T CHM13 truth set (excluding the centromere regions) 500 1000 1500 2000 2500 3000 3500 Regression line: slope: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='75 intercept: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='24 (b) Figure 14: SV quality comparison for common true positive calls in HQ3 and min- imap2 against HG002-to-CHM13 dipcall truth set (excluding the centromere re- gion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (a) Comparison of SV breakpoint accuracy in HQ3 and minimap2 for common true positive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The difference of SV breakpoint is compared to the truth set generated from comparing HG002 haplotype-resolved assembly to T2T CHM13 build and the SV calls in the centromere region are excluded in this truth set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' A smaller difference represents better break- point accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Therefore, slope of the regression line < 1 represents better accuracy of HQ3 than minimap2 on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (b) Comparison of SV length similarity in HQ3 and minimap2 for common true positive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The slope of the regression line < 1 represents better SV length in minimap2 than HQ3 on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' However, this is due to a large density of SVs with length similarity ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='95 in both minimap2 and HQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' For length 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generation dna sequencing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Nature genetics 43, 491 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' [20] Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=', Chu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=', Warren, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' & Birol, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Nanosim: nanopore sequence read simulator based on statistical characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' GigaScience 6, gix010 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' [21] Jeffares, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Transient structural variations have strong effects on quantitative traits and reproductive isolation in fission yeast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Nature communications 8, 1–11 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' [22] Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' A synthetic-diploid benchmark for accurate variant-calling evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Nature methods 15, 595–597 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 27 A Supplementary material A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 Quantization method from QAlign [13] The nucleotide sequences are inferred from the nanopore current signals by basecallers, there- fore, using a Q-mer map to translate the basecalled sequences to the current levels implicitly maintains all of the “equivalent” basecalled sequences that could be inferred from the observed current levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' These current levels can be quantized to an alphabet of finite size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Mathematically, the quantization process is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Let Σ = {A, C, G, T} be the alphabet of nucleotide sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' For a symbol s ∈ Σ, let ¯s be the Watson-Crick complement of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' A string x = s1s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' sn over Σ is called a nucleotide sequence, where |x| = n is the string length and the reverse complement of x is x = s1s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' sn = snsn−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Let p(x) be a list of all Q-mers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Q=6) in the string x, sorted by their occurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' For example, p(x) = k1k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' kn−Q+1 and each Q-mer ki = sisi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' si+Q−1 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' , n − Q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Now, we define f : ΣQ → R as the Q-mer map 1, which is a deterministic function that translates each Q-mer (ki) to the (median) current level (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Now, let C(x) = c1c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' cn−Q+1 be the sequence of the current levels, so that ci = f(ki) for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' , n − Q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The current sequence C(x) can be further quantized into w(x) = q1q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' qn−Q+1 by applying hard thresholding function qi = g(ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The thresholding can be ternary (qi ∈ {0, 1, 2}) for HQ3 (Figure 1c and Supplemental Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' We define w(x) as the quantized reverse complementary of sequence x, so w(x) = w(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' Supplemental Figure 15 explains this process using a toy example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2 Generalization of HQAlign method A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 Initial alignment The nucleotide query x is aligned to a set of nucleotide target sequences t = (t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' , tm) using Minimap2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This is similar to aligning a read to a genome which has several chromosome 1Q-mer map is determined by the chemistry of the nanopore flow cell, and is therefore dataset dependent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=', the Q-mer map for sequencing using R9 flow cell is different from Q-mer map for sequencing using R9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 flow cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The Q-mer maps used in this work are generated by Nanopolish (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='com/jts/nanopolish).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 28 sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' This step identifies the region of interests on the target t, say, tj[si : ei], where tj, j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' , m} represent alignment to one or more target chromosomes that x aligns to, i ∈ {1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' } represent represent one or more alignments to chromosome j, si and ei are the corresponding start and end location of each alignment i on the target tj, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2 Hybrid alignment In this step, the query x is re-aligned to an extended region of interest on the target tj[sq i : eq i] using the QAlign method, where sq i = si − bi and eq i = ei + bi, bi = (1 − fi + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='25)n is an appended extension of the region of interest on target, fi = (ei − si)/n is the fraction of read aligned in initial step, and n is the length of the query x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The nucleotide query x and the nucleotide extended target tj[sq i : eq i] are converted to the quantized query xq, quantized reverse complement query xq and quantized extended target tq j[sq i : eq i], respectively, using the quantization method demonstrated in QAlign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' These quantized sequences are then aligned using modified minimap2 pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 29 ACGTAACGTATTG [ ACGTAA , CGTAAC , GTAACG , TAACGT , AACGTA , ACGTAT , CGTATT , GTATTG ] [ 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='13 , 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='67 , 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='75 , 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='27 , 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='85 , 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='16 , 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='75 , 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='26 ] 12011020 Nucleotide seq List of 6-mers 𝑄-mer map 𝑄-mer map translates 6-mers to median current value List of current levels Q3 seq Hard thresholding: (current level) 58 – 82 à 0 (quantize level) (current level) 82 – 98 à 1 (quantize level) (current level) 98 – 120 à 2 (quantize level) Figure 15: An example for the Quantization method for QAlign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The nucleotide se- quences are first translated to current level sequences using the Q-mer map, and then the (continuous) current level sequences are quantized to finite levels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' three levels for HQ3) by hard thresholding the current levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='3 Accessing HQAlign on github HQAlign requires python 3, and the installation guideline can be found on github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' The software is available at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='com/joshidhaivat/HQAlign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='git usage: python hqalign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='py [-h] -r REF -i READS -o OUTPUT [-t THREADS] [-k KMER] arguments: h, --help show this help message and exit r REF, --ref REF reference genome filename in fasta format i READS, --reads READS directory location of read files in fasta format (with file extension .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='fasta) o OUTPUT, --output OUTPUT location of directory of output files t THREADS, --threads THREADS maximum number of parallel threads (default=4) k KMER, --kmer KMER minimizer length for hybrid step (default=18) 30 Minimap2 (v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='24) Reads data (Nucleotide) Reference Genome (Nucleotide) Sort & merge alignments using samtools Sniffles2 Miniamp2 SV callsets in vcf format Minimap2 alignments (in sam & paf format) Alignments in bam format Quantization to three levels HQAlign with modified Minimap2 (v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='24) HQ3 SV callsets in vcf format Q3 reads Q3 reference HQAlign pipeline Figure 16: Complete pipeline for SV calling using minimap2 and HQAlign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 31 0 1 2 3 4 5 6 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='9 1 Comparison of minimap2 SV calls to same calls in HQ3 Unique calls in minimap2 Common calls in HQ3 and minimap2 412 minimap2 calls that are captured in HQ3 at low SV length similarity 397 minimap2 calls in unique region (a) 0 1 2 3 4 5 6 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2 0.' metadata={'source': 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Unique calls in HQ3 Common calls in HQ3 and minimap2 329 HQ3 calls that are captured in minimap2 at low SV length similarity 539 HQ3 calls in unique region (b) Figure 17: Comparison of SV calls made by minimap2 and HQ3 to other method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (a) For the complementary calls (in blue) and common calls (in red) made by minimap2, this figure compares SV length similarity and distance to nearest SV in HQ3 of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 397 complementary calls made by minimap2 are in unique region, whereas 412 complementary calls in minimap2 are captured in neighboring region (within 1000 bp) in HQ3 but with a low SV length similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (b) For the complementary calls (in blue) and common calls (in red) made by HQ3, this figure compares SV length similarity and distance to nearest SV in minimap2 of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 539 complementary calls made by HQ3 are in unique region, whereas 329 complementary calls in HQ3 are captured in neighboring region (within 1000 bp) in minimap2 but with a low SV length similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 32 0 1 2 3 4 5 6 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='9 1 Comparison of minimap2 SV calls to same calls in HQ3 Unique calls in minimap2 Common calls in HQ3 and minimap2 445 minimap2 calls that are captured in HQ3 at low SV length similarity 290 minimap2 calls in unique region (a) 0 1 2 3 4 5 6 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content='4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (a) For the complementary calls (in blue) and common calls (in red) made by minimap2, this figure compares SV length similarity and distance to nearest SV in HQ3 of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 290 complementary calls made by minimap2 are in unique region, whereas 445 complementary calls in minimap2 are captured in neighboring region (within 1000 bp) in HQ3 but with a low SV length similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' (b) For the complementary calls (in blue) and common calls (in red) made by HQ3, this figure compares SV length similarity and distance to nearest SV in minimap2 of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 179 complementary calls made by HQ3 are in unique region, whereas 347 complementary calls in HQ3 are captured in neighboring region (within 1000 bp) in minimap2 but with a low SV length similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} +page_content=' 33' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE2T4oBgHgl3EQfWgd-/content/2301.03834v1.pdf'} diff --git a/SdE5T4oBgHgl3EQfaQ-N/content/tmp_files/2301.05587v1.pdf.txt b/SdE5T4oBgHgl3EQfaQ-N/content/tmp_files/2301.05587v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a0b7bac57908b2b93f4c2ffc9ad03d00eabf8845 --- /dev/null +++ b/SdE5T4oBgHgl3EQfaQ-N/content/tmp_files/2301.05587v1.pdf.txt @@ -0,0 +1,2307 @@ +Astronomy & Astrophysics manuscript no. main +©ESO 2023 +January 16, 2023 +On the Role of Interplanetary Shocks in Accelerating MeV +Electrons +N. Talebpour Sheshvan1, N. Dresing1, R. Vainio1, A. Afanasiev1 and D. E. Morosan2 +1. Department of Physics and Astronomy, University of Turku, Finland +2. Department of Physics, University of Helsinki, Finland +e-mail: nasrin.talebpoursheshvan@utu.fi +ABSTRACT +Context. One of the sources of solar energetic particle (SEP) events is shocks that are driven by fast coronal mass ejections (CMEs). +They can accelerate SEPs up to relativistic energies and are attributed to the largest SEP events. New studies suggest that CME-driven +shocks can potentially accelerate electrons to MeV energies in the vicinity of the Sun. +Aims. We focus on relativistic electrons associated with strong IP shocks between 2007 and 2019 to determine whether the shocks +can keep accelerating such electrons up to 1 AU distance. +Methods. We have analyzed High Energy Telescope (HET) observations aboard the STEREO spacecraft of potential electron ener- +getic storm particle (ESP) events, characterized by intensity time series that peak at the time of, or close to, the associated CME-driven +shock crossing. We present a new filtering method to assess the statistical significance of particle intensity increases and apply it to +MeV electron observations in the vicinity of interplanetary shocks. We employed a STEREO in-situ shock list, which contained in +total 587 shocks at the two STEREO spacecraft, from which we identified 27 candidate events by visual inspection. +Results. Our method identified nine clear cases, where a significant increase of MeV electrons was found in association with a shock. +Typically, the highest statistical significance was observed in the highest of the three HET energy channels (2.8–4.0 MeV). All nine +cases were associated with shocks driven by interplanetary CMEs that showed large transit speeds, in excess of 900 km s−1. In several +cases multiple shocks were observed within one day of the shock related to the electron increase. +Conclusions. Although electron ESP events at MeV energies are found to be rare at 1 AU our filtering method is not designed to +identify a potential interplanetary shock contribution from distances closer to the Sun. Observations by Parker Solar Probe or Solar +Orbiter, taken during closer approaches to the Sun, will likely provide clarity on interplanetary shock acceleration of electrons. +Key words. Sun: coronal mass ejections (CMEs) – Sun: particle emission - Acceleration of particles - shock waves – interplanetary +medium +1. Introduction +Shock waves propagating in the interplanetary (IP) space are +potential accelerators of energetic particles in the heliosphere. +Coronal mass ejections (CMEs) propagating through the IP +space are the main drivers of IP shocks, in particular during the +solar active years. Gradual solar energetic particle (SEP) events +(see e.g., Reames 1999; Lee et al. 2012; Reames 2013; Reames +et al. 2014; Desai & Giacalone 2016; Bruno et al. 2018) are +thought to be accelerated by CME-driven shocks propagating +through the corona and IP space (Lario et al. 2008). Energetic +storm particle (ESP) events are increases of energetic particle +intensities associated with the passage of transient interplane- +tary shocks (e.g., Bryant et al. 1962; Rao et al. 1967; Lario et al. +2003; Huttunen-Heikinmaa & Valtonen 2009). +The role of IP shocks for proton intensity enhancements has +been investigated in a broad energy range from a few keV to tens +of MeV (e.g., Lario et al. 2005; Dresing et al. 2016b; Mäkelä +et al. 2011; Ameri et al. 2022). The first ESP events were re- +ported by Bryant et al. (1962) and classified by Sarris & van +Allen (1974) in two categories, spike events and classic ESP +events. Spike events are characterized by a sudden rise in proton +intensities lasting only between 5 and 20 minutes. In contrast, +classic ESP events exhibit gradual intensity increases several +hours before the shock passage in agreement with the predic- +tions of the classical diffusive shock acceleration (DSA) theory +(Desai & Giacalone 2016). Lario et al. (2003, 2005) suggested +also other types of ESP events based on the features of the in- +tensity profile like ESP+spike or step-like events and those with +irregular time-intensity profiles not related to the time of shock +crossing. +Although it is well established that protons frequently exhibit +classic ESP events, the role of shocks in electron acceleration, +especially in IP space, is still not clear. Direct observations of +electron acceleration at shocks in IP space are rare (e.g., Simnett +2003; Mitchell et al. 2021). Previous studies have analyzed in- +situ shock crossings at spacecraft situated at 1 AU and found the +shock acceleration efficiency to be very low for electrons at near- +relativistic energies, i.e., around 100 keV (Tsurutani & Lin 1985; +Lario et al. 2005; Mäkelä et al. 2011; Dresing et al. 2016b). The +acceleration efficiency is, however, increasing at lower energies, +i.e., tens of keV (Yang et al. 2019). +Although IP shocks at 1 AU do not show clear ability to +accelerate electrons in-situ at near-relativistic energies, Dresing +et al. (2022) showed that there is a strong correlation between +electron peak intensities observed at Earth’s distance and the +Mach number of the shock wave close to the Sun, especially +at MeV energies. This provides strong evidence for the ability +of coronal shocks to accelerate electrons to relativistic energies +and motivates revisiting relativistic electron observations during +in-situ shock crossings. +Article number, page 1 of 13 +arXiv:2301.05587v1 [astro-ph.SR] 13 Jan 2023 + +A&A proofs: manuscript no. main +In this study, we present observations of MeV electron ESP +events observed with the STEREO mission (e.g., Kaiser et al. +2008) by investigating the entire STEREO dataset for relativistic +electron enhancements connected with in-situ shock crossings. +Our goal is to determine if IP shocks contribute to MeV elec- +tron acceleration. In order to account for acceleration regions +that could reside somewhat away from the local region sampled +by the spacecraft, we extended the time interval of the relativis- +tic electron observations to several hours around the time of the +shock passage. As a result, a set of candidate events is estab- +lished, which we then examine more closely to determine the +effects of IP shocks on MeV electrons close to 1 AU. +The structure of the paper is the following: in §2, we de- +scribe the event selection and a data filtering method developed +for identifying shock-related electron enhancements, in §3 we +present the observations and results of our analysis, and in §4 +we discuss the results and present the conclusions of our study. +2. Event selection and filtering method +To find potential electron ESP events, we used the STEREO in- +situ shock list (Jian et al. 2013). The list starts at the beginning +of the STEREO mission in January 2007 and the latest update is +on April 2019. There are no STEREO-B events after September +2014 because of the loss of contact with the spacecraft. In total, +there are 587 interplanetary shocks in the list we used, of which +340 were detected at STEREO-A and 247 at STEREO-B. +At each of the listed in-situ shock-crossing times, we ana- +lyzed the electron and proton intensities measured by the High +Energy Telescope (HET; von Rosenvinge et al. 2008). HET is +one of four instruments in the Solar Energetic Particle subsys- +tem, which is part of the In-situ Measurements of Particles and +CME Transients investigation (IMPACT; Luhmann et al. 2008) +on STEREO. HET measures the highest energy particles includ- +ing protons in eleven energy channels from 13.6 to >100 MeV +and electrons in three energy channels from 0.7 to 4.0 MeV. In +the following, we denote the electron channels as e1 (0.7-1.4 +MeV), e2 (1.4-2.8 MeV) and e3 (2.8-4.0 MeV). +We first visually scanned the electron observations for inten- +sity enhancements taking place close in time to the shock arrivals +at the spacecraft and, thus, potentially caused by local accelera- +tion of electrons by the shocks. Thus, we found 27 electron ESP +candidates. Because the in-situ shock crossings are often asso- +ciated with high proton and ion fluxes, we needed to be sure +that the selected events are not the result of ion contamination +in the electron channels. For this, we analyzed 1-minute HET +data of the whole year of 2013 (see Appendix B and Fig. B.1). +Our conclusion is that the HET electron enhancements are real +and not due to ion contamination. We also checked the radio ob- +servations of STEREO/SWAVES to exclude the possibility that +the electron enhancements are caused by SEP injections at the +Sun coincidentally occurring closely before the enhancements. +For this, we searched for type III radio bursts, the signature +of accelerated electrons impulsively injected into interplanetary +space (Reid & Ratcliffe 2014), occurring about 10-30 minutes +before the observed electron enhancement of selected events. +Apart from one event (6 Nov 2013, see a more detailed analysis +in section 3), none of our candidates turned out to be associated +with a coincident solar injection. +In our simple scanning method it was not always easy +to identify electron intensity enhancements potentially associ- +ated with local shock acceleration because they are usually +mixed with enhancements of the corresponding SEP events (e.g., +Kouloumvakos et al. 2015). For that reason, we introduced a fil- +tering method for the intensities: +∆I(ti; N, M) = I(ti) − 1 +M +i−1−N +� +j=i−M−N +I(t j), +(1) +where M is the number of points (time bins) constituting the +averaging time window and N is the number of points giving the +time lag of the averaging window before each time ti. Thus, the +method uses the running averaging window with variable length +and lag. In the analysis, we took N = 180 for all the cases, which +corresponds to a constant three-hour lag for one-minute data. +The parameter M was given values of 15, 30, 60, 120, 240 and +480, which correspond to a temporal window varying from 15 +min to 8 hours for one-minute data. +The applied filtering method allows one to identify more +clearly local (in time) variations of the particle intensity. The fil- +tered data are shown in the six bottom panels of Fig. 1(a), which +also gives an example the analysis carried out for all our 27 elec- +tron ESP event candidates. In the intensity time series, one can +see intensity enhancements (in various energy channels, for both +electrons and protons) associated with the shock and superposed +on the ongoing SEP event. The filtering of the intensities con- +veniently shows the time interval when the intensity variation is +strongest. +However, Fig. 1(a) also shows that if one wants to compare +the filtered intensities corresponding to different energy chan- +nels (to find, e.g., which energy channel has the strongest rela- +tive enhancement), Eq. (1) is not convenient due to lower back- +ground levels of higher energy channels. Therefore, we modi- +fied Eq. (1) by normalizing the filtered intensity ∆I by the "local +background" intensity, i.e., the intensity value of the running av- +erage window: +∆Inorm(ti; N, M) = +I(ti) +1 +M +�i−1−N +j=i−M−N I(tj) +− 1. +(2) +Figure 1(b) shows the normalized filtered (NF) intensities ∆Inorm +for electrons in the considered example, while Figure 1(c) +presents both electrons and protons. +In order to determine if an intensity enhancement around +the shock-crossing time is significant compared to its preceding +intensity level, we calculate the mean and the standard devia- +tion (SD) of the NF data inside the background window, using +the six different averaging window lengths (M1-M6, plotted in +the lower six panels of the figures) for each of the three elec- +tron channels. We then compared the NF data with their cor- +responding variable background level and calculate the z-score, +i.e., the number of SDs above the background level. We applied +the z-score to identify statistically significant electron enhance- +ments by requiring that at least three consecutive NF data points +within a six-hour window surrounding the shock-crossing time +are above the mean + 2 SDs threshold level. From this we identi- +fied which energy channel and averaging window length yielded +the most significant electron enhancements. We then returned +to the intensity-time profile of the chosen electron channel and +identified the time of its peak intensity Ipeak within a six-hour +window surrounding the shock time. Since not all the events have +a distinct peak, we divided them into three categories: plateau +(Fig. A.2), peak (Fig. 3), and plateau + peak (Fig. 4). +Naturally, the NF data also show a strong response at the be- +ginning of the associated SEP events. These intensity enhance- +ments also increase the background levels used in our method, +once the background window moves over them. +Article number, page 2 of 13 + +N. Talebpour Sheshvan, N. Dresing, R. Vainio , A. Afanasiev and D. E. Morosan : On the Role of Interplanetary Shocks in Accelerating MeV +Electrons +(a) +(b) +(c) +Fig. 1: Analysis of the shock crossing on 29 January 2012 observed by STEREO-A. (a) From top to bottom: proton and electron +intensity-time profiles as measured by HET, magnetic field vector magnitude and components as measured by IMPACT/MAG, +STEREO/WAVES dynamic spectrum, and the rest are filtered data as obtained with the fixed time lag parameter (N = 180) and +increasing span parameter M. (b) Zoom-in around the time of shock number 125 passage with the electron NF data. (c) Zoom-in +around the time of shock number 125 passage with both the electron and proton NF data. The black vertical dashed line indicates the +time of the shock passage over the spacecraft, and the orange dashed line marks the time of the peak intensity Ipeak in the electron +energy channel having the strongest response in terms of the NF data. Each shaded area in (b) shows the running mean + 2 SD level +of the corresponding running background window for the electron energy channel having the strongest response in terms of the NF +data. +In the detailed analyses of example events, we also +make use of CME information obtained from the SOHO +LASCO CME Catalog (http://cdaw.gsfc.nasa.gov/CME_ +list; e.g., Yashiro et al. 2004; Gopalswamy et al. 2009, 2010). +3. Observations and Results +The filtering technique (described in §2), applied to the visu- +ally identified candidate events in STEREO/HET observations, +enabled us to list IP shocks potentially accelerating electrons +to MeV energies. Out of the 27 candidate IP shock events ob- +served by STEREO A and B, nine (∼1.5%) turned out to have +significant relativistic electron intensity enhancements. The first +and second column of Table 1 list the event date, shock num- +ber 1, and the observing spacecraft (A or B), respectively. Two +numbers are indicated in parentheses in the third column; the +first one gives the number of shocks that occurred in the two +days prior to our main shock, and the second number provides +the number of shocks that occurred in the two days following +our main shock. The shock transit speed calculated using the +start time of the corresponding type III radio burst observed +by STEREO/WAVES and the CME arrival time 2 is listed in +1 https://stereo-ssc.nascom.nasa.gov/pub/ins_data/ +impact/level3/IPs.pdf +2 https://stereo-ssc.nascom.nasa.gov/pub/ins_data/ +impact/level3/ICMEs.pdf +the fourth column. The shock parameters, including the mag- +netosonic Mach number, shock normal angle, and shock mag- +netic compression ratio, given in the fifth, sixth, and seventh +columns, respectively, were taken from the STEREO IP shock +list3. The last four columns contain information regarding the +shock-associated electron enhancements. Column (8) gives the +energy channel with the highest z-score in the NF data among +the three electron channels. Column (9) gives the time delay be- +tween the intensity peak time and the shock crossing time (neg- +ative value indicates that the intensity peaks before the shock- +crossing time). The shape of the intensity-time profile around +the shock crossing (visually evaluated) is given in column (10), +and in the last column of the Table [1], the background window +length parameter M corresponding to the highest z-score in the +related electron channel, in the 6 hours window around the time +of shock crossing, is presented. +Applying the normalization to the filtered data allows us to +identify the energy channel with the strongest response on the +events. In seven of these events the most significant signal be- +longs to the e3 channel of HET in the range of 2.8-4.0 MeV +(Table 1). In six cases (∼67%) the peak intensity is observed up- +stream of the shock, with five of them occurring very close to the +time to the shock crossing (less than 20 minutes). In four cases +(shock numbers 87, 126, 151, and 246, see Fig. 4, A.3), the MeV +3 https://stereo-ssc.nascom.nasa.gov/pub/ins_data/ +impact/level3/IPs.pdf +Article number, page 3 of 13 + +STEREO-A/ HET / 2012-01-29 +124 +102 +101 +100 +50 +[B| +(lu) +0 +101 +100 +(MHz) +10-1 +10-2 +20 +N=180, M=15 +20 +N=180, M=30 +0 +20 +N=180, M=60 +N=180, M=120 +10 +N +-10 +20 +N=180, M=240 +N +0 +N=180, M=480 +20 +N +01-28 +01-28 +01-28 +01-29 +01-29 +01-29 +01-29 +01-30 +01-30 +06:00 +12:00 +18:00 +00:00 +06:00 +12:00 +18:00 +00:00 +06:00 +Date / Time in year 2012STEREO-A/ HET / 2012-01-29 +0.7-1.4 MeV e +7. +102 +-sr-s-MeV) +1.4-2.8 MeV e +2.8-4.0 MeV e +13.6-15.1 MeVp +101 +15.0-17.1 MeV.p +-) +17.0-19.3 MeV p +20.8-23.8 MeV p +100 +125 +50 F +(lu) +10 +N=180, M=15 +5 +10 +N=180, M=30 +5 +10 +N=180, M=60 +Alnom +10 +N=180, M=120 +5 +10 +N=180, M=240 +5 +10 +N=180, M=480 +e3:mean+2std +wou +5 +01-29 +01-29 +01-29 +01-29 +01-29 +01-29 +01-29 +10:00 +11:00 +12:00 +13:00 +14:00 +15:00 +16:00 +Date / Time in year 2012STEREO-A /HET / 2012-01-29 +102 +101 +100 +50 +0 +101 +100 +10-1 +10-2 +40 +N=180, M=15 +20 +40 +N=180, M=30 +20 +48 +N=180, M=60 +20 +40 +N=180, M=120 +20 +40 +N=180, M=240 +20 +40 +N=180, M=480 +20 +01-29 +01-29 +01-29 +01-29 +01-29 +01-29 +01-29 +11:30 +12:00 +12:30 +13:00 +13:30 +14:00 +14:30 +Date / Time in year 2012A&A proofs: manuscript no. main +Table 1: Parameters of the nine electron ESP events. Associated shock parameters from https://stereo-ssc.nascom.nasa. +gov/pub/ins_data/impact/level3/IPs.pdf. +Date +Sh num./SC +Nsh1 +Vt [km/s]2 +Mms +θBn[◦] +rB3 +er4 +∆t [min]5 +shape +M6 +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +(9) +(10) +(11) +2011-06-05 +87 / A +(1,0)* +1749 +1.54 +52.60 +1.57 +e1 ++66 +peak +60 +2012-01-29 +125 / A +(0,0) +917 +2.00 +87.70 +2.15 +e3 +−13 +peak +480 +2012-03-08 +126 / B +(1,1)* +1147 +1.48 +58.70 +1.60 +e3 +−92 +plateau +480 +2012-05-28 +141 / A +(0,0) +1321 +2.85 +74.80 +2.70 +e3 +−4 +peak +480 +2012-07-23 +151 / A +(0,0) +2099 +2.46 +45.50 +2.17 +e3 +−1 +plateau+peak +480 +2013-11-06 +198 / B +(1,1) +989 +1.92 +64.30 +2.02 +e3 ++31 +peak +15 +2014-09-25 +246 / B +(2,1) +2114 +1.56 +55.30 +1.60 +e1 ++132 +peak +480 +2017-07-24 +316 / A +(0,2) +1197 +2.00 +47.92 +1.92 +e3 +0 +plateau +240 +2017-09-19 +322 / A +(0,2) +985 +2.59 +44.90 +2.33 +e3 +−8 +peak +15 +1The first (second) number in each bracket indicates how many shocks were observed within two days before (after) the corresponding shock +indicated in column (2). +2 Transit speed of the associated CME. +3 Magnetic compression ratio of the shock. +4 The electron energy channel showing the strongest signal in terms of of NF data. +5 ∆t between the peak intensity time and the shock-crossing time. Negative numbers refer to peaks in the upstream region and positive numbers +indicate peaks in downstream region of the shock. +6 Best averaging time window for each event. +*SIR shock occurs one day before the shock crossing time. +electron channels show overall higher intensities than the deka- +MeV proton channels around the time of the shock crossings. +However, in the remaining five cases (shocks number 125, 141, +198, 316 and 322) the overall proton intensities are higher. In two +events (shocks number 125 and 246), the signal of the NF data +of the proton channels are stronger than the those of the electron +channels (Fig. 1 and A.3 (c)). However, in the rest of events the +response of the MeV electrons, as measured by the NF data, is +stronger than that of the protons. The mean value of the mag- +netosonic Mach number of the nine analyzed in-situ shocks is +2.04, and is varying from 1.48 to 2.85. These are not outstanding +values among IP shocks. The shock normal angles θBn vary from +45.5° to 87.7°, i.e., from oblique to quasi-perpendicular. The ma- +jority of the shocks are oblique shocks. Only three shocks are +quasi-perpendicular with shock normal angles larger than 60°. +All the events in Table 1 are associated with interplanetary +CME (ICME)-driven shocks. Having applied different window +length from 15 to 480 minutes we found that the strongest sig- +nals in the NF data are usually found for large window length of +240 or 480 minutes. In the two days before and two days after the +shock passage time, we found other shocks associated with some +of our events, which is illustrated in Fig. 2. For example, shocks +number 125 and 316 have one shock before the on the same day. +The shock number 141 has one shock after, and shock 246 has +two shocks before on the same day. In three events (shocks num- +ber 87, 126, 198), one shock occurred a day before; two of those +are shocks related to stream interaction regions (SIRs). There +was no previous shock in the five events (Fig. 2). +At the time of the in-situ shock crossing, radio spectrograms +observed at the same STEREO spacecraft reveal another feature, +which is present in all of the nine events: an increase in the dy- +namic spectrum simultaneous in a broad range of frequencies. +The feature is most likely quasi-thermal noise extending from +the kHz range up to ∼1 MHz. To gain a better understanding of +this attribute, we investigated the STEREO list of shocks, and +we found that this is a very common trait on the downstream of +the shock crossing. Therefore, it is not a unique feature of the +electron ESP events studied here. +In the following we will discuss some of the most interesting +events in more detail. +Event on 28 May 2012 – shock number 141 +This event (Fig. 3) is an example of a peak-like electron ESP +event and it stands out from the others in terms of its very strong +response in the e3 energy channel. LASCO-C2 recorded a fast +Halo-CME with a linear speed of 1966 km/s at 2:24 UT on 27 +May 2012.4 +The STEREO/WAVES dynamic spectrum shows a type III +radio burst on 26 May 2012 at 20:48 UT (third panel from top +of Fig. 3 (a)). In addition, a type II radio burst, a signature of +a CME-driven shock accelerating electron beams (e.g., Nelson +& Melrose 1985), begins at 20:50 UT and continues until 23:20 +UT between 16000-300 kHz. The first increase in particle in- +tensities occurred simultaneously with the eruption on 26 May. +A significant change of the intensity time profile occurs around +6:00 UT on 27 May 2012, when the fluxes begin to rise again +after a continuous decrease and reach their maximum value 4 +minutes before the shock time, in the upstream region. This is +most likely due to the ICME-driven shock, whose contribution +gets significantly stronger around this time (see Fig. 3 near the +black vertical dashed line marked with shock number 141). +The obliquity θBn of the in-situ shock is almost 75° and the +Mach number is 2.85. The NF data of the electron channels uti- +lizing various window lengths are displayed in the six lower pan- +els of Fig. 3. Significant changes in the SEP intensities start to +show up three hours before the shock, and the fluxes of all en- +ergy channels starts to rise, which also appears as signal changes +in the NF data. The signal in the e3 channel is the strongest com- +pared to the other electron energy channels (blue line in Fig. 3 +plots) and even greater than those of the proton channels (Fig. 3 +(c)), despite the fact that protons have a higher intensity than +electrons in the time-intensity profile (top panel). +Between the other windows lengths, the highest value of the +z-score at the time of shock crossing belongs to M = 480. The +energetic particle time profiles in the top panel of Fig. 3 show +4 http://cdaw.gsfc.nasa.gov/CME_list +Article number, page 4 of 13 + +N. Talebpour Sheshvan, N. Dresing, R. Vainio , A. Afanasiev and D. E. Morosan : On the Role of Interplanetary Shocks in Accelerating MeV +Electrons + + +Shock number / SC +87 / A +124–125 / A +126 / B +141–142 / A +151 / A +198 / B +244–245–246 / B +316–317 / A +322 / A +(in one day) +323 +318 +247 +199 +321 +315 +140 +125 +1 day +SIR +ICME +123 +127 +88 +- 5 day ++20 day +126 +- 20 day ++7 day +143 ++8 day +- 7 day +86 + 149 150 +197 +243 +- 5 day +152 +Fig. 2: All the numbers represent the shock number from the STEREO list (https://stereo-ssc.nascom.nasa.gov/pub/ins_ +data/impact/level3/IPs.pdf). All of the ICME events in the blue central column occurred simultaneously with the relevant +shock passage from our list of nine events. The nine numbers in bold print with underline indicate the events on our list. The SIR +shocks from the list represented by the number of shocks inside the red box. Two way arrows indicate a day and ∓ day denotes a +duration of more than a day between the occurrence of two shocks. +an intensity decay immediately after the shock crossing, ob- +served in both electrons and protons. The magnetic field vari- +ations in this event indicate that the ICME driving the shock and +related to the CME associated with the SEP event has a magnetic +cloud-like structure with a gradually rotating and enhanced-in- +magnitude magnetic field. However, the field shows a strong de- +pletion in magnetic field magnitude around 9:00 UT, which may +indicate the presence of a pre-existing ICME ahead of the driv- +ing ICME that is related to the fast halo CME. The sheath re- +gion of the driving ICME has a complex magnetic structure and +could even have a partly closed magnetic topology. This, how- +ever, needs a more thorough dedicated study in the future. +Event on 23 July 2012 – shock number 151 +This event (Fig. 4) has been studied in detail by several authors +(e.g., Russell et al. 2013; Temmer & Nitta 2015; Riley et al. +2016) as it is one of the most energetic eruptions observed dur- +ing the space era. We do not provide a detailed analysis of this +event but examine the key observational properties in relation to +our event sample. The event is an example of a plateau + peak- +like shaped electron ESP event, with electron fluxes higher than +proton fluxes for the majority of the time-intensity profile. The +LASCO CME catalog reports a fast CME with a speed of 2003 +km/s at 2:24 UT on 23 July 2012, followed by a clear type II ra- +dio burst (third panel from top Fig. 4 (a)), which continued until +21:40 UT visible by both STEREO A and B (not shown). HET +measured a very gradual SEP event associated with a type III +burst at 01:45 UT on 23 July 2012 . The enhancement of particle +intensities associated with the fast forward ICME-driven shock +observed by STEREO A at 20:55 UT on 23 July 2012 reached its +peak value 1 minute before the shock time, in the upstream re- +gion. It is an oblique shock with θBn = 45◦ and a Mach number of +2.46 based on the shock list (but we note that many of the earlier +detailed analyses give somewhat different values for the obliq- +uity and strength of the shock). The electron flux increases four +orders of magnitude during this event making it an exceptionally +intense event. By zooming in around the ESP event (Fig. 4 (b) +and (c)), we see a pronounced plateau-shaped intensity increase +between 19:15 to 22:56 with the time of the shock crossing in +the middle. During this time, the magnetic field gradually starts +to weaken and only a small jump in magnetic field magnitude +is observed at the shock (2nd panel from top). In the NF data, +the HET e3 channel (2.8–4.0 MeV) exhibits the strongest shock +response. The M = 480 window length had the highest value by +computing the z-score around the shock crossing in six different +window lengths. The end of the plateau-like shape of intensities +around 23:00 UT on 23 July is accompanied by the beginning of +an ICME, which has a highly compressed magnetic field reach- +ing values beyond 100 nT. The ICME has again a complicated +structure that may consist of two individual flux ropes (Russell +et al. 2013; Riley et al. 2016), first of which would be related to +a pre-existing CME in the solar wind overtaken by the fast erup- +tion. This implies that the strong decrease of fluxes at the onset +of the ICME would be related to a change in the topology of the +field. +Event on 6 November 2013 – shock number 198 +Shock number 198 in Fig. 5, which occurred on 6 November +2013, is unique in terms of being the only MeV electron ESP +event in our sample without a previous SEP event (Chiappetta +et al. 2021). According to the SOHO LASCO CME Catalog, the +associated fast Halo-CME occurred with a linear speed of 1040 +km/s at 5:12 UT on 4 November 2013. Both STEREO/WAVES +recorded a strong type II radio burst extending to very low fre- +quency (not shown). Magnetic field components increased in the +first hours of 5 Nov 2013 in the upstream region at the same time +as the quasi-thermal noise signature in the radio spectrogram. +About 40 minutes before the shock crossing a small type III ra- +dio burst appeared in the radio spectrogram. This new solar event +might have injected SEPs at the right time to provide seed parti- +Article number, page 5 of 13 + +A&A proofs: manuscript no. main +cles for the in-situ shock on 6 Nov leading to electron intensity +peaks close to the shock crossing time (31 min) on the down- +stream side. The e2 and e3 channels responded in a similar way +as indicated by the NF data, but the response of the e1 channel +is weaker. The magnetic field associated with the ICME driving +this event is less intense than in the other two example events. It +is characterized by the presence of large-amplitude fluctuations +throughout the event. Also in this case, the downstream of the +shock seems to host a compressed magnetic field region between +the shock and the driving ICME related to the fast halo CME +(from about 02:40 to 05:00 UT), which coincides with the start +of the decrease of particle fluxes indicating a possible change in +field topology. The structure of the field and the duration of the +compressed region does not, however, seem to be consistent with +a double flux-rope structure like in the 23 July 2012 event. +Shock number 199 – example of an event missed by the fil- +tering method +Shock number 198 is followed by a gradual SEP event that oc- +curred on 7 Nov 2013 (see Fig. 6 (a)) in conjunction with a Halo +CME, resulting in a type II radio burst in the dynamic spectrum. +The SEP event was also studied by Dresing et al. (2016a), who +reported that STEREO B was situated inside a magnetic cloud +of a pre-event ICME during the start of the SEP event. +The 7 Nov 2013 CME drove a shock that passed STEREO B +on 8 Nov (shock number 199). The SEP event has a long dura- +tion and a plateau-type profile, also in electron channels e1–e3. +This could be an indication of a prolonged electron injection that +could be related to continuous acceleration at the shock. How- +ever, the signal of the NF data does not show a significant re- +sponse of the electron channels for the associated IP shock cross- +ing of shock number 199 (Fig. 6 b). This implies that the shock +does not maintain its ability to accelerate electrons up to 1 AU. +4. Discussion and Conclusion +We scanned the STEREO mission for interplanetary shocks +that show local acceleration signatures of relativistic electrons. +Nine events were identified by applying a novel filtering method +that identifies significant peaks in time-intensity profiles of +HET energy channels. All shocks were associated with ICMEs. +The events were divided in three classes based on their time- +intensity profiles, i.e., plateau-like, peak-like and plateau+peak- +like events. The identified electron ESP events had variable elec- +tron to proton ratios, some being proton dominated while some +were dominated by electrons in the STEREO/HET energy range. +The nine events that show electron acceleration signatures +are all related to very fast shocks with transit speeds exceeding +900 km/s. The transit speed was recently identified by Ameri +et al. (2022) as the best correlated shock property with the ∼10 +MeV proton intensities at the shock in ESP events. A high transit +speed may imply that a shock has been strong all its way from +the Sun to the observer, allowing it to maintain its acceleration +efficiency for an extended amount of time. +Three ESP events were analyzed in more detail. Two of them +were associated with SEP events and one was an isolated ESP +event. The ones associated with SEP events were more intense. +Their associated ICMEs had complex magnetic structure with +indications of interactions in the interplanetary space between +pre-existing ICME structures and the fast eruption driving the +event. The isolated ESP event is not as strong as the ones asso- +ciated with SEPs. It, too, had a complex ICME driver, although +the peak magnitude of the interplanetary magnetic field was not +as high as in the SEP-event-related events. Although it lacked an +SEP event related to the eruption, the isolated event had a type +III burst occurring a few hours before the shock passage just be- +fore the start of rise of the energetic particle fluxes. This solar +event could have provided seed particles for the shock and thus +facilitated the acceleration of electrons to energies detected in +HET energy range. +While it performs well with ESP-like events, our filtering +method is unable to detect more gradual contributions of shocks +to the energetic electron fluxes. Some events (e.g., shock no. +199) show plateau-like but gradually decaying intensities un- +til the shock passage, and then faster decays after the passage +of the leading edge of the ICME. In such cases, shocks may +still have significantly contributed to electron acceleration as the +shock propagates from the corona to 1 AU. Equipped with the +new instruments onboard Solar Orbiter and Parker Solar Probe +that reach close distances to the Sun, such events may appear as +much more clear-cut ESP events. +In conclusion, while ESP events accelerating electrons to rel- +ativistic energies are not very common at 1 AU, there are still a +number of cases that show signatures of local acceleration by the +shock and/or its sheath region up to several MeVs. The next so- +lar maximum with fast CMEs and new observational capabilities +will likely bring more clarity to the issue of electron acceleration +by interplanetary shocks. +Acknowledgements. We acknowledge financial support from the European +Union’s Horizon 2020 research and innovation programme under grant agree- +ment No. 101004159 (SERPENTINE) and the support of Academy of Finland +(FORESAIL, grants 312357 and 336809). SOHO LASCO CME catalog is gener- +ated and maintained at the CDAW Data Center by NASA and The Catholic Uni- +versity of America in cooperation with the Naval Research Laboratory. SOHO is +a project of international cooperation between ESA and NASA. +References +Ameri, D., Valtonen, E., Al-Sawad, A., & Vainio, R. 2022, Advances in Space +Research +Bruno, A., Bazilevskaya, G. A., Boezio, M., et al. 2018, ApJ, 862, 97 +Bryant, D. A., Cline, T. L., Desai, U. D., & McDonald, F. B. 1962, J. Geo- +phys. 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Pogorelov, +J. Spann, & O. Verkhoglyadova, 191–194 +Kaiser, M. L., Kucera, T. A., Davila, J. M., et al. 2008, Space Sci. Rev., 136, 5 +Kouloumvakos, A., Nindos, A., Valtonen, E., et al. 2015, A&A, 580, A80 +Lario, D., Aran, A., & Decker, R. B. 2008, Space Weather, 6, S12001 +Lario, D., Ho, G. C., Decker, R. B., et al. 2003, AIP Conference Proceedings, +679, 640 +Lario, D., Hu, Q., Ho, G. C., et al. 2005, in ESA Special Publication, Vol. 592, +Solar Wind 11/SOHO 16, Connecting Sun and Heliosphere, ed. B. Fleck, +T. H. Zurbuchen, & H. Lacoste, 81 +Lee, M. A., Mewaldt, R. A., & Giacalone, J. 2012, Space Sci. Rev., 173, 247 +Luhmann, J. G., Curtis, D. W., Schroeder, P., et al. 2008, Space Sci. Rev., 136, +117 +Mäkelä, P., Gopalswamy, N., Akiyama, S., Xie, H., & Yashiro, S. 2011, Journal +of Geophysical Research (Space Physics), 116, A08101 +Article number, page 6 of 13 + +N. Talebpour Sheshvan, N. Dresing, R. Vainio , A. Afanasiev and D. E. Morosan : On the Role of Interplanetary Shocks in Accelerating MeV +Electrons +(a) +(b) +(c) +Fig. 3: SEP event on 27 May 2012 and corresponding ESP event on 28 May 2012 observed by STEREO A. (a) from top to bottom: +time profile of proton and electron intensities at different energy channels as indicated in the legend in the top panel of (b), the +evolution of magnetic field magnitude and its components in RTN coordinates, STEREO/WAVES dynamic spectrum. The lower six +panels show the normalized filtered data of the three electron channels using a fixed time lag parameter of (N = 180) and increasing +window lengths M. Figures (b) and (c) are the same format as 1. The boxes show the N and M value of each panel and the gray one +indicates the the one yielding the most significant signal of the NF data. +(a) +(b) +(c) +Fig. 4: Summary of electron and proton observations by the STEREO/HET instrument on 23 July 2012. Figures (a), (b) and (c) +have the same format as Fig. 3 +Article number, page 7 of 13 + +STEREO-A / HET / 2012-07-23 +105 +104 +103 +102 +151 +25 +0 +25 +101 +100 +10-1 +10-2 +N=180, M=15 +40 +20 +0 +N=180, M=30 +40 +20 +0 +N=180, M=60 +40 +02 +0 +N=180, M=120 +40 +20 +0 +N=180. M=240 +40 +20 +0 +N=180, M=480 +40 +20 +07-23 +07-23 +07-23 +07-23 +07-23 +07-23 +07-23 +07-23 +07-23 +07-23 +19:00 +19:30 +20:00 +20:30 +21:00 +21:30 +22:00 +22:30 +23:00 +23:30 +Date / Time in year 2012STEREO A/ HET / 2012-05 +102 +142 +101 +100 +10-1 +141 +10-2 +[BI +(lu) ug +25 +R +0 +-25 +N +101 +f (MHz) +100 +10-1 +10-2 +N=180, M=15 +0 +100 +N=180, M=30 +wou +50 +0 +N=180, M=60 +0 +100 +N=180, M=120 +50 +0 +N=180, M=240 +0 +N=180, M=480 +0 +05-27 +05-27 +05-27 +05-27 +05-28 +05-28 +05-28 +05-28 +05-29 +00:00 +06:00 +12:00 +18:00 +00:00 +06:00 +12:00 +18:00 +00:00 +Date / Time in year 2012STEREO-A / HET / 2012-05-28 +102 +0.7-1.4 MeV e +1.4-2.8 MeV e +2.8-4.0 MeV e +101 +13.6-15.1 MeV p +15.0-17.1 MeV p +17.0-19.3 MeV p +100 +20.8-23.8 MeV p +141 +25 +25 +N=180, M=15 +50 +0 +N=180, M=30 +50 +N=180, M=60 +50 +0 +N=180, M=120 +50 +0 +N=180, M=240 +50 +0 +N=180, M=480 +50 +e3:mean+2std +05-27 +05-28 +05-28 +05-28 +05-28 +05-28 +05-28 +05-28 +05-28 +23:00 +00:00 +01:00 +02:00 +03:00 +04:00 +05:00 +06:00 +07:00 +Date / Time in year 2012STEREO-A / HET / 2012-05-28 +102 +101 +141 +100 +25 +0 +-25 +101 +100 +10-1 +10-2 +N=180, M=15 +40 +20 +:180,M=30 +40 +20 +=180, M=60 +40 +20 +I=180,M=120 +40 +20 +=180, M=240 +50 +25 +=180, M=480 +50 +25 +05-28 +05-28 +05-28 +05-28 +05-28 +05-28 +05-28 +05-28 +02:00 +02:15 +02:30 +02:45 +03:00 +03:15 +03:30 +03:45 +Date / Time in year 2012STEREO-A / HET / 2012-07 +104 +102 +100 +151 +100 +B +() +R +0 +N +101 +100 +10-2 +N=180, M=15 +0 +N=180, M=30 +0 +N=180,M=60 +N=180, M=120 +N=180, M=240 +400 +N=180, M=480 +07-23 +07-23 +07-23 +07-23 +07-23 +07-24 +07-24 +07-24 +07-24 +07-24 +04:00 +08:00 +12:00 +16:00 +20:00 +00:00 +04:00 +08:00 +12:00 +16:00 +Date / Time in year 2012STEREO-A / HET / 2012-07-23 +105 +0.7-1.4 MeV e +1.4-2.8 MeV e +104 +2.8-4.0 MeV e +13.6-15.1 MeV p +103 +15.0-17.1 MeV p +17.0-19.3 MeV p +20.8-23.8 MeV p +102 +151 +25 +-25 +N=180, M=15 +40 +20 +0 +N=180, M=30 +40 +20 +0 +N=180, M=60 +40 +20 +0 +N=180, M=120 +40 +20 +0 +N=180, M=240 +40 +20 +0 +N=180, M=480 +40 +20 +e3:mean+2std +M +0 +07-23 +07-23 +07-23 +07-23 +07-23 +07-23 +18:00 +19:00 +20:00 +21:00 +22:00 +23:00 +Date / Time in year 2012A&A proofs: manuscript no. main +Fig. 5: Same format as Fig. 3 showing STEREO B/HET observations of the event associated with the ICME-driven shock on 6 Nov +2013. +Fig. 6: Same format as Fig. 3 but showing STEREO B/HET observations the event associated with the ICME-driven shock on 8 +Nov 2013. +Mitchell, J. G., Nolfo, G. A. D., Hill, M. E., et al. 2021, The Astrophysical Jour- +nal, 919, 119 +Nelson, G. J. & Melrose, D. B. 1985, in IN: Solar radiophysics: Studies of emis- +sion from the sun at metre wavelengths (A87-13851 03-92). Cambridge and +New York, Cambridge University Press, 1985, p. 333-359., ed. D. J. McLean +& N. R. Labrum, 333–359 +Rao, U. R., McCracken, K. G., & Bukata, R. P. 1967, Journal of Geophysical +Research (1896-1977), 72, 4325 +Reames, D. V. 1999, Space Sci. Rev., 90, 413 +Reames, D. V. 2013, Space Sci. Rev., 175, 53 +Reames, D. V., Cliver, E. W., & Kahler, S. W. 2014, Sol. Phys., 289, 3817 +Reid, H. A. S. & Ratcliffe, H. 2014, Research in Astronomy and Astrophysics, +14, 773 +Riley, P., Caplan, R. M., Giacalone, J., Lario, D., & Liu, Y. 2016, ApJ, 819, 57 +Russell, C. T., Mewaldt, R. A., Luhmann, J. G., et al. 2013, ApJ, 770, 38 +Sarris, E. T. & van Allen, J. A. 1974, J. Geophys. Res., 79, 4157 +Article number, page 8 of 13 + +STEREO-B / HET / 2013-11 +102 +198 +(cm2-sr-s-Mev)- +101 +100 +10- +10-2 +25 +[BI +(l) g +R +0 +N +-25 +101 +100 +(MHz) +10-2 +N=180, M=15 + 25 +0 +N=180, M=30 +25 +0 +N=180, M=60 + 25 +0 +N=180, M=120 +25 +N=180, M=240 + 25 +0 +N=180, M=480 +25 +0 +11-04 +11-05 +11-05 +11-06 +11-06 +11-07 +12:00 +00:00 +12:00 +00:00 +12:00 +00:00STEREO-B / HET / 2013-11-06 +0.7-1.4 MeV e +100 +1.4-2.8 MeV e +2.8-4.0 MeV e +13.6-15.1 MeV p +10-1 +198 +15.0-17.1 MeV p +17.0-19.3 MeV p +20.8-23.8 MeV p +10-2 +25 +0 +25E +N=180, M=15 +50 +N=180, M=30 +50 +N=180, M=60 +50 +N=180, M=120 +50 +N=180, M=240 +50 +N=180, M=480 +50 +MMM +e3:mean+2std +11-06 +11-06 +11-06 +11-06 +11-06 +11-06 +11-06 +11-06 +11-06 +00:00 +00:30 +01:00 +01:30 +02:00 +02:30 +03:00 +03:30 +04:00STEREO-B / HET / 2013-11-06 +100 +10-1 +198 +10-2 +25 +0E +-25 +101 +100 +10-1 +10-2 +50 +N=180, M=15 +25 +50 +N=180, M=30 +25 +0 +50 +N=180, M=60 +25 +50 +N=180, M=120 +25 +50 +N=180, M=240 +25 +50 +N=180, M=480 +25 +11-06 +11-06 +11-06 +11-06 +11-06 +11-06 +11-06 +11-06 +11-06 +00:00 +00:30 +01:00 +01:30 +02:00 +02:30 +03:00 +03:30 +04:00STEREO-B / HET / 2013-11 +102 +101 +100 +10 +199 +10-2 +25 +[BI +() g +R +0 +N +-25 +101 +100 +(MHz) +10-2 +400 +N=180, M=15 +N +0 +N=180, M=30 +0 +N=180, M=60 +408 +N=180, M=120 +0 +N=180, M=240 +0 +N=180, M=480 +0 +11-07 +11-07 +11-08 +11-08 +11-08 +11-08 +11-09 +11-09 +12:00 +18:00 +00:00 +06:00 +12:00 +18:00 +00:00 +06:00STEREO-B / HET / 2013-11-08 +102 +0.7-1.4 MeV e +1.4-2.8 MeV e +101 +2.8-4.0 MeV e +13.6-15.1 MeV p +100 +15.0-17.1 MeV p +20.8-23.8 MeV p +10-1 +199 +10F +-10 +N=180, M=15 +L +N=180, M=30 +A! +1N=180, M=60 +N=180, M=120 +Ld +N=180, M=240 +1 +N=180, M=480 +e3:mean+2std +1 +11-08 +11-08 +11-08 +11-08 +11-08 +11-08 +11-08 +11-08 +11:30 +12:00 +12:30 +13:00 +13:30 +14:00 +14:30 +15:00STEREO-B / HET / 2013-11-08 +102 +101 +100 +199 +10-1 +25 +0 +-25 +101 +100 +10-1 +10-2 +N=180, M=15 +M +N=180, M=30 +N=180, M=60 +N=180, M=120 +N=180, M=240 +N=180, M=480 +0m +11-08 +11-08 +11-08 +11-08 +11-08 +11-08 +11-08 +11-08 +11-0811-08 +09:00 +10:00 +11:00 +12:00 +13:00 +14:00 +15:00 +16:00 +17:00 +18:00N. Talebpour Sheshvan, N. Dresing, R. Vainio , A. Afanasiev and D. E. Morosan : On the Role of Interplanetary Shocks in Accelerating MeV +Electrons +Simnett, G. M. 2003, Sol. Phys., 213, 387 +Temmer, M. & Nitta, N. V. 2015, Sol. Phys., 290, 919 +Tsurutani, B. T. & Lin, R. P. 1985, J. Geophys. Res., 90, 1 +von Rosenvinge, T. T., Reames, D. V., Baker, R., et al. 2008, Space Sci. Rev., +136, 391 +Wraase, S., Heber, B., Böttcher, S., et al. 2018, A&A, 611, A100 +Yang, L., Wang, L., Li, G., et al. 2019, The Astrophysical Journal, 875, 104 +Yashiro, S., Gopalswamy, N., Michalek, G., et al. 2004, Journal of Geophysical +Research (Space Physics), 109, A07105 +Article number, page 9 of 13 + +A&A proofs: manuscript no. main +Appendix A: Additional STEREO/HET observations +of electron ESP events +For each event listed in Table 1 we show a figure in the for- +mat of Fig. 3 presenting a long duration time profile of the SEP +event and corresponding ESP particle intensities measured by +STEREO/HET, the magnetic field magnitude and its compo- +nents measured by the IMPACT/MAG instrument in RTN co- +ordinates and a STEREO/WAVES dynamic spectrum. We also +show the normalized filtered data with the fixed value of three- +hour time lag (N) for all the cases and six given values for a +temporal window of background intensity from 15 minutes to 8 +hours (M). The figures contain the zoom-in profile around the +shock crossing time and the shaded area of the mean plus two +times of SD of the moving background window (the color of the +shaded area is based on the electron energy channel showing the +strongest signal in terms of of NF data). In addition, NF data for +the proton channels have been added for comparison. The black +and orange dashed lines on the plots represent the time of shock +crossing and peak intensity time of selected energy channel of +electron, respectively. +Appendix B: A test for proton contamination in the +electron energy channels +During periods of high proton intensities, such as in ESP events, +the detection of small electron intensities can be difficult or even +contaminated by the protons, which is a known feature in ener- +getic particle instruments such as STEREO/SEPT (e.g., Wraase +et al. 2018). To determine if there is proton contamination of +the electron measurements used in this study we plot 2D his- +tograms of deka-MeV proton intensities as a function of 2.8–4.0 +MeV electron intensities in Fig B.1. We use 1-minute data of the +whole year of 2013. None of the three proton channels shows a +one-to-one correlation with the electron intensities. Furthermore, +the low or even missing electron intensities at times when high +proton intensities are observed (upper left corner), supports the +assumption that proton contamination does not play a significant +role in the electron channels. +Article number, page 10 of 13 + +N. Talebpour Sheshvan, N. Dresing, R. Vainio , A. Afanasiev and D. E. Morosan : On the Role of Interplanetary Shocks in Accelerating MeV +Electrons +Fig. A.1: Same format as Fig. 3 but showing STEREO A/HET observations of an SIR-associated shock on 4 June 2011 and a +ICME-driven shock on 5 June 2011. +Fig. A.2: Same format as Fig. 3 but showing STEREO B/HET observations of an SIR shock on 7 Mar 2012 and a ICME-driven +shock on 8 Mar 2012. +Article number, page 11 of 13 + +STEREO-A/ HET /2011-06 +103 +86 +leV)-1 +101 +10 +87 +15 +/B/ +Brtn +25 +N +101 +B +10 +19 +(MH +10- +10 +50 +N=180, M=15 +0 +N=180, M=30 +N=180, M=60 +N=180, M=120 +N=180, M=240 += 20 +N=180, M=480 +06-04 +06-05 +06-05 +06-06 +06-06 +12:00 +00:00 +12:00 +00:00 +12:00 +Date / Time in year 2011STEREO-A/ HET / 2011-06-05 +0.7-1.4 MeV e +102 +1.4-2.8 MeV e +W +2.8-4.0 MeV e +13.6-15.1 MeVp +15.0-17.1 MeV p +17.0-19.3 MeV p +101 +20.8-23.8 MeV p +87 +10 +-10 +N=180, M=15 +2 +N=180, M=30 +N=180, M=60 +2 +N=180, M=120 +M +N=180, M=240 +N=180, M=480 +0 +el:mean+2std +06-05 +06-05 +06-05 +06-05 +06-05 +06-05 +06-05 +16:00 +17:00 +18:00 +19:00 +20:00 +21:00 +22:00 +Date / Time in year 2011STEREO-A / HET /2011-06-05 +102 +M +87 +10 +0 +-10 +101 +100 +10-1 +10-2 +2 +N=180, M=15 +N=180, M=30 +0W +N=180, M=60 +0W +N=180, M=120 +0W +N=180, M=240 +N=180, M=480 +06-05 +06-05 +06-05 +06-05 +06-05 +06-05 +06-05 +16:00 +17:00 +18:00 +19:00 +20:00 +21:00 +22:00 +Date / Time in year 2011STEREO-B / HET / 2012-03 +102 +(cm?-sr-s-MeV) +101 +100 +125 +126 +25 +[B +Brtn (nT) +R +Y +0 +25 +N +101 +100 +(ZHW) +10-1 +f +10-2 +100 +N=180, M=15 +50 +0 +100 +N=180, M=30 +50 +0 +100 +N=180, M=60 +Alnorm +50 +0 +N=180, M=120 +50 +0 +N=180, M=240 +0 +N=180, M=480 +0 +03-07 +03-07 +03-07 +03-07 +03-08 +03-08 +03-08 +03-08 +03-09 +00:00 +06:00 +12:00 +18:00 +00:00 +06:00 +12:00 +18:00 +00:00 +Date / Time in year 2012STEREO-B / HET / 2012-03-08 +0.7-1.4'MeV e +102 +1.4-2.8 MeV e +2.8-4.0 MeV e +13.6-15.1 MeV p +101 +15.0-17.1 MeV p +126 +17.0-19.3 MeV p +20.8-23.8 MeV p +100 +25 +25 +N=180, M=15 +0 +N=180, M=30 +N=180, M=60 +0 +N=180, M=120 +0 +N=180, M=240 +0 +e3:mean+2std +N=180, M=480 +0 +03-08 +03-08 +03-08 +03-08 +03-08 +03-08 +03-08 +03-08 +03-08 +03-08 03-09 +04:00 +06:00 +08:00 +10:00 +12:00 + 14:00 +16:00 +18:00 +20:00 +22:00 00:00 +Date / Time in year 2012STEREO-B / HET / 2012-03-08 +102 +101 +126 +WWWL +100 +25 +0 +-25 +101 +100 +10-1 +10-2 +N=180, M=15 +N=180, M=30 +0 +N=180, M=60 +-P +1 +N=180, M=120 +1 +01 +N=180, M=240 +1 +N=180, M=480 +0 +-1 +03-08 +03-08 +03-08 +03-08 +03-08 +03-08 +03-08 +03-08 +10:00 +11:00 +12:00 +13:00 +14:00 +15:00 +16:00 +17:00 +Date / Time in year 2012A&A proofs: manuscript no. main +Fig. A.3: Same format as Fig. 3 but showing STEREO B/HET observations of three ICME-driven shocks in one day on 25 Sep +2014. +Fig. A.4: Same format as Fig. 3 but showing STEREO A/HET observations of two ICME-driven shocks on 24 Sep 2017. +Article number, page 12 of 13 + +STEREO-B / HET / 2014-09-25 +101 +100 +10-1 +246 +244 +245 +10-2 +50 +[BI +() g +R +0 +-50 +N +101 +f (MHz) +100 +10-1 +10-2 +100 +N=180,M=15 +wou +50 +100 +N=180, M=30 +50 +0 +100 +N=180, M=60 +50 +0 +100 +N=180, M=120 +Alnom +50 +0 +100 +N=180, M=240 +Alnom +50 +0 +100 +N=180, M=480 +50 +09-24 +09-25 +09-25 +09-25 +09-25 +09-25 +09-25 +09-25 +09-25 +09-26 +21:00 +00:00 +03:00 +06:00 +09:00 +12:00 +15:00 +18:00 +21:00 +00:00 +Date / Time in year 2014STEREO-B / HET / 2014-09-25 +0.7-1.4 MeV e +1.4-2.8 MeV.e +W +2.8-4.0 MeV.e +101. +13.6-15.1 MeV p +15.0-17.1 MeVp +17.0-19.3 MeV p +20.8-23.8 MeV p +100 +246 +50 +-50 +N=180, M=15 +N=180, M=30 +HUM +N=180, M=60 +AAN +N=180, M=120 +N=180, M=240 +N=180, M=480 +el:mean+2std +WNY +09-25 +09-25 +09-25 +09-25 +09-25 +09-25 +09-25 +14:00 +15:00 +16:00 +17:00 +18:00 +19:00 +20:00 +Date / Time in year 2014STEREO-B / HET / 2014-09 +245 +101 +246 +100 +50 +-50 +101 +100 +10-1 +10-2 +N=180, M=15 +5 +N=180, M=30 +N=180, M=60 +N=180, M=120 +5 +N=180, M=240 +5 +N=180, M=480 +09-25 +09-25 +09-25 +09-25 +09-25 +09-25 +09-25 +14:00 +15:00 +16:00 +17:00 +18:00 +19:00 +20:00 +Date / Time in year 2014STEREO-A / HET / 2017-07 +102 +101 +100 +316 +10-1 +317 +10-2 +50 +[B +Brtn (nT) +0 +-50 +N +101 +(MHz) +100 +4 +10-1 +10-2 +40 +N=180, M=15 +uou +20 +N +0 +40 +N=180, M=30 +40 +N=180, M=60 +nom +50 +N=180, M=120 +0 +N=180, M=240 +0 +100 +N=180, M=480 +Alnom +50 +07-23 +07-23 +07-23 +07-24 +07-24 +07-24 +07-24 +07-25 +07-25 +07-25 +06:00 +12:00 +18:00 +00:00 +06:00 +12:00 +18:00 +00:00 +06:00 +12:00 +Date / Time in year 2017STEREO-A/ HET / 2017-07-24 +0.7-1.4 MeV e +102 +1.4-2.8 MeV e +2.8-4.0 MeV e +13.6-15.1 MeV p +wMmmn +NNAAM +101 +SWM +15.0-17.1 MeV p +w +17.0-19.3 MeYp +20.8-23.8 MeV p +100 +316 +317 +10 +N=180,M=15 +N=180, M=30 +N=180, M=60 +W +N=180, M=120 +0 +N=180, M=240 +N=180, M=480 +e3:mean +t2sto +07-24 +07-24 +07-24 +07-24 +07-24 +07-24 +07-24 +10:00 +11:00 +12:00 +13:00 +14:00 +15:00 +16:00 +Date / Time in year 2017STEREO-A / HET / 2017-07-24 +102 +101 +WW +NENWNY +316 +1% +0 +119 +100 +10-1 +10-2 +N=180, M=15 +N=180, M=30 +0 +N=180, M=60 +N=180, M=120 +N=180. M=240 +N=180, M=480 +M +0 +07-24 +07-24 +07-24 +07-24 +07-24 +07-24 +07-24 +10:00 +11:00 +12:00 +13:00 +14:00 +15:00 +16:00 +Date / Time in year 2012N. Talebpour Sheshvan, N. Dresing, R. Vainio , A. Afanasiev and D. E. Morosan : On the Role of Interplanetary Shocks in Accelerating MeV +Electrons +(a) +(b) +(c) +Fig. A.5: Same format as Fig. 3 but showing STEREO A/HET observations of an ICME-driven shock on 19 Sep 2017 at 2:56 with +prominent plasma signatures on the downstream side of the shock. +Fig. B.1: 2D histograms of varying proton versus 2.8–4.0. MeV electron intensities measured by STEREO-A/HET using 1-minute +data of the whole year 2013. The plots show the logarithm of the intensities (excluding all zero intensities). The color scale (marking +the number of observations in each bin) is logarithmic. The black solid line marks unity. The left, middle, and right panels show +13.6–15.1 MeV, 14.9– 17.1 MeV, and 17.0–19.3 MeV protons, respectively. +Article number, page 13 of 13 + +STEREO-A / HET / 2017-09 +102 +-1 +101 +100 +322 +25 +() g +B +R +0E +-25 +N +101 +100 +(MHz) +10-1 +f +10-2 +N=180, M=15 +N=180, M=30 +N=180, M=60 +N=180, M=120 +0w +N=180, M=240 +N=180, M=480 +09-17 +09-17 +09-18 +09-18 +09-18 +09-18 +09-19 +09-19 +09-19 +09-19 +12:00 +18:00 +00:00 +06:00 +12:00 +18:00 +00:00 +06:00 +12:00 +18:00 +Date / Time in year 2017STEREO-A/ HET / 2017-09-19 +102 +0.7-1.4 MeV e +1.4-2.8 MeV e +101 +2.8-4.0 MeV e +EAA +13.6-15.1 MeV p +15.0-17.1 MeVp. +100 +W +17.0-19.3 MeV p +20.8-23.8 MeV p +10-1 +322 +25 +-25 +10 +N=180, M=15 +10 +N=180, M=30 +0K +10 +N=180, M=60 +10 +N=180, M=120 +0 +10 +N=180, M=240 +10 +e3:mean+2std +N=180, M=480 +09-19 +09-19 +09-19 +09-19 +09-19 +09-19 +09-19 +00:00 +01:00 +02:00 +03:00 +04:00 +05:00 +06:00 +Date / Time in year 2017STEREO-A/ HET / 2017-09-19 +102 +101 +100 +322 +10-1 +25 +-25 +101 +100 +10-1 +10-2 +20 +N=180, M=15 +10 +28 +N=180, M=30 +10 +28 +N=180, M=60 +10 +28 +N=180, M=120 +10 +28 +N=180, M=240 +10 +28 +N=180, M=480 +10 +1 +09-19 +09-19 +09-19 +09-19 +09-19 +01:00 +02:00 +03:00 +04:00 +05:00 +Date / Time in year 2017STEREOAHET1min2013 +2.0 +E104 +1.5 +103 +P +1.0 +.MeV +g10(13.6-15.1 +0.5 +E 102 +0.0 +601 +0.5 +101 +-1.0 +1.5 +100 +1.5 +-1.0 +0.5 +0.0 +0.5 +log10 (2.8-4 MeV e)STEREOAHET1min2013 +2.0 +E104 +1.5 +1.0 +E 103 + (14.9-17.1 MeV p +0.5 +0.0 +102 +10 +-0.5 +601 +-1.0 +101 +-1.5 +100 +1.5 +-1.0 +0.5 +0.0 +0.5 +log10 (2.8-4 MeV e)STEREOAHET1min2013 +1.5 +1.0 +E 103 +(17.0-19.3MeVp +0.5 +0.0 +102 +log10 +-0.5 +1.0 +101 +-1.5 +100 +-1.5 +-1.0 +0.5 +0.0 +0.5 +log10 (2.8-4 MeV e) \ No newline at end of file diff --git a/SdE5T4oBgHgl3EQfaQ-N/content/tmp_files/load_file.txt b/SdE5T4oBgHgl3EQfaQ-N/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c11f1c7707bcc0fa45d2e323bc37f6212ce898bf --- /dev/null +++ b/SdE5T4oBgHgl3EQfaQ-N/content/tmp_files/load_file.txt @@ -0,0 +1,1093 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf,len=1092 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' main ©ESO 2023 January 16, 2023 On the Role of Interplanetary Shocks in Accelerating MeV Electrons N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Talebpour Sheshvan1, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Dresing1, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Vainio1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Afanasiev1 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Morosan2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Department of Physics and Astronomy, University of Turku, Finland 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Department of Physics, University of Helsinki, Finland e-mail: nasrin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='talebpoursheshvan@utu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='fi ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' One of the sources of solar energetic particle (SEP) events is shocks that are driven by fast coronal mass ejections (CMEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' They can accelerate SEPs up to relativistic energies and are attributed to the largest SEP events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' New studies suggest that CME-driven shocks can potentially accelerate electrons to MeV energies in the vicinity of the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' We focus on relativistic electrons associated with strong IP shocks between 2007 and 2019 to determine whether the shocks can keep accelerating such electrons up to 1 AU distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' We have analyzed High Energy Telescope (HET) observations aboard the STEREO spacecraft of potential electron ener- getic storm particle (ESP) events, characterized by intensity time series that peak at the time of, or close to, the associated CME-driven shock crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' We present a new filtering method to assess the statistical significance of particle intensity increases and apply it to MeV electron observations in the vicinity of interplanetary shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' We employed a STEREO in-situ shock list, which contained in total 587 shocks at the two STEREO spacecraft, from which we identified 27 candidate events by visual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Our method identified nine clear cases, where a significant increase of MeV electrons was found in association with a shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Typically, the highest statistical significance was observed in the highest of the three HET energy channels (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' All nine cases were associated with shocks driven by interplanetary CMEs that showed large transit speeds, in excess of 900 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In several cases multiple shocks were observed within one day of the shock related to the electron increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Although electron ESP events at MeV energies are found to be rare at 1 AU our filtering method is not designed to identify a potential interplanetary shock contribution from distances closer to the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Observations by Parker Solar Probe or Solar Orbiter, taken during closer approaches to the Sun, will likely provide clarity on interplanetary shock acceleration of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Sun: coronal mass ejections (CMEs) – Sun: particle emission - Acceleration of particles - shock waves – interplanetary medium 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Introduction Shock waves propagating in the interplanetary (IP) space are potential accelerators of energetic particles in the heliosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Coronal mass ejections (CMEs) propagating through the IP space are the main drivers of IP shocks, in particular during the solar active years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Gradual solar energetic particle (SEP) events (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Reames 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Reames 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Reames et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Desai & Giacalone 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Bruno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2018) are thought to be accelerated by CME-driven shocks propagating through the corona and IP space (Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Energetic storm particle (ESP) events are increases of energetic particle intensities associated with the passage of transient interplane- tary shocks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Bryant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 1962;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 1967;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Huttunen-Heikinmaa & Valtonen 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The role of IP shocks for proton intensity enhancements has been investigated in a broad energy range from a few keV to tens of MeV (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Dresing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2016b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Mäkelä et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Ameri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The first ESP events were re- ported by Bryant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' (1962) and classified by Sarris & van Allen (1974) in two categories, spike events and classic ESP events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Spike events are characterized by a sudden rise in proton intensities lasting only between 5 and 20 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In contrast, classic ESP events exhibit gradual intensity increases several hours before the shock passage in agreement with the predic- tions of the classical diffusive shock acceleration (DSA) theory (Desai & Giacalone 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' (2003, 2005) suggested also other types of ESP events based on the features of the in- tensity profile like ESP+spike or step-like events and those with irregular time-intensity profiles not related to the time of shock crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Although it is well established that protons frequently exhibit classic ESP events, the role of shocks in electron acceleration, especially in IP space, is still not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Direct observations of electron acceleration at shocks in IP space are rare (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Simnett 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Previous studies have analyzed in- situ shock crossings at spacecraft situated at 1 AU and found the shock acceleration efficiency to be very low for electrons at near- relativistic energies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', around 100 keV (Tsurutani & Lin 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Mäkelä et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Dresing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2016b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The acceleration efficiency is, however, increasing at lower energies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', tens of keV (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Although IP shocks at 1 AU do not show clear ability to accelerate electrons in-situ at near-relativistic energies, Dresing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' (2022) showed that there is a strong correlation between electron peak intensities observed at Earth’s distance and the Mach number of the shock wave close to the Sun, especially at MeV energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' This provides strong evidence for the ability of coronal shocks to accelerate electrons to relativistic energies and motivates revisiting relativistic electron observations during in-situ shock crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Article number, page 1 of 13 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='05587v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='SR] 13 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' main In this study, we present observations of MeV electron ESP events observed with the STEREO mission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Kaiser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2008) by investigating the entire STEREO dataset for relativistic electron enhancements connected with in-situ shock crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Our goal is to determine if IP shocks contribute to MeV elec- tron acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In order to account for acceleration regions that could reside somewhat away from the local region sampled by the spacecraft, we extended the time interval of the relativis- tic electron observations to several hours around the time of the shock passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' As a result, a set of candidate events is estab- lished, which we then examine more closely to determine the effects of IP shocks on MeV electrons close to 1 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The structure of the paper is the following: in §2, we de- scribe the event selection and a data filtering method developed for identifying shock-related electron enhancements, in §3 we present the observations and results of our analysis, and in §4 we discuss the results and present the conclusions of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Event selection and filtering method To find potential electron ESP events, we used the STEREO in- situ shock list (Jian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The list starts at the beginning of the STEREO mission in January 2007 and the latest update is on April 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' There are no STEREO-B events after September 2014 because of the loss of contact with the spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In total, there are 587 interplanetary shocks in the list we used, of which 340 were detected at STEREO-A and 247 at STEREO-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' At each of the listed in-situ shock-crossing times, we ana- lyzed the electron and proton intensities measured by the High Energy Telescope (HET;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' von Rosenvinge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' HET is one of four instruments in the Solar Energetic Particle subsys- tem, which is part of the In-situ Measurements of Particles and CME Transients investigation (IMPACT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Luhmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2008) on STEREO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' HET measures the highest energy particles includ- ing protons in eleven energy channels from 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='6 to >100 MeV and electrons in three energy channels from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='7 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In the following, we denote the electron channels as e1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4 MeV), e2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV) and e3 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' We first visually scanned the electron observations for inten- sity enhancements taking place close in time to the shock arrivals at the spacecraft and, thus, potentially caused by local accelera- tion of electrons by the shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Thus, we found 27 electron ESP candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Because the in-situ shock crossings are often asso- ciated with high proton and ion fluxes, we needed to be sure that the selected events are not the result of ion contamination in the electron channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' For this, we analyzed 1-minute HET data of the whole year of 2013 (see Appendix B and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Our conclusion is that the HET electron enhancements are real and not due to ion contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' We also checked the radio ob- servations of STEREO/SWAVES to exclude the possibility that the electron enhancements are caused by SEP injections at the Sun coincidentally occurring closely before the enhancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' For this, we searched for type III radio bursts, the signature of accelerated electrons impulsively injected into interplanetary space (Reid & Ratcliffe 2014), occurring about 10-30 minutes before the observed electron enhancement of selected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Apart from one event (6 Nov 2013, see a more detailed analysis in section 3), none of our candidates turned out to be associated with a coincident solar injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In our simple scanning method it was not always easy to identify electron intensity enhancements potentially associ- ated with local shock acceleration because they are usually mixed with enhancements of the corresponding SEP events (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Kouloumvakos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' For that reason, we introduced a fil- tering method for the intensities: ∆I(ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' N, M) = I(ti) − 1 M i−1−N � j=i−M−N I(t j), (1) where M is the number of points (time bins) constituting the averaging time window and N is the number of points giving the time lag of the averaging window before each time ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Thus, the method uses the running averaging window with variable length and lag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In the analysis, we took N = 180 for all the cases, which corresponds to a constant three-hour lag for one-minute data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The parameter M was given values of 15, 30, 60, 120, 240 and 480, which correspond to a temporal window varying from 15 min to 8 hours for one-minute data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The applied filtering method allows one to identify more clearly local (in time) variations of the particle intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The fil- tered data are shown in the six bottom panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 1(a), which also gives an example the analysis carried out for all our 27 elec- tron ESP event candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In the intensity time series, one can see intensity enhancements (in various energy channels, for both electrons and protons) associated with the shock and superposed on the ongoing SEP event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The filtering of the intensities con- veniently shows the time interval when the intensity variation is strongest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' However, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 1(a) also shows that if one wants to compare the filtered intensities corresponding to different energy chan- nels (to find, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', which energy channel has the strongest rela- tive enhancement), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' (1) is not convenient due to lower back- ground levels of higher energy channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Therefore, we modi- fied Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' (1) by normalizing the filtered intensity ∆I by the "local background" intensity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', the intensity value of the running av- erage window: ∆Inorm(ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' N, M) = I(ti) 1 M �i−1−N j=i−M−N I(tj) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' (2) Figure 1(b) shows the normalized filtered (NF) intensities ∆Inorm for electrons in the considered example, while Figure 1(c) presents both electrons and protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In order to determine if an intensity enhancement around the shock-crossing time is significant compared to its preceding intensity level, we calculate the mean and the standard devia- tion (SD) of the NF data inside the background window, using the six different averaging window lengths (M1-M6, plotted in the lower six panels of the figures) for each of the three elec- tron channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' We then compared the NF data with their cor- responding variable background level and calculate the z-score, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', the number of SDs above the background level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' We applied the z-score to identify statistically significant electron enhance- ments by requiring that at least three consecutive NF data points within a six-hour window surrounding the shock-crossing time are above the mean + 2 SDs threshold level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' From this we identi- fied which energy channel and averaging window length yielded the most significant electron enhancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' We then returned to the intensity-time profile of the chosen electron channel and identified the time of its peak intensity Ipeak within a six-hour window surrounding the shock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Since not all the events have a distinct peak, we divided them into three categories: plateau (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='2), peak (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3), and plateau + peak (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Naturally, the NF data also show a strong response at the be- ginning of the associated SEP events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' These intensity enhance- ments also increase the background levels used in our method, once the background window moves over them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Article number, page 2 of 13 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Talebpour Sheshvan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Dresing, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Vainio , A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Afanasiev and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Morosan : On the Role of Interplanetary Shocks in Accelerating MeV Electrons (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 1: Analysis of the shock crossing on 29 January 2012 observed by STEREO-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' (a) From top to bottom: proton and electron intensity-time profiles as measured by HET, magnetic field vector magnitude and components as measured by IMPACT/MAG, STEREO/WAVES dynamic spectrum, and the rest are filtered data as obtained with the fixed time lag parameter (N = 180) and increasing span parameter M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' (b) Zoom-in around the time of shock number 125 passage with the electron NF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' (c) Zoom-in around the time of shock number 125 passage with both the electron and proton NF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The black vertical dashed line indicates the time of the shock passage over the spacecraft, and the orange dashed line marks the time of the peak intensity Ipeak in the electron energy channel having the strongest response in terms of the NF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Each shaded area in (b) shows the running mean + 2 SD level of the corresponding running background window for the electron energy channel having the strongest response in terms of the NF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In the detailed analyses of example events, we also make use of CME information obtained from the SOHO LASCO CME Catalog (http://cdaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='gov/CME_ list;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Yashiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Gopalswamy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2009, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Observations and Results The filtering technique (described in §2), applied to the visu- ally identified candidate events in STEREO/HET observations, enabled us to list IP shocks potentially accelerating electrons to MeV energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Out of the 27 candidate IP shock events ob- served by STEREO A and B, nine (∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='5%) turned out to have significant relativistic electron intensity enhancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The first and second column of Table 1 list the event date, shock num- ber 1, and the observing spacecraft (A or B), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Two numbers are indicated in parentheses in the third column;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' the first one gives the number of shocks that occurred in the two days prior to our main shock, and the second number provides the number of shocks that occurred in the two days following our main shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The shock transit speed calculated using the start time of the corresponding type III radio burst observed by STEREO/WAVES and the CME arrival time 2 is listed in 1 https://stereo-ssc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='nascom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='gov/pub/ins_data/ impact/level3/IPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='pdf 2 https://stereo-ssc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='nascom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='gov/pub/ins_data/ impact/level3/ICMEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='pdf the fourth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The shock parameters, including the mag- netosonic Mach number, shock normal angle, and shock mag- netic compression ratio, given in the fifth, sixth, and seventh columns, respectively, were taken from the STEREO IP shock list3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The last four columns contain information regarding the shock-associated electron enhancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Column (8) gives the energy channel with the highest z-score in the NF data among the three electron channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Column (9) gives the time delay be- tween the intensity peak time and the shock crossing time (neg- ative value indicates that the intensity peaks before the shock- crossing time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The shape of the intensity-time profile around the shock crossing (visually evaluated) is given in column (10), and in the last column of the Table [1], the background window length parameter M corresponding to the highest z-score in the related electron channel, in the 6 hours window around the time of shock crossing, is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Applying the normalization to the filtered data allows us to identify the energy channel with the strongest response on the events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In seven of these events the most significant signal be- longs to the e3 channel of HET in the range of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0 MeV (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In six cases (∼67%) the peak intensity is observed up- stream of the shock, with five of them occurring very close to the time to the shock crossing (less than 20 minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In four cases (shock numbers 87, 126, 151, and 246, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 4, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='3), the MeV 3 https://stereo-ssc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='nascom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='gov/pub/ins_data/ impact/level3/IPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='pdf Article number, page 3 of 13 STEREO-A/ HET / 2012-01-29 124 102 101 100 50 [B| (lu) 0 101 100 (MHz) 10-1 10-2 20 N=180, M=15 20 N=180, M=30 0 20 N=180, M=60 N=180, M=120 10 N 10 20 N=180, M=240 N 0 N=180, M=480 20 N 01-28 01-28 01-28 01-29 01-29 01-29 01-29 01-30 01-30 06:00 12:00 18:00 00:00 06:00 12:00 18:00 00:00 06:00 Date / Time in year 2012STEREO-A/ HET / 2012-01-29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4 MeV e 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 102 sr-s-MeV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV e 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0 MeV e 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='6-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeVp 101 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='p ) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='3 MeV p 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV p 100 125 50 F (lu) 10 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=15 5 10 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=30 5 10 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=60 Alnom 10 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=120 5 10 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 5 10 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 e3:mean+2std wou 5 01-29 01-29 01-29 01-29 01-29 01-29 01-29 10:00 11:00 12:00 13:00 14:00 15:00 16:00 Date / Time in year 2012STEREO-A /HET / 2012-01-29 102 101 100 50 0 101 100 10-1 10-2 40 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=15 20 40 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=30 20 48 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=60 20 40 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=120 20 40 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 20 40 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 20 01-29 01-29 01-29 01-29 01-29 01-29 01-29 11:30 12:00 12:30 13:00 13:30 14:00 14:30 Date / Time in year 2012A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' main Table 1: Parameters of the nine electron ESP events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Associated shock parameters from https://stereo-ssc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='nascom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' gov/pub/ins_data/impact/level3/IPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Date Sh num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='/SC Nsh1 Vt [km/s]2 Mms θBn[◦] rB3 er4 ∆t [min]5 shape M6 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) 2011-06-05 87 / A (1,0)* 1749 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='54 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='57 e1 +66 peak 60 2012-01-29 125 / A (0,0) 917 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='00 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='15 e3 −13 peak 480 2012-03-08 126 / B (1,1)* 1147 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='48 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='60 e3 −92 plateau 480 2012-05-28 141 / A (0,0) 1321 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='85 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='70 e3 −4 peak 480 2012-07-23 151 / A (0,0) 2099 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='46 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='17 e3 −1 plateau+peak 480 2013-11-06 198 / B (1,1) 989 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='92 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='02 e3 +31 peak 15 2014-09-25 246 / B (2,1) 2114 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='56 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='60 e1 +132 peak 480 2017-07-24 316 / A (0,2) 1197 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='00 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='92 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='92 e3 0 plateau 240 2017-09-19 322 / A (0,2) 985 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='59 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='90 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='33 e3 −8 peak 15 1The first (second) number in each bracket indicates how many shocks were observed within two days before (after) the corresponding shock indicated in column (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2 Transit speed of the associated CME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3 Magnetic compression ratio of the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 4 The electron energy channel showing the strongest signal in terms of of NF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 5 ∆t between the peak intensity time and the shock-crossing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Negative numbers refer to peaks in the upstream region and positive numbers indicate peaks in downstream region of the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 6 Best averaging time window for each event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' SIR shock occurs one day before the shock crossing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' electron channels show overall higher intensities than the deka- MeV proton channels around the time of the shock crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' However, in the remaining five cases (shocks number 125, 141, 198, 316 and 322) the overall proton intensities are higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In two events (shocks number 125 and 246), the signal of the NF data of the proton channels are stronger than the those of the electron channels (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='3 (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' However, in the rest of events the response of the MeV electrons, as measured by the NF data, is stronger than that of the protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The mean value of the mag- netosonic Mach number of the nine analyzed in-situ shocks is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='04, and is varying from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='48 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' These are not outstanding values among IP shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The shock normal angles θBn vary from 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='5° to 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='7°, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', from oblique to quasi-perpendicular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The ma- jority of the shocks are oblique shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Only three shocks are quasi-perpendicular with shock normal angles larger than 60°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' All the events in Table 1 are associated with interplanetary CME (ICME)-driven shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Having applied different window length from 15 to 480 minutes we found that the strongest sig- nals in the NF data are usually found for large window length of 240 or 480 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In the two days before and two days after the shock passage time, we found other shocks associated with some of our events, which is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' For example, shocks number 125 and 316 have one shock before the on the same day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The shock number 141 has one shock after, and shock 246 has two shocks before on the same day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In three events (shocks num- ber 87, 126, 198), one shock occurred a day before;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' two of those are shocks related to stream interaction regions (SIRs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' There was no previous shock in the five events (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' At the time of the in-situ shock crossing, radio spectrograms observed at the same STEREO spacecraft reveal another feature, which is present in all of the nine events: an increase in the dy- namic spectrum simultaneous in a broad range of frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The feature is most likely quasi-thermal noise extending from the kHz range up to ∼1 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' To gain a better understanding of this attribute, we investigated the STEREO list of shocks, and we found that this is a very common trait on the downstream of the shock crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Therefore, it is not a unique feature of the electron ESP events studied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In the following we will discuss some of the most interesting events in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Event on 28 May 2012 – shock number 141 This event (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3) is an example of a peak-like electron ESP event and it stands out from the others in terms of its very strong response in the e3 energy channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' LASCO-C2 recorded a fast Halo-CME with a linear speed of 1966 km/s at 2:24 UT on 27 May 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4 The STEREO/WAVES dynamic spectrum shows a type III radio burst on 26 May 2012 at 20:48 UT (third panel from top of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In addition, a type II radio burst, a signature of a CME-driven shock accelerating electron beams (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Nelson & Melrose 1985), begins at 20:50 UT and continues until 23:20 UT between 16000-300 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The first increase in particle in- tensities occurred simultaneously with the eruption on 26 May.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' A significant change of the intensity time profile occurs around 6:00 UT on 27 May 2012, when the fluxes begin to rise again after a continuous decrease and reach their maximum value 4 minutes before the shock time, in the upstream region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' This is most likely due to the ICME-driven shock, whose contribution gets significantly stronger around this time (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3 near the black vertical dashed line marked with shock number 141).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The obliquity θBn of the in-situ shock is almost 75° and the Mach number is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The NF data of the electron channels uti- lizing various window lengths are displayed in the six lower pan- els of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Significant changes in the SEP intensities start to show up three hours before the shock, and the fluxes of all en- ergy channels starts to rise, which also appears as signal changes in the NF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The signal in the e3 channel is the strongest com- pared to the other electron energy channels (blue line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3 plots) and even greater than those of the proton channels (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3 (c)), despite the fact that protons have a higher intensity than electrons in the time-intensity profile (top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Between the other windows lengths, the highest value of the z-score at the time of shock crossing belongs to M = 480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The energetic particle time profiles in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3 show 4 http://cdaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='gov/CME_list Article number, page 4 of 13 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Talebpour Sheshvan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Dresing, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Vainio , A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Afanasiev and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Morosan : On the Role of Interplanetary Shocks in Accelerating MeV Electrons Shock number / SC 87 / A 124–125 / A 126 / B 141–142 / A 151 / A 198 / B 244–245–246 / B 316–317 / A 322 / A (in one day) 323 318 247 199 321 315 140 125 1 day SIR ICME 123 127 88 5 day +20 day 126 20 day +7 day 143 +8 day 7 day 86 149 150 197 243 5 day 152 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2: All the numbers represent the shock number from the STEREO list (https://stereo-ssc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='nascom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='gov/pub/ins_ data/impact/level3/IPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='pdf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' All of the ICME events in the blue central column occurred simultaneously with the relevant shock passage from our list of nine events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The nine numbers in bold print with underline indicate the events on our list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The SIR shocks from the list represented by the number of shocks inside the red box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Two way arrows indicate a day and ∓ day denotes a duration of more than a day between the occurrence of two shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' an intensity decay immediately after the shock crossing, ob- served in both electrons and protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The magnetic field vari- ations in this event indicate that the ICME driving the shock and related to the CME associated with the SEP event has a magnetic cloud-like structure with a gradually rotating and enhanced-in- magnitude magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' However, the field shows a strong de- pletion in magnetic field magnitude around 9:00 UT, which may indicate the presence of a pre-existing ICME ahead of the driv- ing ICME that is related to the fast halo CME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The sheath re- gion of the driving ICME has a complex magnetic structure and could even have a partly closed magnetic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' This, how- ever, needs a more thorough dedicated study in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Event on 23 July 2012 – shock number 151 This event (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 4) has been studied in detail by several authors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Temmer & Nitta 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Riley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2016) as it is one of the most energetic eruptions observed dur- ing the space era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' We do not provide a detailed analysis of this event but examine the key observational properties in relation to our event sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The event is an example of a plateau + peak- like shaped electron ESP event, with electron fluxes higher than proton fluxes for the majority of the time-intensity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The LASCO CME catalog reports a fast CME with a speed of 2003 km/s at 2:24 UT on 23 July 2012, followed by a clear type II ra- dio burst (third panel from top Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 4 (a)), which continued until 21:40 UT visible by both STEREO A and B (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' HET measured a very gradual SEP event associated with a type III burst at 01:45 UT on 23 July 2012 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The enhancement of particle intensities associated with the fast forward ICME-driven shock observed by STEREO A at 20:55 UT on 23 July 2012 reached its peak value 1 minute before the shock time, in the upstream re- gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' It is an oblique shock with θBn = 45◦ and a Mach number of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='46 based on the shock list (but we note that many of the earlier detailed analyses give somewhat different values for the obliq- uity and strength of the shock).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The electron flux increases four orders of magnitude during this event making it an exceptionally intense event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' By zooming in around the ESP event (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 4 (b) and (c)), we see a pronounced plateau-shaped intensity increase between 19:15 to 22:56 with the time of the shock crossing in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' During this time, the magnetic field gradually starts to weaken and only a small jump in magnetic field magnitude is observed at the shock (2nd panel from top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In the NF data, the HET e3 channel (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0 MeV) exhibits the strongest shock response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The M = 480 window length had the highest value by computing the z-score around the shock crossing in six different window lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The end of the plateau-like shape of intensities around 23:00 UT on 23 July is accompanied by the beginning of an ICME, which has a highly compressed magnetic field reach- ing values beyond 100 nT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The ICME has again a complicated structure that may consist of two individual flux ropes (Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Riley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2016), first of which would be related to a pre-existing CME in the solar wind overtaken by the fast erup- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' This implies that the strong decrease of fluxes at the onset of the ICME would be related to a change in the topology of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Event on 6 November 2013 – shock number 198 Shock number 198 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 5, which occurred on 6 November 2013, is unique in terms of being the only MeV electron ESP event in our sample without a previous SEP event (Chiappetta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' According to the SOHO LASCO CME Catalog, the associated fast Halo-CME occurred with a linear speed of 1040 km/s at 5:12 UT on 4 November 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Both STEREO/WAVES recorded a strong type II radio burst extending to very low fre- quency (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Magnetic field components increased in the first hours of 5 Nov 2013 in the upstream region at the same time as the quasi-thermal noise signature in the radio spectrogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' About 40 minutes before the shock crossing a small type III ra- dio burst appeared in the radio spectrogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' This new solar event might have injected SEPs at the right time to provide seed parti- Article number, page 5 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' main cles for the in-situ shock on 6 Nov leading to electron intensity peaks close to the shock crossing time (31 min) on the down- stream side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The e2 and e3 channels responded in a similar way as indicated by the NF data, but the response of the e1 channel is weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The magnetic field associated with the ICME driving this event is less intense than in the other two example events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' It is characterized by the presence of large-amplitude fluctuations throughout the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Also in this case, the downstream of the shock seems to host a compressed magnetic field region between the shock and the driving ICME related to the fast halo CME (from about 02:40 to 05:00 UT), which coincides with the start of the decrease of particle fluxes indicating a possible change in field topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The structure of the field and the duration of the compressed region does not, however, seem to be consistent with a double flux-rope structure like in the 23 July 2012 event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Shock number 199 – example of an event missed by the fil- tering method Shock number 198 is followed by a gradual SEP event that oc- curred on 7 Nov 2013 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 6 (a)) in conjunction with a Halo CME, resulting in a type II radio burst in the dynamic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The SEP event was also studied by Dresing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' (2016a), who reported that STEREO B was situated inside a magnetic cloud of a pre-event ICME during the start of the SEP event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The 7 Nov 2013 CME drove a shock that passed STEREO B on 8 Nov (shock number 199).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The SEP event has a long dura- tion and a plateau-type profile, also in electron channels e1–e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' This could be an indication of a prolonged electron injection that could be related to continuous acceleration at the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' How- ever, the signal of the NF data does not show a significant re- sponse of the electron channels for the associated IP shock cross- ing of shock number 199 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 6 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' This implies that the shock does not maintain its ability to accelerate electrons up to 1 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Discussion and Conclusion We scanned the STEREO mission for interplanetary shocks that show local acceleration signatures of relativistic electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Nine events were identified by applying a novel filtering method that identifies significant peaks in time-intensity profiles of HET energy channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' All shocks were associated with ICMEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The events were divided in three classes based on their time- intensity profiles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', plateau-like, peak-like and plateau+peak- like events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The identified electron ESP events had variable elec- tron to proton ratios, some being proton dominated while some were dominated by electrons in the STEREO/HET energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The nine events that show electron acceleration signatures are all related to very fast shocks with transit speeds exceeding 900 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The transit speed was recently identified by Ameri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' (2022) as the best correlated shock property with the ∼10 MeV proton intensities at the shock in ESP events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' A high transit speed may imply that a shock has been strong all its way from the Sun to the observer, allowing it to maintain its acceleration efficiency for an extended amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Three ESP events were analyzed in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Two of them were associated with SEP events and one was an isolated ESP event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The ones associated with SEP events were more intense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Their associated ICMEs had complex magnetic structure with indications of interactions in the interplanetary space between pre-existing ICME structures and the fast eruption driving the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The isolated ESP event is not as strong as the ones asso- ciated with SEPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' It, too, had a complex ICME driver, although the peak magnitude of the interplanetary magnetic field was not as high as in the SEP-event-related events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Although it lacked an SEP event related to the eruption, the isolated event had a type III burst occurring a few hours before the shock passage just be- fore the start of rise of the energetic particle fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' This solar event could have provided seed particles for the shock and thus facilitated the acceleration of electrons to energies detected in HET energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' While it performs well with ESP-like events, our filtering method is unable to detect more gradual contributions of shocks to the energetic electron fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Some events (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', shock no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 199) show plateau-like but gradually decaying intensities un- til the shock passage, and then faster decays after the passage of the leading edge of the ICME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In such cases, shocks may still have significantly contributed to electron acceleration as the shock propagates from the corona to 1 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Equipped with the new instruments onboard Solar Orbiter and Parker Solar Probe that reach close distances to the Sun, such events may appear as much more clear-cut ESP events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In conclusion, while ESP events accelerating electrons to rel- ativistic energies are not very common at 1 AU, there are still a number of cases that show signatures of local acceleration by the shock and/or its sheath region up to several MeVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The next so- lar maximum with fast CMEs and new observational capabilities will likely bring more clarity to the issue of electron acceleration by interplanetary shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' We acknowledge financial support from the European Union’s Horizon 2020 research and innovation programme under grant agree- ment No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 101004159 (SERPENTINE) and the support of Academy of Finland (FORESAIL, grants 312357 and 336809).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' SOHO LASCO CME catalog is gener- ated and maintained at the CDAW Data Center by NASA and The Catholic Uni- versity of America in cooperation with the Naval Research Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' SOHO is a project of international cooperation between ESA and NASA.' 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Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', 173, 247 Luhmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Curtis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Schroeder, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2008, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', 136, 117 Mäkelä, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Gopalswamy, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Akiyama, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Xie, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', & Yashiro, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2011, Journal of Geophysical Research (Space Physics), 116, A08101 Article number, page 6 of 13 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Talebpour Sheshvan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Dresing, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Vainio , A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Afanasiev and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Morosan : On the Role of Interplanetary Shocks in Accelerating MeV Electrons (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3: SEP event on 27 May 2012 and corresponding ESP event on 28 May 2012 observed by STEREO A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' (a) from top to bottom: time profile of proton and electron intensities at different energy channels as indicated in the legend in the top panel of (b), the evolution of magnetic field magnitude and its components in RTN coordinates, STEREO/WAVES dynamic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The lower six panels show the normalized filtered data of the three electron channels using a fixed time lag parameter of (N = 180) and increasing window lengths M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Figures (b) and (c) are the same format as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The boxes show the N and M value of each panel and the gray one indicates the the one yielding the most significant signal of the NF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 4: Summary of electron and proton observations by the STEREO/HET instrument on 23 July 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Figures (a), (b) and (c) have the same format as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3 Article number, page 7 of 13 STEREO-A / HET / 2012-07-23 105 104 103 102 151 25 0 25 101 100 10-1 10-2 N=180, M=15 40 20 0 N=180, M=30 40 20 0 N=180, M=60 40 02 0 N=180, M=120 40 20 0 N=180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 40 20 0 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 40 20 07-23 07-23 07-23 07-23 07-23 07-23 07-23 07-23 07-23 07-23 19:00 19:30 20:00 20:30 21:00 21:30 22:00 22:30 23:00 23:30 Date / Time in year 2012STEREO A/ HET / 2012-05 102 142 101 100 10-1 141 10-2 [BI (lu) ug 25 R 0 25 N 101 f (MHz) 100 10-1 10-2 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=15 0 100 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=30 wou 50 0 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=60 0 100 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=120 50 0 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 0 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 0 05-27 05-27 05-27 05-27 05-28 05-28 05-28 05-28 05-29 00:00 06:00 12:00 18:00 00:00 06:00 12:00 18:00 00:00 Date / Time in year 2012STEREO-A / HET / 2012-05-28 102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4 MeV e 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV e 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0 MeV e 101 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='6-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeV p 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeV p 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='3 MeV p 100 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV p 141 25 25 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=15 50 0 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=30 50 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=60 50 0 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=120 50 0 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 50 0 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 50 e3:mean+2std 05-27 05-28 05-28 05-28 05-28 05-28 05-28 05-28 05-28 23:00 00:00 01:00 02:00 03:00 04:00 05:00 06:00 07:00 Date / Time in year 2012STEREO-A / HET / 2012-05-28 102 101 141 100 25 0 25 101 100 10-1 10-2 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=15 40 20 :180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='M=30 40 20 =180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=60 40 20 I=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='M=120 40 20 =180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 50 25 =180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 50 25 05-28 05-28 05-28 05-28 05-28 05-28 05-28 05-28 02:00 02:15 02:30 02:45 03:00 03:15 03:30 03:45 Date / Time in year 2012STEREO-A / HET / 2012-07 104 102 100 151 100 B () R 0 N 101 100 10-2 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=15 0 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=30 0 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='M=60 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=120 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 400 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 07-23 07-23 07-23 07-23 07-23 07-24 07-24 07-24 07-24 07-24 04:00 08:00 12:00 16:00 20:00 00:00 04:00 08:00 12:00 16:00 Date / Time in year 2012STEREO-A / HET / 2012-07-23 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4 MeV e 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV e 104 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0 MeV e 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='6-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeV p 103 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeV p 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='3 MeV p 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV p 102 151 25 25 N=180, M=15 40 20 0 N=180, M=30 40 20 0 N=180, M=60 40 20 0 N=180, M=120 40 20 0 N=180, M=240 40 20 0 N=180, M=480 40 20 e3:mean+2std M 0 07-23 07-23 07-23 07-23 07-23 07-23 18:00 19:00 20:00 21:00 22:00 23:00 Date / Time in year 2012A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 5: Same format as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3 showing STEREO B/HET observations of the event associated with the ICME-driven shock on 6 Nov 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 6: Same format as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3 but showing STEREO B/HET observations the event associated with the ICME-driven shock on 8 Nov 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Mitchell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Nolfo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Hill, M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2013, ApJ, 770, 38 Sarris, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' & van Allen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 1974, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', 79, 4157 Article number, page 8 of 13 STEREO-B / HET / 2013-11 102 198 (cm2-sr-s-Mev)- 101 100 10- 10-2 25 [BI (l) g R 0 N 25 101 100 (MHz) 10-2 N=180, M=15 25 0 N=180, M=30 25 0 N=180, M=60 25 0 N=180, M=120 25 N=180, M=240 25 0 N=180, M=480 25 0 11-04 11-05 11-05 11-06 11-06 11-07 12:00 00:00 12:00 00:00 12:00 00:00STEREO-B / HET / 2013-11-06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4 MeV e 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV e 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0 MeV e 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='6-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeV p 10-1 198 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeV p 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='3 MeV p 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV p 10-2 25 0 25E N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=15 50 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=30 50 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=60 50 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=120 50 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 50 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 50 MMM e3:mean+2std 11-06 11-06 11-06 11-06 11-06 11-06 11-06 11-06 11-06 00:00 00:30 01:00 01:30 02:00 02:30 03:00 03:30 04:00STEREO-B / HET / 2013-11-06 100 10-1 198 10-2 25 0E 25 101 100 10-1 10-2 50 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=15 25 50 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=30 25 0 50 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=60 25 50 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=120 25 50 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 25 50 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 25 11-06 11-06 11-06 11-06 11-06 11-06 11-06 11-06 11-06 00:00 00:30 01:00 01:30 02:00 02:30 03:00 03:30 04:00STEREO-B / HET / 2013-11 102 101 100 10 199 10-2 25 [BI () g R 0 N 25 101 100 (MHz) 10-2 400 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=15 N 0 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=30 0 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=60 408 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=120 0 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 0 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 0 11-07 11-07 11-08 11-08 11-08 11-08 11-09 11-09 12:00 18:00 00:00 06:00 12:00 18:00 00:00 06:00STEREO-B / HET / 2013-11-08 102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4 MeV e 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV e 101 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0 MeV e 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='6-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeV p 100 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeV p 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV p 10-1 199 10F 10 N=180, M=15 L N=180, M=30 A!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 1N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=60 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=120 Ld N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 1 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 e3:mean+2std 1 11-08 11-08 11-08 11-08 11-08 11-08 11-08 11-08 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00STEREO-B / HET / 2013-11-08 102 101 100 199 10-1 25 0 25 101 100 10-1 10-2 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=15 M N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=30 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=60 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=120 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 0m 11-08 11-08 11-08 11-08 11-08 11-08 11-08 11-08 11-0811-08 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Talebpour Sheshvan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Dresing, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Vainio , A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Afanasiev and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Morosan : On the Role of Interplanetary Shocks in Accelerating MeV Electrons Simnett, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M.' metadata={'source': 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2008, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', 136, 391 Wraase, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Heber, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Böttcher, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2018, A&A, 611, A100 Yang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Li, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2019, The Astrophysical Journal, 875, 104 Yashiro, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Gopalswamy, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Michalek, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2004, Journal of Geophysical Research (Space Physics), 109, A07105 Article number, page 9 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' main Appendix A: Additional STEREO/HET observations of electron ESP events For each event listed in Table 1 we show a figure in the for- mat of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3 presenting a long duration time profile of the SEP event and corresponding ESP particle intensities measured by STEREO/HET, the magnetic field magnitude and its compo- nents measured by the IMPACT/MAG instrument in RTN co- ordinates and a STEREO/WAVES dynamic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' We also show the normalized filtered data with the fixed value of three- hour time lag (N) for all the cases and six given values for a temporal window of background intensity from 15 minutes to 8 hours (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The figures contain the zoom-in profile around the shock crossing time and the shaded area of the mean plus two times of SD of the moving background window (the color of the shaded area is based on the electron energy channel showing the strongest signal in terms of of NF data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' In addition, NF data for the proton channels have been added for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The black and orange dashed lines on the plots represent the time of shock crossing and peak intensity time of selected energy channel of electron, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Appendix B: A test for proton contamination in the electron energy channels During periods of high proton intensities, such as in ESP events, the detection of small electron intensities can be difficult or even contaminated by the protons, which is a known feature in ener- getic particle instruments such as STEREO/SEPT (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=', Wraase et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' To determine if there is proton contamination of the electron measurements used in this study we plot 2D his- tograms of deka-MeV proton intensities as a function of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0 MeV electron intensities in Fig B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' We use 1-minute data of the whole year of 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' None of the three proton channels shows a one-to-one correlation with the electron intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Furthermore, the low or even missing electron intensities at times when high proton intensities are observed (upper left corner), supports the assumption that proton contamination does not play a significant role in the electron channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Article number, page 10 of 13 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Talebpour Sheshvan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Dresing, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Vainio , A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Afanasiev and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Morosan : On the Role of Interplanetary Shocks in Accelerating MeV Electrons Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1: Same format as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3 but showing STEREO A/HET observations of an SIR-associated shock on 4 June 2011 and a ICME-driven shock on 5 June 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='2: Same format as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3 but showing STEREO B/HET observations of an SIR shock on 7 Mar 2012 and a ICME-driven shock on 8 Mar 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Article number, page 11 of 13 STEREO-A/ HET /2011-06 103 86 leV)-1 101 10 87 15 /B/ Brtn 25 N 101 B 10 19 (MH 10- 10 50 N=180, M=15 0 N=180, M=30 N=180, M=60 N=180, M=120 N=180, M=240 = 20 N=180, M=480 06-04 06-05 06-05 06-06 06-06 12:00 00:00 12:00 00:00 12:00 Date / Time in year 2011STEREO-A/ HET / 2011-06-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4 MeV e 102 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV e W 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0 MeV e 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='6-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeVp 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeV p 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='3 MeV p 101 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV p 87 10 10 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=15 2 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=30 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=60 2 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=120 M N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 0 el:mean+2std 06-05 06-05 06-05 06-05 06-05 06-05 06-05 16:00 17:00 18:00 19:00 20:00 21:00 22:00 Date / Time in year 2011STEREO-A / HET /2011-06-05 102 M 87 10 0 10 101 100 10-1 10-2 2 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=15 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=30 0W N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=60 0W N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=120 0W N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 06-05 06-05 06-05 06-05 06-05 06-05 06-05 16:00 17:00 18:00 19:00 20:00 21:00 22:00 Date / Time in year 2011STEREO-B / HET / 2012-03 102 (cm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='-sr-s-MeV) 101 100 125 126 25 [B Brtn (nT) R Y 0 25 N 101 100 (ZHW) 10-1 f 10-2 100 N=180, M=15 50 0 100 N=180, M=30 50 0 100 N=180, M=60 Alnorm 50 0 N=180, M=120 50 0 N=180, M=240 0 N=180, M=480 0 03-07 03-07 03-07 03-07 03-08 03-08 03-08 03-08 03-09 00:00 06:00 12:00 18:00 00:00 06:00 12:00 18:00 00:00 Date / Time in year 2012STEREO-B / HET / 2012-03-08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content="4'MeV e 102 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV e 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0 MeV e 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='6-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeV p 101 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeV p 126 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='3 MeV p 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV p 100 25 25 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=15 0 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=30 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=60 0 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=120 0 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 0 e3:mean+2std N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 0 03-08 03-08 03-08 03-08 03-08 03-08 03-08 03-08 03-08 03-08 03-09 04:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 Date / Time in year 2012STEREO-B / HET / 2012-03-08 102 101 126 WWWL 100 25 0 25 101 100 10-1 10-2 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=15 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=30 0 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=60 P 1 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=120 1 01 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 1 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 0 1 03-08 03-08 03-08 03-08 03-08 03-08 03-08 03-08 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Date / Time in year 2012A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='3: Same format as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3 but showing STEREO B/HET observations of three ICME-driven shocks in one day on 25 Sep 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4: Same format as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3 but showing STEREO A/HET observations of two ICME-driven shocks on 24 Sep 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Article number,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' page 12 of 13 STEREO-B / HET / 2014-09-25 101 100 10-1 246 244 245 10-2 50 [BI () g R 0 50 N 101 f (MHz) 100 10-1 10-2 100 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='M=15 wou 50 100 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=30 50 0 100 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=60 50 0 100 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=120 Alnom 50 0 100 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 Alnom 50 0 100 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 50 09-24 09-25 09-25 09-25 09-25 09-25 09-25 09-25 09-25 09-26 21:00 00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 00:00 Date / Time in year 2014STEREO-B / HET / 2014-09-25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4 MeV e 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='e W 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='e 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='6-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeV p 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeVp 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='3 MeV p 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV p 100 246 50 50 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=15 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=30 HUM N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=60 AAN N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=120 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 el:mean+2std WNY 09-25 09-25 09-25 09-25 09-25 09-25 09-25 14:00 15:00 16:00 17:00 18:00 19:00 20:00 Date / Time in year 2014STEREO-B / HET / 2014-09 245 101 246 100 50 50 101 100 10-1 10-2 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=15 5 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=30 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=60 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=120 5 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 5 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 09-25 09-25 09-25 09-25 09-25 09-25 09-25 14:00 15:00 16:00 17:00 18:00 19:00 20:00 Date / Time in year 2014STEREO-A / HET / 2017-07 102 101 100 316 10-1 317 10-2 50 [B Brtn (nT) 0 50 N 101 (MHz) 100 4 10-1 10-2 40 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=15 uou 20 N 0 40 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=30 40 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=60 nom 50 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=120 0 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 0 100 N=180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=480 Alnom 50 07-23 07-23 07-23 07-24 07-24 07-24 07-24 07-25 07-25 07-25 06:00 12:00 18:00 00:00 06:00 12:00 18:00 00:00 06:00 12:00 Date / Time in year 2017STEREO-A/ HET / 2017-07-24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4 MeV e 102 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV e 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0 MeV e 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='6-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeV p wMmmn NNAAM 101 SWM 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeV p w 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='3 MeYp 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV p 100 316 317 10 N=180,M=15 N=180, M=30 N=180, M=60 W N=180, M=120 0 N=180, M=240 N=180, M=480 e3:mean t2sto 07-24 07-24 07-24 07-24 07-24 07-24 07-24 10:00 11:00 12:00 13:00 14:00 15:00 16:00 Date / Time in year 2017STEREO-A / HET / 2017-07-24 102 101 WW NENWNY 316 1% 0 119 100 10-1 10-2 N=180, M=15 N=180, M=30 0 N=180, M=60 N=180, M=120 N=180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' M=240 N=180, M=480 M 0 07-24 07-24 07-24 07-24 07-24 07-24 07-24 10:00 11:00 12:00 13:00 14:00 15:00 16:00 Date / Time in year 2012N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Talebpour Sheshvan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Dresing, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Vainio , A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Afanasiev and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Morosan : On the Role of Interplanetary Shocks in Accelerating MeV Electrons (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='5: Same format as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 3 but showing STEREO A/HET observations of an ICME-driven shock on 19 Sep 2017 at 2:56 with prominent plasma signatures on the downstream side of the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1: 2D histograms of varying proton versus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' MeV electron intensities measured by STEREO-A/HET using 1-minute data of the whole year 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The plots show the logarithm of the intensities (excluding all zero intensities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The color scale (marking the number of observations in each bin) is logarithmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The black solid line marks unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' The left, middle, and right panels show 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='6–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeV, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='9– 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeV, and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0–19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='3 MeV protons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' Article number, page 13 of 13 STEREO-A / HET / 2017-09 102 1 101 100 322 25 () g B R 0E 25 N 101 100 (MHz) 10-1 f 10-2 N=180, M=15 N=180, M=30 N=180, M=60 N=180, M=120 0w N=180, M=240 N=180, M=480 09-17 09-17 09-18 09-18 09-18 09-18 09-19 09-19 09-19 09-19 12:00 18:00 00:00 06:00 12:00 18:00 00:00 06:00 12:00 18:00 Date / Time in year 2017STEREO-A/ HET / 2017-09-19 102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4 MeV e 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV e 101 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0 MeV e EAA 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='6-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeV p 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='1 MeVp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content=' 100 W 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='0-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='3 MeV p 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE5T4oBgHgl3EQfaQ-N/content/2301.05587v1.pdf'} +page_content='8 MeV p 10-1 322 25 25 10 N=180,' 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+Department of Mathematics and Statistics +Kanpur 208016, India +email: suprio@iitk.ac.in +Subhra Sankar Dhar +IIT Kanpur +Department of Mathematics and Statistics +Kanpur 208106, India +email: subhra@iitk.ac.in +Abstract +In this article, we study the test for independence of two random elements X and +Y lying in an infinite dimensional space H (specifically, a real separable Hilbert space +equipped with the inner product ⟨., .⟩H). In the course of this study, a measure of as- +sociation is proposed based on the sup-norm difference between the joint probability +density function of the bivariate random vector (⟨l1, X⟩H, ⟨l2, Y ⟩H) and the prod- +uct of marginal probability density functions of the random variables ⟨l1, X⟩H and +⟨l2, Y ⟩H, where l1 ∈ H and l2 ∈ H are two arbitrary elements. It is established that +the proposed measure of association equals zero if and only if the random elements +are independent. In order to carry out the test whether X and Y are independent or +not, the sample version of the proposed measure of association is considered as the +test statistic after appropriate normalization, and the asymptotic distributions of the +test statistic under the null and the local alternatives are derived. The performance +of the new test is investigated for simulated data sets and the practicability of the +1 +arXiv:2301.00375v1 [math.ST] 1 Jan 2023 + +test is shown for three real data sets related to climatology, biological science and +chemical science. +Keywords: Climate, Measure of Association, Projection, Separable Hilbert Space. +1 +Introduction +1.1 +Key Ideas and Literature Review +For univariate and multivariate data, there have been several attempts to test whether two +or more random variables or vectors are independent or not in various situations (see, e.g., +Blum et al. (1961), Szekely et al. (2007), Genest et al. (1961), Einmahl and Van Keilegom +(2008), Dette et al. (2013), Bergsma and Dassios (2014), Dhar et al. (2016), Han et al. +(2017), Dhar et al. (2018), Drton et al. (2020), Chatterjee (2021), Berrett et al. (2021), +She et al. (2022b), She et al. (2022a) and a few references therein). To the best of our +knowledge, all these tests are based on a certain measure of association, which equals +zero if and only if the finite dimensional random variables or vectors are independent, +and consequently, the tests based on those measures of associations lead to consistent +tests. Therefore, one first needs to develop a measure of association having such necessary +and sufficient relation with independence in the infinite dimensional setting. Once such +a relation is found, in principle, it allows us to formulate a reasonable test statistic for +checking whether two random elements are independent or not in infinite dimensional +space. This article addresses this issue in the subsequent sections. +Let us first discuss the utility of the framework in terms of infinite dimensional spaces. +In many interdisciplinary subjects like Biology, Economics, Climatology or Finance, we +recently come across many data sets, where the dimension of the data is much larger than +the sample size of the data set, and in majority of the cases, the standard multivariate +techniques cannot be implemented because of their high dimensionalities relative to the +sample sizes. In order to overcome this problem, one can embed such data into a suit- +able infinite dimensional space. For instance, functional data (see Ramsay and Silverman +(2002), Ferraty and Vieu (2006)) is an infinite dimensional data, and one can analyse the +2 + +functional data using the techniques adopted for infinite dimensional data. In this work, +we consider random elements lying in a real separable Hilbert space. The separability of +the space allows us to write the random elements in the countably infinite orthonormal +expansion of a suitable basis, making it easier to handle the theoretical issues. In the con- +text of measure of association or the test for independence of infinite dimensional random +elements, we would like to mention that only a few limited numbers of articles are available +in the literature on this topic, and the contributions of the major ones are described here. +Almost a decade ago, Lyons (2013) (see also Lyons (2018) and Lyons (2021)) studied +the distance covariance, which is proposed by Szekely et al. (2007) for finite dimensional +random vectors, in a certain metric spaces. +However, Lyons (2013) did not study the +corresponding testing of hypothesis problems based on the proposed measure. +Before +the aforesaid work, Cuesta-Albertos et al. (2006) studied the random projection based +goodness of fit test and two sample tests for the infinite dimensional element, which can be +applied on the test for independence of the infinite dimensional random elements as well. +During the almost same period, Gretton et al. (2007) proposed a new methodology based +on the Hilbert-Schmidt independence criterion (HSIC) for testing independence of two +finite dimensional random vectors, and it can too be applied for the Hilbert space valued +random elements. However, none of these articles used the density functions of the random +variables obtained by the certain projections of the random elements lying in infinite +dimensional space. This article proposes a methodology for testing the independence of +random elements defined in infinite dimensional space based on the probability density +functions of projections of said random elements. +1.2 +Contributions +We consider two random elements X and Y lying in a real separable Hilbert space H +(equipped with the inner product ⟨., .⟩H) and propose a methodology of testing indepen- +dence between X and Y . This is the first major contribution of this article. The method- +ology is developed based on sup-norm distance between the joint probability density of +the bivariate random vector (⟨l1, X⟩H, ⟨l2, Y ⟩H) and the product of marginal probability +density functions of the random variables ⟨l1, X⟩H and ⟨l2, Y ⟩H, where l1 ∈ H and l2 ∈ H +3 + +are two arbitrary elements. This measure of association equals with zero for all l1 ∈ H and +l2 ∈ H if and only if X and Y are independent random elements, and it is non-negative as +well. +The second major contribution is the following. For a given data, a sample version +of the measure with appropriate normalization is proposed, which is considered as the +test statistic for testing independence of infinite dimensional random elements, and the +asymptotic distribution of the sample version under null and local/contiguous alternatives +is derived after appropriate normalization. These theoretical results enable us to study the +consistency and the asymptotic power of the corresponding test. Unlike other tests, since +the test statistic used in this methodology depends on the choice of the kernel and the +associated bandwidth, the performance of the proposed test can be enhanced by suitable +choices of the kernel and the bandwidth. +The third major contribution of the work is the implementation of the test. As the +test statistic is based on the supremum over the infinite choices of infinite dimensional +random elements l1 and l2, the exact computation of the test statistic is intractable for a +given data. To overcome this problem, l1 and l2 are chosen over certain finite collection +of possibilities of the choices, and it is shown that this method approximates the actual +statistic when the number of possible choices of l1 and l2 are sufficiently large. Overall, it +is established that the approximated test is easily implementable, and it gives satisfactory +results in analysing real data as well. +1.3 +Challenges +In the course of this work, we overcome a few mathematical challenges. The first challenge +involves the interpretation of independence between two infinite dimensional random el- +ements, and it is resolved using the concept of projection towards all possible directions +(see Lemma 2.1). Using Proposition 2.2, we further reduce the problem by relying on +vectors of unit length, rather that those with arbitrary lengths. This procedure reduces +the complexity of the optimization problem associated with infinite dimensional projection +vector to a large extent. The second challenge concerns the optimization problem asso- +ciated with time parameters involved with the test statistic. To overcome this issue, we +4 + +approximate the test statistic in two steps: first, we use a finite dimensional approximation +for any random vector on an orthonormal basis and then, we compute the test statistic +on a sufficiently large number of time points chosen over a sufficiently large interval. Us- +ing advanced techniques of analysis, it is shown that the later version of the test statistic +approximates arbitrarily well the original test statistic (see Proposition 3.1 and Lemma +6.8). The third challenge is related to asymptotic distribution of the test statistic. As the +key term of the test statistic involves a triangular array and a handful number of complex +terms, the asymptotic distribution of the key term follows after careful use of CLT associ- +ated with triangular array (see the proofs of Lemmas 6.2, 6.3, 6.4, 6.5 and 6.7). Finally, +the asymptotic distribution of the test statistic follows from Cram´er–Wold theorem and +continuous mapping theorem (see van der Vaart (1998)). +1.4 +Applications +We analyse well known climate data, which consists of daily temperature and precipitation +at thirty five locations in Canada averaged over 1960 to 1994 (see Ramsay and Silverman +(2005)). Note that the dimension of the data equals 365, which is much larger than the +sample size (i.e., 35) of the data, and embedding such a high dimensional data into a +specific Hilbert space is legitimate enough. From the point of view of climate science, it is +of interest to know how far the temperature depends on the precipitation at a given time +point. The analysis of this real data using the proposed methodology gives some insight +about the long standing issue in climatology. +Along with it, two more data are also analysed to show the practicability of the proposed +methodology. One data set is well-known Berkeley growth data, which consists the heights +of 39 boys and 54 girls measured at 31 time points between the ages 1 and 18 years, +and from medical science’s point of view, it is of interest to know whether the growth (in +terms of height) of the boys and the girls are independent or not. We address it using +our proposed test. Another data set is Coffee data, which contains spectroscopy readings +taken at 286 wavelength values for 14 samples of each of the two varieties of coffee, namely, +Arabica and Robusta. As coffee is one of the most popular beverages, one may be interested +to know whether the chemical features of the Arabica coffee and the Robusta coffee are +5 + +independent or not, and this issue is also addressed using our test. +1.5 +Organization of the Article +The article is organized as follows. Section 2 introduces the basic concepts of separable +Hilbert space valued random elements, and Section 2.1 proposes the criterion for checking +independence between two random elements. Afterwards, for a given data, the estimation +of the criterion is studied in Section 2.2, and Section 2.2.2 formulates the test statistic. The +asymptotic properties of the test statistic and associated test are investigated in Section 2.3, +and Section 2.3.1 studies the performance of the proposed test in terms of the asymptotic +power for various examples. The performance of the proposed test for finite sample sizes is +studied in Section 3, and real data analysis is carried out in Section 4. Section 5 contains +some concluding remarks, and finally, Sections 6 and 7 contain all technical details and +additional numerical results, respectively. +2 +Methodology +We first formulate the statement of the problem and subsequently construct an appropriate +test statistics. Let H denote a real separable Hilbert space equipped with the inner product +⟨., .⟩H and the norm ||.||H = +� +⟨., .⟩H, and suppose that (Ω, A, P) is a probability space. Let +us now consider an H × H valued bivariate random element (X, Y ), which is a measurable +mapping from (Ω, A, P) into H×H equipped with its Borel σ-algebra B(H×H) generated +by the open sets of H × H. +In other words, for any Borel set U1 ∈ H and U2 ∈ H, +(X−1(U1), Y −1(U2)) ∈ A. We now want to check whether the random elements X and Y +are independent or not, and in this regard, a new criterion for checking independence is +proposed here. +2.1 +Proposed Criterion for Independence +We here propose a criteria for checking independence based on certain one-dimensional +projections of the random elements X and Y lying on a separable Hilbert space H. In +6 + +order to formulate the criterion, a useful lemma is stated here. +Lemma 2.1 Let X and Y be two H valued random elements on a probability space (Ω, A, P), +where H is a real separable Hilbert space. Then H-valued X and H-valued Y are two in- +dependent random elements if and only if the real valued random variable ⟨l1, X⟩H and the +real valued random variable ⟨l2, Y ⟩H are independent for every l1 ∈ H and l2 ∈ H. In other +words, +P(⟨l1 , X⟩H ∈ A, ⟨l2 , Y ⟩H ∈ B) = P(⟨l1 , X⟩H ∈ A)P(⟨l2 , Y ⟩H ∈ B) +for all Borel subsets A and B of R if and only if +P(X ∈ C, Y ∈ D) = P(X ∈ C)P(Y ∈ D) +for all Borel subsets C and D of H. +The assertion in Lemma 2.1 indicates that P(A∩B) = PX(A)PY (B) for all A ∈ A and +B ∈ A, where PX and PY are marginal measures associated with X and Y , respectively of +the product measure P if and only if Q(C ∩ D) = QX(C)QY (D), where Q is the product +measure of (⟨l1, X⟩H, ⟨l2, Y ⟩H), QX is the measure associated with ⟨l1, X⟩H, and QY is +the measure associated with ⟨l2, Y ⟩H, respectively. Here C ∈ B(R) and D ∈ B(R) are +two arbitrary Borel sets. This fact indicates that if ⟨l1, X⟩H and ⟨l2, Y ⟩H are absolutely +continuous random variables, then the joint probability density function of ⟨l1, X⟩H and +⟨l2, Y ⟩H equals with the product of marginal probability density functions of ⟨l1, X⟩H and +⟨l2, Y ⟩H for all l1 and l2. Writing with notation, suppose that f(⟨l1,X⟩H,⟨l2,Y ⟩H)(., .), f⟨l1,X⟩H(.) +and f⟨l2,Y ⟩H(.) are the joint probability density, marginal probability density functions of +(⟨l1, X⟩H, ⟨l2, Y ⟩H), ⟨l1, X⟩H and ⟨l2, Y ⟩H, respectively. Then X and Y are independent +random elements if and only if +f(⟨l1,X⟩H,⟨l2,Y ⟩H)(s, t) − f⟨l1,X⟩H(s)f⟨l2,Y ⟩H(t) = 0 +(2.1) +for all s ∈ R, t ∈ R, l1 ∈ H and l2 ∈ H. The identity in (2.1) motivates us to formulate a +7 + +criterion for checking independence between X and Y is the following : +T(R1, R2) := +sup +||l1||H≤R1,||l2||H≤R2 +sup +s∈R,t∈R +|f(⟨l1,X⟩H,⟨l2,Y ⟩H)(s, t) − f⟨l1,X⟩H(s)f⟨l2,Y ⟩H(t)|, +(2.2) +where R1 and R2 are two arbitrary large positive real numbers. Using (2.1), we have the +following: +Proposition 2.1 T(R1, R2) = 0 if and only if X and Y are H-valued independent random +elements, where T(R1, R2) is as defined in (2.2). In other words, T(R1, R2) is degenerate +at 0 if and only if X and Y are H-valued independent random elements. +However, note that finding the supremum over {l1 : ||l1||H ≤ R1} and {l2 : ||l2||H ≤ R2} +is not easily tractable in practice, and to overcome it, one can find the supremum over the +boundary of the unit sphere, i.e., {l1 : ||l1||H = 1} and {l2 : ||l2||H = 1} defined in H, and +this equivalence is asserted in Proposition 2.2. +Proposition 2.2 Let +T := +sup +||l1||H=1,||l2||H=1 +sup +s∈R,t∈R +|f(⟨l1,X⟩H,⟨l2,Y ⟩H)(s, t) − f⟨l1,X⟩H(s)f⟨l2,Y ⟩H(t)|. +(2.3) +Then +T(R1, R2) = 0 ⇔ T = 0, +where T(R1, R2) is the same as defined in (2.2). +Finally, the following theorem characterizes the independence of X and Y based on equality +of T with zero. +Theorem 2.1 Let (X, Y ) be a bivariate random element taking values on H × H, where +H is a real separable Hilbert space. Then, X and Y are independent if and only if T = 0, +where T is defined in (2.3). +The assertion in Theorem 2.1 implies that the testing independence between the random +elements X and Y defined in H is equivalent to testing T = 0, and hence, in order to test +8 + +whether T = 0 or not (OR X and Y are independent random elements or not) for a given +data, one needs to have an appropriate estimator of T. +2.2 +Estimation of T +Let (X1, Y1), . . . , (Xn, Yn) be an i.i.d sequence of random elements, and they are identically +distributed with (X, Y ). In order to estimate T, one first needs to find l1 and l2 from +the unit sphere in H; see (2.3). +However, in (2.3), l1 and l2 appear in the joint and +marginal probability density functions and as such, we end up with a two fold issue. First +one involves an infinite-dimensional optimization in l1 and l2 and the second one is the +estimation of the corresponding joint and marginal probability density functions. +The +following may be one of the procedures. +2.2.1 +Approximation of l1 and l2 +In view of the fact that H is a real separable Hilbert space, and since l1 ∈ H and l2 ∈ H, +we have +l1 = +∞ +� +i=1 +l1,iei and l2 = +∞ +� +i=1 +l2,iei, +where for a fixed i, ei = (0, . . . , 0 +� �� � +i−1 +, 1, 0, . . .) is an infinite dimensional basis in H, and l1,i +and l2,i (i = 1, 2, . . .) are the coefficients of the orthonormal expansion of l1 and l2. Note +that +∞ +� +i=1 +l2 +1,i = 1, and +∞ +� +i=1 +l2 +2,i = 1, +and +����� +�����l1 − +M +� +i=1 +l1,iei +����� +����� +2 +H +→ 0 and +����� +�����l2 − +M +� +i=1 +l2,iei +����� +����� +2 +H +→ 0 +as M → ∞ as H is a real separable Hilbert space (see, e.g., Rudin (1991)). Using this +fact, we propose an approximate choices of lj (j = 1 and 2) as +ˆlM +j += +M +� +i=1 +lj,iei, +(2.4) +9 + +where M is a sufficiently large positive integer. Afterwards, to compute the criterion T or +estimate T, we compute the supremum over (lj,1, . . . , lj,M) for j = 1 and 2, where +M +� +i=1 +l2 +j,i = 1, +and this optimization problem can approximately be solved using polar transformation. +For j = 1 and 2, let us consider the following transformation into the spherical coordi- +nate system. +lj,1 = cos θj,1, +lj,2 = sin θj,1 cos θj,2, +lj,3 = sin θj,1 sin θj,2 cos θj,3, +... +lj,M−1 = sin θj,1 sin θj,2 . . . sin θj,M−2 cos θj,M−1, +lj,M = sin θj,1 sin θj,2 . . . sin θj,M−2 sin θj,M−1, +(2.5) +where for i = 1, . . . , M − 2, θj,i ∈ (− π +2, π +2) and θj,M−1 ∈ (−π, π). Observe that for j = 1 +and 2, +M +� +i=1 +l2 +j,i = 1. In practice, in the course of implementing the test, we consider K +equally spaced choices of θj,i from (− π +2, π +2) for i = 1, . . . , M − 2, and similarly, K equally +spaced θj,M−1 are chosen from (−π, π). Using this grid searching of θj,i (j = 1 and 2, and +i = 1, . . . , M − 1), suppose that the supremum associated with l1 and l2 involved in T (see +(2.3)) attains at some θ(K) +j.i for j = 1 and 2, and i = 1, . . . , M, and corresponding associated +lj,i are denoted by l(K) +j,i . Finally, the approximated choices of lj is considered as +ˆlK +j = +M +� +i=1 +l(K) +j,i ei. +(2.6) +Observe that +M +� +i=1 +(l(K) +j,i )2 = 1 for j = 1 and 2. The following proposition guarantees that +ˆlK +j +is approximate lj (j = 1 and 2) arbitrarily well. Note that ˆlK +j +(j = 1 and 2) depend +on M as well. However, for notational convenience, we do not explicitly write M in the +expression of ˆlK +j . +Proposition 2.3 For j = 1 and 2, +��� +���ˆlK +j − lj +��� +��� +H → 0 almost surely as M → ∞ and +K → ∞, where ˆlK +j is defined in (2.6). +10 + +Observe that ˆlK +j +is also unknown in practice as the joint and marginal probability +density functions involved in T (see (2.3)) are unknown. Therefore, as indicated from +Proposition 2.3 that ˆlK +j is a good approximation of lj (j = 1 and 2), and since computing +ˆlK +j +is much less complex relative to computing lj (j = 1 and 2), we will work on ˆlK +j +in +formulating the test statistic and studying its properties. +2.2.2 +Formulation of Test Statistic +As we said earlier, formally speaking we want to test H0 : X ⊥ Y against H1 : X ̸⊥ Y , +where X and Y are two random elements lying in a real separable Hilbert space H, and it +follows from Theorem 2.1 that it is equivalent to test H∗ +0 : T = 0 against H∗ +1 : T ̸= 0, where +T is defined in (2.3). In order to test H∗ +0 against H∗ +1, one needs to have an appropriate +estimator of T, and as an initial step in the course of this work, Section 2.2.1 studied how +to approximate l1 and l2 involved in T (see (2.3)), and the approximated lj (j = 1 and 2) +are denoted by ˆlK +j defined in (2.6). +Here one first needs to project the infinite dimensional data into one dimensional +space through ˆlK +1 and ˆlK +2 . Let us fist consider the data (X1, Y1), . . . , (Xn, Yn), which be- +long to the same probability space of (X, Y ), and suppose that the transformed data are +(⟨ˆlK +1 , X1⟩H, ⟨ˆlK +2 , Y1⟩H), . . . , (⟨ˆlK +1 , Xn⟩H, ⟨ˆlK +2 , Yn⟩H), where the projection vectors ˆlK +1 and ˆlK +2 +are unknown in practice as we discussed at the end of Section 2.2.1. In order to identify +the optimum ˆlK +1 and ˆlK +2 for a given data, we need to estimate the joint probability density +function f(⟨l1,X⟩H,⟨l2,Y ⟩H)(., .) and the marginal probability density functions f⟨l1,X⟩H(.) and +f⟨l2,Y ⟩H(.). We describe the estimation of the probability density functions in the following. +Let k : R2 → (0, ∞) be a bivariate function such that +� +k(x, y)dxdy = 1, k1 : R → +(0, ∞) and k2 : R → (0, ∞) are such that +� +k1(x)dx = 1 and +� +k2(x)dx = 1. Note that +k, k1 (and k2) can be considered as the kernels involved in the estimator of the bivariate +and univariate probability density functions, respectively. For details on kernel density +estimation, the readers may refer to Silverman (1998), and a few more technical conditions +on the kernels will be described at the appropriate places. Now, suppose that Tn is an +appropriate estimator T based on the kernel density estimators k, k1 and k2, which can be +formulated as follows. +11 + +Tn = +sup +θj,i∈(− π +2 , π +2 ) +j=1,2;i=1,...,M−1 +θj,M−2∈(−π,π) +j=1,2 +s∈R,t∈R +����� +1 +nh +3 +2n +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +� +− +1 +n2h2 +n +n +� +i,j=1 +k1 +� +⟨ˆlK +1 , Xi⟩H − s +hn +� +k2 +� +⟨ˆlK +2 , Yj⟩H − t +hn +������ , +(2.7) +where hn is a sequence of bandwidth such that hn → 0 as n → ∞. Note that ˆlK +j involves +θj,i (see (2.5) and (2.6)), and for this reason, supremum taken over ˆlK +j +eventually boils +down to the supremum taken over θj,i (j = 1 and 2, i = 1, . . . , M) in the expression of Tn +(see (2.7)). The asymptotic properties of Tn are studied in the subsequent section. +2.3 +Asymptotic Properties of Tn +Recall that the testing of hypothesis problem H0 : X ⊥ Y against H1 : X ̸⊥ Y is equivalent +to testing of hypothesis problem H∗ +0 : T = 0 against H∗ +1 : T ̸= 0. In order to check whether +for a given data T = 0 or not, the test statistic Tn (see (2.7)) is formulated in Section 2.2.2, +and in order to carry out the test based on Tn, the distributional feature of Tn is required. +However, due to complex formulation of Tn, the derivation of exact distribution of Tn may +not be tractable, and to overcome it, we here derive the asymptotic distribution of Tn. We +first assume the following technical conditions and then state the asymptotic distribution +of Tn under local alternatives, and the asymptotic distribution under null hypothesis (i.e., +H0 or H∗ +0). +(A1) The sequence of bandwidth {hn}n≥1 is such that hn → 0 as n → ∞, nh2 +n → c (for +some constant c ∈ R) as n → ∞, and nhn → ∞ as n → ∞. +(A2) The partial derivatives of the joint probability density function f of the bivariate +random vector (⟨l1, X⟩H, ⟨l2, Y ⟩H) exist, i.e., +∂f(⟨l1,X⟩H,⟨l2,Y ⟩H)(z1,z2) +∂z1 +and +∂f(⟨l1,X⟩H,⟨l2,Y ⟩H)(z1,z2) +∂z2 +exist. +(A3) The random elements X and Y are norm bounded, i.e., ||X||H and ||Y ||H are bounded +random variables. Moreover, the joint probability density function of ⟨l1, X⟩H and ⟨l2, Y ⟩H +is uniformly bounded. +12 + +(A4) The first and the second derivatives of the probability density functions of the random +variables ⟨l1, X⟩H and ⟨l2, Y ⟩H are uniformly bounded. +(A5) For i = 1 and 2, the kernels ki : R → A ⊂ R+ are such that +� +ki(m)dm = 1, +� +mki(m)dm < ∞, +� +k2 +i (m)dm < ∞, +� +m2ki(m)dm < ∞ and +� +m{ki(m)}2dm < ∞. Here +A is a bounded set. +(A6) The bivariate kernel k : R2 → A ⊂ R+ is such that +� +m1k(m1, m2)dm1dm2 < ∞, +� +m2k(m1, m2)dm1dm2 < ∞, +� +m1{k(m1, m2)}2dm1dm2 < ∞ and +� +m2{k(m1, m2)}2dm1dm2 < +∞. Here A is a bounded set. +Remark 2.1 Condition (A1) indicates that for various choices of hn, the asymptotic re- +sults described in Theorem 2.2 holds, and such conditions are common across the literature +in kernel density estimation (see, e.g., Silverman (1998)). Condition (A2) is satisfied when +the directional derivatives of the joint probability density function of ||X||H and ||Y ||H exist, +and these are indeed realistic assumptions. Further, when the infinite dimensional random +elements X and Y are norm bounded, and the joint probability density function of the +random variables ⟨l1, X⟩H and ⟨l2, Y ⟩H is uniformly bounded (see Proposition 6.1), then +the marginal probability density functions of the random variables ⟨l1, X⟩H and ⟨l2, Y ⟩H +are also bounded. For instance, for well-known Gaussian process, the conditions (A3) and +(A4) hold. Condition (A5) implies certain mild restriction on the tail behaviour of the +chosen kernels. However, for compact support of the kernel, there is no need to put restric- +tions on the tail behaviour of the chosen kernels as the integrations involving kernels are +finite as long as the the kernels are continuous. Condition (A6) asserted some Integrability +assumptions on the chosen bivariate kernel. In this case also, if the support of the chosen +bivariate kernel is compact, the continuity of the chosen bivariate kernel is good enough to +satisfy the integrability assumptions in condition (A6). +13 + +Theorem 2.2 Let us denote +T G,L +n += +sup +θj,i∈(− π +2 , π +2 ) +j=1,2;i=1,...,M−1 +θj,M−2∈(−π,π) +j=1,2 +s∈{s1,...,sL} +t∈{t1,...,tL} +����� +1 +nh +3 +2n +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +� +− +1 +n2h2 +n +n +� +i,j=1 +k1 +� +⟨ˆlK +1 , Xi⟩H − s +hn +� +k2 +� +⟨ˆlK +2 , Yj⟩H − t +hn +������ , +(2.8) +where si = −G + i 2G +L and ti = −G + i 2G +L for i = 1, . . . , L are equi-distributied points in +[−G, G], and Tn is the same as defined in (2.7). Then, under (A1), (A5) and (A6), +� +nhn +� +lim +G→∞ lim +L→∞(Tn − T G,L +n +) +� +p→ 0 +as n → ∞. +Moreover, under conditions (A1)–(A6) and when T = +λ +√nhn for some λ > 0, √nhnT G,L +n +− +λ converges weakly to the distribution of +sup +s,t∈{−G+i 2G +L |i=1,...,L} +{|Zs,t|} as n → ∞, K → ∞ +and M → ∞ (see (2.6) for the definitions of K and M), where Zs,t is a random variable +associated with normal distribution with mean += −√c +� +fZ1(s)f +′ +Z2(t) +� +mk2(m)dm + fZ2(t)f +′ +Z1(s) +� +mk1(m)dm +� +and variance += fZ1,Z2(s, t) +� +k2(m1, m2)dm1dm2+ {fZ1(s)}2fZ2(t) +� +{k2(m)}2+fZ1(s){fZ2(t)}2 +� +{k1(m)}2dm. +Here c = lim +n→∞ nh2 +n and (Z1, Z2) = (⟨l1, X⟩H, ⟨l2, Y ⟩H). +Remark 2.2 The first part in the statement of Theorem 2.2 indicates that T G,L +n +is an +appropriate approximation of Tn for sufficiently large G and L after normalized by √nhn. +Here G controls the length of the interval of the time parameters, and L controls the number +of time points after discretization. The motivation of constructing T G,L +n +will be discussed +in details in Section 3.1. Moreover, with the same normalization factor √nhn, T G,L +n +− λ +14 + +converges weakly to a certain random variable associated with the functional of multivariate +normal distribution. All together, one can approximately compute the asymptotic power of +the test based on Tn using the assertion in Theorem 2.2 +Remark 2.3 Observe that Theorem 2.2 is stated when T = +λ +√nhn, and since √nhn → ∞ +as n → ∞ (see Condition (A1)), it indicates that the statement of this theorem is valid +for local alternatives. As a result, the assertion of this theorem enables us to compute the +approximated asymptotic power of the test based on Tn. Further, as a special case, when +λ = 0, the asymptotic distribution of Tn under null hypothesis (i.e., T = 0) follows from +Theorem 2.2, and it is formally stated in Corollary 2.4. +Corollary 2.4 Under conditions (A1)-(A6) and when T = 0 (i.e., under null hypothesis), +√nhnT G,L +n +converges weakly to the distribution of +sup +s,t∈{−G+i 2G +L |i=1,...,L} +{|Zs,t|} as n → ∞, +K → ∞ and M → ∞, where Zs,t denotes a random variable associated with normal +distribution with mean and variance, which are the same as described in the statement of +Theorem 2.2. +Corollary 2.4 asserts the asymptotic distribution T G,L +n +under the null hypothesis, and +therefore, one can approximately estimate the critical value of the test based on Tn using +the asymptotic distribution of T G,L +n +stated in Corollary 2.4 as Tn and T G,L +n +can be made +arbitrary close to each other for sufficiently large G and L (see the first part of Theorem +2.2). All in all, one can explicitly express the power of the test based on Tn when T = t0 +using Corollary 2.4 and Theorem 2.2. In notation, if ˆcα is such that PH0[√nhnTn > ˆcα] → α +as n → ∞, the the power of the test based on Tn under H1 (i.e., when T = t0) is given by +PH1 +�� +nhnTn > ˆcα +� += P +�� +nhn(Tn − t0) > ˆcα − +� +nhnt0 +� +, +(2.9) +and the asymptotic power of the test based on Tn under local alternatives (e.g., T = +λ +√nhn +as mentioned in Theorem 2.2) is given by +P� +T= +λ +√nhn +� �� +nhnTn > ˆcα +� += P +�� +nhnTn − λ > ˆcα − λ +� +. +(2.10) +15 + +Moreover, using (2.9), one can establish the consistency of the test based on Tn, which is +stated in Proposition 2.4. +Proposition 2.4 Let ˆcα is such that PH0[√nhnTn > ˆcα] → α as n → ∞, where α ∈ (0, 1) +is a preassigned constant. Then, the test based on Tn, which rejects H0 if √nhnTn ≥ ˆcα, +is a consistent test. In other words, PH1[√nhnTn ≥ ˆcα] → 1 as n → ∞. +Proposition 2.4 asserts that the test based on Tn is expected to perform well in terms of +power for a sufficiently large sample. Next, we study the performance of the test based on +Tn through the asymptotic power study. +2.3.1 +Asymptotic Power Study +Recall that Theorem 2.2 states the asymptotic distribution of T G,L +n +under local alternatives, +i.e., when T = +λ +√nhn (λ ≥ 0), and one can compute the asymptotic power of the test based +on Tn for various choices of the random processes in separable Hilbert space and λ. In the +course of study, the critical value with α% level of significance, i.e., ˆcα is computed using +the result described in Corollary 2.4, and using the obtained ˆcα, one can approximate the +asymptotic power of the test based on Tn using (2.10). +We consider following three examples : +Example 1 : Let (X, Y ) be L2([0, 1]) × L2([0, 1]) valued random element. Suppose that +X(t) (t ∈ [0, 1]) has the same distribution as U exp(t), where U follows uniform distribution +over [0, 1], and Y (t) = ||X||L2([0,1]) × B(t), where B(t) (t ∈ [0, 1]) is a Gaussian process +with E(B(t)) = 0 and E(B(t)B(s)) = min(t, s) for all t ∈ [0, 1] and s ∈ [0, 1]. +Example 2 : Let (X, Y ) be L2([0, 1]) × L2([0, 1]) valued random element. Suppose that +X(t) (t ∈ [0, 1]) has the same distribution as U exp(t), where U follows uniform distribution +over [0, 1], and Y (t) = ||X||L2([0,1]) × B1(t), where B1(t) (t ∈ [0, 1]) is fractional Brownian +motion with Hurst index = .25, i.e., E(B1(t)) = 0 and E(B1(t)B1(s)) = 1 +2(|t|0.5 + |s|0.5 − +|t − s|0.5) for all t ∈ [0, 1] and s ∈ [0, 1]. +Example 3 : Let (X, Y ) be L2([0, 1]) × L2([0, 1]) valued random element. Suppose that +X(t) (t ∈ [0, 1]) has the same distribution as U exp(t), where U follows uniform distribution +over [0, 1], and Y (t) = ||X||L2([0,1]) × B2(t), where B2(t) (t ∈ [0, 1]) is fractional Brownian +16 + +0 +2 +4 +6 +8 +10 +0.1 +0.2 +0.3 +0.4 +Example 1 +λ +Asymptotic Power +0 +2 +4 +6 +8 +10 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Example 2 +λ +Asymptotic Power +0 +2 +4 +6 +8 +10 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Example 3 +λ +Asymptotic Power +Figure 1: Asymptotic power of the proposed test for Examples 1, 2 and 3 at 5% level of +significance when λ varies from 0 to 10. +motion with Hurst index = .75, i.e., E(B2(t)) = 0 and E(B2(t)B2(s)) = 1 +2(|t|1.5 + |s|1.5 − +|t − s|1.5) for all t ∈ [0, 1] and s ∈ [0, 1]. +Note that in order to compute the asymptotic power of the test, one first needs to +compute the critical value, which can be estimated using the result stated in Corollary +2.4, i.e., when T = 0, and it is equivalent to the fact that X and Y are indepen- +dent random elements in H (see Theorem 2.1). +Now, in order to satisfy this statisti- +cal independence between X and Y , we first consider X(t) and Y (t) are two indepen- +dent random elements in L2([0, 1]) associated with standard Brownian motion. +After- +wards, the terms involving the joint and the marginal probability density functions (i.e., +f(⟨l1,X⟩H,⟨l2,Y ⟩H)(., .), f⟨l1,X⟩H(.), and f⟨l2,Y ⟩H(.)) appeared in the mean and the variance of +the asymptotic distribution of Tn under H0 (see Corollary 2.4) are estimated by bivariate +and marginal kernel density function. For the bivariate kernel density function, we consider +the two fold product of the marginal kernel density function. In this study, for estimating +the marginal probability density function, we use the well-known Epanechnikov Kernel +because of its optimal properties (see, e.g., Silverman (1998)). Finally, α% (denoted as ˆcα) +critical value is computed as the (1 − α)-th quantile of the normal distribution described +in Corollary 2.4. Precisely speaking, +ˆcα = G−1 +H0(α), +(2.11) +17 + +where GH0 is the cumulative distribution function of √nhnT G,L +n +(see Corollary 2.4 for the +explicit form of GH0). In the study, we consider G = 20 and L = 10. +Now, for Examples 1, 2 and 3, the approximated asymptotic powers of the test based +on Tn are illustrated in the diagrams in Figure 1 for various choices of λ. In general, it +is observed from all three diagrams that the asymptotic power of the test based on Tn +increases as λ increases. The increasing feature of asymptotic power with respect to λ for +all three examples has been seen in view of the fact that X and Y are not independent +random elements and as the value of T is deviating more from the null hypothesis as λ is +deviating from zero. Moreover, among Examples 1, 2 and 3, observe that the asymptotic +powers for Examples 2 and 3 are more than that for Example 1 since in Examples 2 and 3, +Y (t) at various time points t ∈ [0, 1] are correlated whereas in Example 1, Y (t) at various +time points t ∈ [0, 1] are uncorrelated. +3 +Finite Sample Level and Power Study +The asymptotic power study in Section 2.3.1 indicates that the power of the test based on +Tn is fairly good when the sample size is large enough. Besides, Proposition 2.4 asserts +the consistency of the test, which also indicates that the proposed test will perform well +in terms of power for a sufficiently large sample. Overall, these facts only guarantee the +expected good performance of the proposed test when the sample size is large enough. +Here, we now want to see the performance of the proposed test when the sample size is +small or moderately small/large, and in order to carry out this study, one needs to compute +Tn for a given data, which is described in the following. +3.1 +Computation of Tn +Observe that the exact distribution of Tn is intractable because of its complex structure, +and for that reason, one cannot compute the critical value from the inverse of the exact dis- +tribution of Tn. In order to overcome this issue, one requires to approximate the sampling +distribution of Tn using Monte-Carlo simulation, and in this procedure, the computation +of Tn for a given sample is requisite. Further, for the same reason provided above, one +18 + +needs to compute Tn for a given sample for computing the power. +Recall Tn from (2.7) : +Tn += +sup +θj,i∈(− π +2 , π +2 ) +j=1,2;i=1,...,M−2 +θj,M−1∈(−π,π) +j=1,2 +s∈R,t∈R +����� +1 +nh +3 +2n +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +� +− +1 +n2h2n +n� +i,j=1 +k1 +� +⟨ˆlK +1 ,Xi⟩H−s +hn +� +k2 +� +⟨ˆlK +2 ,Yj⟩H−t +hn +������ . +Note that in the expression in Tn, the supremum is taken over finitely many choices of θj,i +(j = 1 and 2, i = 1, . . . , M −1), which are involved in ˆlK +1 and ˆlK +2 and over s ∈ R and t ∈ R. +As R in an interval with infinite length, the computation of supremum on s and t over R +becomes unmanageable, and in order to overcome this issue, we take a sufficiently large +interval [−G, G] and do the grid search on {s1, . . . , sL} and {t1, . . . , tL}, where si ∈ [−G, G] +and ti ∈ [−G, G] (i = 1, . . . , L) are equally spaced points. This approximation procedure +makes the computation doable in practice since domain of both variables s and t become +finite. Therefore, in practice, we will compute the following. +T G,L +n += +sup +θj,i∈(− π +2 , π +2 ) +j=1,2;i=1,...,M−1 +θj,M−2∈(−π,π) +j=1,2 +s∈{s1,...,sL} +t∈{t1,...,tL} +����� +1 +nh +3 +2n +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +� +− +1 +n2h2 +n +n +� +i,j=1 +k1 +� +⟨ˆlK +1 , Xi⟩H − s +hn +� +k2 +� +⟨ˆlK +2 , Yj⟩H − t +hn +������ , +(3.12) +where G ∈ R and L ∈ Z+ are sufficiently large. The proposition 3.1 along with the first +part of Theorem 2.2 guide our choice of T G,L +n +as an approximation of Tn. +Proposition 3.1 For all n ≥ 1, under (A5) and (A6), lim +G→∞ lim +L→∞(Tn − T G,L +n +) +a.s. += 0, where +Tn is defined in (2.7), and T G,L +n +is defined in (3.12). +19 + +3.1.1 +Estimated Level and Power +Here, we discuss how one can estimate the level and the power of the test based on Tn. +In this study, we choose M = 10 and K = 20 as we have observed that any larger values +than the aforementioned chosen values does not much change the result. For details about +the variables M and K, recall Proposition 2.3 and the discussion before the proposition. +Besides, as Proposition 3.1 indicates, the choices of G and L are also an issue of concern. +After having a thorough investigation, it is found that G = 20 and L = 10 are reasonably +good choices in terms of computational complexity and the precision of this finite approx- +imation. Throughout the study, we consider k1 and k2 in the expression in Tn (see (2.7)) +as Epanechnikov kernel (see Silverman (1998)) for its optimal property, and the bivariate +kernel k is considered as the two folds product of Epanechnikov kernels. As the bandwidth, +we choose hn = cn− 1 +6, where c is estimated by cross validation technique (see, e.g., Duong +and Hazelton (2005) and Hall et al. (1992)). In the course of this study, the experiments +are carried out for n = 20, 50 and 100. +In order to carry out the critical value of the test, we generate two independent data +sets X = (X1, . . . , Xn) and Y = (Y1, . . . , Yn) from standard Brownian motion, and replicate +this experiment 1000 times. Note that as X and Y are independent, the null hypothesis +H0 : X ⊥ Y (⇔ T = 0) is satisfied. +Now, let Tn,i be the value of Tn for the i-th +replicate (i = 1, . . . , 1000), and these 1000 values of Tn,i provide the estimated distribution +of Tn. Therefore, (1 − α)-th (α ∈ (0, 1)) quantile of Tn,i (i = 1, . . . , 1000), denoted as +ˆcα can be considered as the estimated critical value. Now, in order to estimate the level +of significance, we generate two independent data sets, namely, X +′ = (X +′ +1, . . . , X +′ +n) and +Y +′ = (Y +′ +1, . . . , Y +′ +n) and replicate this experiment M times. Suppose that Tn,i is the value of +Tn for the i-th replication, and the estimated level is defined as +1 +M +M +� +i=1 +1(Tn,i>ˆcα). Next, in +order to estimate the power of the test, we generate two dependent data sets, namely, X +′′ = +(X +′′ +1 , . . . , X +′′ +n) and Y +′′ = (Y +′′ +1 , . . . , Y +′′ +n ) and replicate this experiment M +′ times. Suppose +that Tn,j is the value of Tn for the j-th replication, and the estimated power is defined as +1 +M′ +M +′ +� +j=1 +1(Tn,j>ˆcα). +To estimate the level, we consider the following three examples. +20 + +Example 4 : Two independent data sets X = {X1, . . . , Xn} and Y = {Y1, . . . , Yn} are +generated from standard Brownian motion. +Example 5 : Two independent data sets X = {X1, . . . , Xn} and Y = {Y1, . . . , Yn} are +generated from fractional Brownian motion with Hurst index = 0.25. +Example 6 : Two independent data sets X = {X1, . . . , Xn} and Y = {Y1, . . . , Yn} are +generated from fractional Brownian motion with Hurst index = 0.75. +model +n = 20 +n = 50 +n = 100 +n = 500 +Example 4 (α = 5%) +0.062 +0.060 +0.057 +0.051 +Example 4 (α = 10%) +0.121 +0.120 +0.114 +0.107 +Example 5 (α = 5%) +0.063 +0.058 +0.054 +0.052 +Example 5 (α = 10%) +0.122 +0.121 +0.110 +0.106 +Example 6 (α = 5%) +0.064 +0.062 +0.060 +0.056 +Example 6 (α = 10%) +0.125 +0.124 +0.119 +0.111 +Table 1: The estimated size of the proposed test for different sample sizes n. The levels of +significance, i.e., α are 5% and 10%. +In order to generate the data from the fractional/standard Brownian motion, we generate +the data corresponding to a certain large dimensional multivariate normal distribution. +The estimated levels for Examples 4, 5 and 6 are reported in Table 1 for the sample sizes +n = 20, 50, 100 and 500 at 5% and 10% level of significance. The reported values indicate +that the estimated level never deviated more than 1.5% from desired level of significance +(i.e., 5% and 10%) when the sample sizes are 20 and 50 and not deviated more than 1% +when the sample sizes are 100 and 500. +To estimate the power, we generate the data (X1, . . . , Xn) and (Y1, . . . , Yn) from the +distributions as described in Examples 1, 2 and 3 in Section 2.3.1. At 5% and 10% level +of significance, the estimated powers for those examples are reported in Table 2 when the +sample sizes are 20, 50 and 100. In all these cases, it indicates the power increases as +the sample size increases. Besides, as we observed in the asymptotic power study (see +Section 2.3.1), the proposed test becomes more powerful when the data is generated from +the distribution described in Examples 2 and 3 than that of Example 1, and this fact is +expected since the correlation structure involved in the fractional Brownian motion having +Hurst parameter ̸= 0.5. +21 + +model +n = 20 +n = 50 +n = 100 +n = 500 +Example 1 (α = 5%) +0.342 +0.506 +0.678 +0.791 +Example 1 (α = 10%) +0.399 +0.601 +0.800 +0.882 +Example 2 (α = 5%) +0.445 +0.624 +0.785 +0.911 +Example 2 (α = 10%) +0.503 +0.632 +0.867 +0.949 +Example 3 (α = 5%) +0.427 +0.641 +0.805 +0.902 +Example 3 (α = 10%) +0.517 +0.666 +0.841 +0.956 +Table 2: The estimated power of the proposed test for different sample sizes n. The level +of significance, i.e., α are 5% and 10%. +A few more simulation studies are carried out in Appendix B. +4 +Real Data Analysis +Here, we implement our proposed test on a real data, which consists of daily temperature +and precipitation at n = 35 locations in Canada averaged over 1960 to 1994. Ramsay and +Silverman (2005) studied this data set in the context of functional principal component +analysis, and the soft version of the data may be found in https://climate.weather.gc. +ca/historical_data/search_historic_data_e.html. The dimension of the data equals +365, which is much larger than the sample size = 35 of the data, and therefore, embedding +such a high dimensional data into a specific Hilbert space is legitimate enough. For this +data set, suppose that Xi (i = 1, . . . , 35) is the average daily temperature for each day of +the year at the i-th location, and Yi is the base 10 logarithm of the corresponding average +precipitation. Here we consider X and Y as L2([0, 1])-valued random elements, where 365 +days are considered as the equally spaced 365 time points over [0, 1]. The curves associated +with X and Y are demonstrated in the diagrams in Figure 2. +In order to check whether X and Y are independent random elements or not, we +compute the p-value of the test based on Tn using Bootstrap resampling procedure. At the +beginning, to compute Tn (precisely speaking, T G,L +n +), we choose tuning parameters M = 10, +K = 20, G = 20 and L = 10 as this issue has been discussed at the beginning of Section +3.1.1. Similarly, we choose the bandwidth hn = cn− 1 +6, where c is estimated by well-known +cross validation technique (see, e.g., Duong and Hazelton (2005)), and as said before, k1 +22 + +Figure 2: +The left diagram plots the average daily temperature and the right diagram plots +the average daily precipitation. Source : https: // fda. readthedocs. io/ en/ latest/ +auto_ examples +23 + +Canadian Weather +Arctic +Atlantic +Continental +Pacific +20 +10 +temperature (C) +0 +-10 +-20 +-30 +0 +50 +100 +150 +200 +250 +300 +350 +dayCanadian Weather +16 +14 +12 +precipitation (mm.) +10 +00 +6 +4 +0 +50 +100 +150 +200 +250 +300 +350 +dayand k2 are chosen as Epanechnikov kernel, and the bivariate kernel k is considered as the +two folds product of Epanechnikov kernels. Based on these aforesaid choices, we generate +500 Bootstrap resamples with size = 35 from the original data (X1, Y1), . . . , (X35, Y35), +and suppose that t0 is the value of Tn for the original data (X1, Y1), . . . , (X35, Y35). Let +tj (j = 1, . . . , 500) be the value of Tn for the j-th resample, and the p-value is defined as +1 +500 +500 +� +j=1 +1(tj>t0). This methodology gives us the p-value as 0.078, which indicates the data +does not favour the null hypothesis, i.e., the temperature curve and the precipitation curve +in those 35 locations in Canada over the period from 1960 to 1994 are not statistically +independent at 8% level of significance. Even from the climate science point of view, it +is expected that the temperature and the amount of precipitation are supposed to have +some dependence structure, and hence, the result shown by the proposed test is reasonable +enough. +For this data, as the temperature and the precipitation are dependent according to our +proposed test, one may be interested to know the estimated size and power of the test for +this data set. In order to compute estimated size and power, we combine (X1, . . . , X35) +and (Y1, . . . , Y35) and mix these 70 observations randomly. Let us denote the new sample +as Z = (Z1, . . . , Z70). Now, partition Z into two parts, namely, Z1 = (Z1, . . . , Z35) and +Z2 = (Z36, . . . , Z70), and note that Z1 and Z2 are two independent data sets as these are +constructed by random mixing of the combined sample of X and Y. Next, from Z1 and Z2, +by Bootstrap methodology, we generate M many resamples Z1,i and Z2,i (i = 1, . . . , M) +and compute the test statistic Tn for each resample. Suppose that ti is the value of Tn +for the i-th resample, and then, (1 − α)-th quantile of (t1, . . . , tM) is considered as the +estimated critical value (denoted as ˆcα) at α% level of significance. In order to estimate +the size of the test, we generate M1 many resamples Z1,i and Z2,i (i = 1, . . . , M1), and the +estimated size can be defined as +1 +M1 +M1 +� +j=1 +1(tj>ˆcα), where tj is the value of Tn for j-th resample +(j = 1, . . . , M1). Next, in order to estimate the power, using Bootstrap methodology, we +generate M2 many resamples from the original data (X1, . . . , X35) and (Y1, . . . , Y35), and +note that in each resample, X-data and Y -data are statistically dependent since original +(X1, . . . , X35) and (Y1, . . . , Y35) are statistically dependent data sets. +Suppose that tk +(k = 1, . . . , M2) is the value of Tn for the k-th resample, and the estimated power can +24 + +be defined as +1 +M2 +M1 +� +k=1 +1(tk>ˆcα). In this study, we consider M1 = M2 = 500 and estimate the +power and the size of the test when α = 0.05. Using all these choices, we obtain the +estimated size = 0.058 and the estimated power = 0.723. Overall, these facts indicate that +the proposed test can achieve the nominal level of the test and poses good power when the +random elements are statistically dependent. +Two more real data are analysed using the proposed methodology in Appendix B. +5 +Concluding Remarks +This article studies the test for independence of two random elements taking values on an +infinite dimensional space. In this test, the test statistic is formulated based on the sup +norm (i.e., L∞ norm) distance between the joint probability density function of two dimen- +sional certain projection of bivariate random element and product of marginal probability +density functions of corresponding marginal random variables. The asymptotic distribu- +tion of the test statistic under certain local alternatives have been derived, and it has been +observed that the proposed test performs well for various choices of local alternatives. Be- +sides, the usefulness of the test is shown on well-known data sets, and simulation studies +also indicate that the proposed test performs well under different scenarios. +It follows from the proof of Theorem 2.2 that the main crux of the proof is the derivation +of the pointwise asymptotic properties of +����� +1 +nh +3 +2n +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +� +− +1 +n2h2 +n +n +� +i,j=1 +k1 +� +⟨ˆlK +1 , Xi⟩H − s +hn +� +k2 +� +⟨ˆlK +2 , Yj⟩H − t +hn +������ , +and afterwards, the asymptotic distribution of T G,L +n +follows from some arguments related +continuous mapping theorem (see, e.g., Serfling (1980)). In this context, we would like to +point out that one may consider any other appropriate norm like Lp norm, when p ∈ [1, ∞). +The study of performance of the test based on Lp norm may be of interest for future +research. +Another issue may need to be discussed, which is related to proposed criterion T (see +25 + +(2.3)). One may argue whether T can be thought of as a measure of association or not. +Note that under some technical conditions, one can argue that T ≤ M, where M > 0 is a +constant. Hence, it may be appropriate to consider T +′ = +T +M as a measure of association. +Observe that T +′ ∈ [0, 1], and for any two random elements X and Y , T +′(X, Y ) = 0 if and +only if X and Y are independent random elements, which follows from the assertion in +Theorem 2.1. In order to establish T +′ as a measure of association, one needs to characterize +the case when T +′(X, Y ) = 1, i.e., in other words, the readers may be interested to know +the perfect dependence structure between the random elements X and Y , which is not +done in this work. The characterization of perfect dependence between X and Y when +T +′(X, Y ) = 1 may be of another interest for future research. +Acknowldgement: The first author acknowledges the research grant DST/INSPIRE/04/2017/002835, +Government of India, and the second author is thankful to the research grants MTR/2019/000039 +and CRG/2022/001489, Government of India. +6 +Appendix A : Technical Details +Proof of Lemma 2.1: Let X and Y be independent, and fix l1 and l2 ∈ H. Observe +that the mappings ⟨l1 , ·⟩H : H → R and ⟨l2 , ·⟩H : H → R defined by x �→ ⟨l1 , x⟩H and +x �→ ⟨l2 , x⟩H are measurable. +Now, for all Borel subsets A and B of R, we have +P(⟨l1 , X⟩H ∈ A, ⟨l2 , Y ⟩H ∈ B) = P(X ∈ ⟨l1 , ·⟩−1 +H (A), Y ∈ ⟨l1 , ·⟩−1 +H (B)) += P(X ∈ ⟨l1 , ·⟩−1 +H (A))P(Y ∈ ⟨l1 , ·⟩−1 +H (B)) +(since X and Y are independent) += P(⟨l1 , X⟩H ∈ A)P(⟨l2 , Y ⟩H ∈ B). +Therefore, the two random variables ⟨l1 , X⟩H and ⟨l2 , Y ⟩H are independent. +Conversely, suppose that ⟨l1 , X⟩H and ⟨l2 , Y ⟩H are independent for all l1 and l2 ∈ H. +26 + +Since, H is separable, consider an orthonormal basis {en : n = 1, 2, · · · }. Observe that +X +H= +∞ +� +n=1 +⟨X , en⟩H en, +Y +H= +∞ +� +n=1 +⟨Y , en⟩H en. +Here A +H= B denotes ||A − B||H = 0 for any A ∈ H and B ∈ H. Since the two families of +real valued random variables {⟨X , en⟩H : n = 1, 2, · · · } and {⟨Y , en⟩H : n = 1, 2, · · · } are +independent, so are X and Y . This completes the proof. +□ +The following lemma is useful in proving Proposition 2.1. +Lemma 6.1 Let (Z, W) denote an R2 valued random vector, and fix non-zero real num- +bers α and β. Then, (Z, W) is absolutely continuous if and only if (αZ, βW) is absolutely +continuous. Moreover, Z and W are independent if and only if αZ and βW are indepen- +dent. +Proof of Lemma 6.1: Consider the transformation (x, y)t ∈ R2 �→ (αx, βy)t ∈ R2. +Observe that the transformation is one-to-one and onto, and the Jacobian of the transfor- +mation is a non-zero constant. Hence, the first statement related to absolute continuity +follows. +To prove the second part, let A and B be Borel subsets of R. Then, 1 +αA (elementwise +product) and 1 +βB (elementwise product) are also Borel subsets of R. Now, +αZ and βW are independent +⇐⇒ P(αZ ∈ A, βW ∈ B) = P(αZ ∈ A) P(βW ∈ B), for all Borel subsets A, B +⇐⇒ P(Z ∈ 1 +αA, W ∈ 1 +βB) = P(Z ∈ 1 +αA) P(W ∈ 1 +βB), for all Borel subsets A, B +⇐⇒ P(Z ∈ A′, W ∈ B′) = P(Z ∈ A′) P(W ∈ B′), for all Borel subsets A′, B′ +⇐⇒ Z and W are independent +This completes the proof. +□ +Proof of Proposition 2.1: Let X and Y be independent. Then, by Lemma 2.1, ⟨l1, X⟩H +and ⟨l2, Y ⟩H are also independent, for all l1 and l2 ∈ H. In particular, the joint proba- +bility density function f⟨l1 , X⟩H,⟨l2 , Y ⟩H equals the product of marginal probability density +27 + +functions f⟨l1 , X⟩H and f⟨l2 , Y ⟩H. Hence, T(R1, R2) = 0. +Conversely, suppose that T(R1, R2) = 0. Then, for all l1 and l2 ∈ H with ∥l1∥H ≤ R1 +and ∥l2∥H ≤ R2, the joint probability density function f⟨l1 , X⟩H,⟨l2 , Y ⟩H equals the product +of marginal probability density functions f⟨l1 , X⟩H and f⟨l2 , Y ⟩H. Therefore, ⟨l1, X⟩H and +⟨l2, Y ⟩H are independent. +Now, consider ¯l1, ¯l2 ∈ H with ∥¯l1∥H > R1 and ∥¯l2∥H > R2. Since +��� +R1 +∥¯l1∥H¯l1 +��� +H = R1 and +��� +R2 +∥¯l2∥H¯l2 +��� +H = R2, independence of ⟨ +R1 +∥¯l1∥H¯l1, X⟩H and ⟨ +R2 +∥¯l2∥H¯l2, Y ⟩H, which follows from the +hypothesis. Finally, the independence of ⟨¯l1, X⟩H and ⟨¯l2, Y ⟩H follows from Lemma 6.1. +The argument is completed by applying Lemma 2.1. +□ +Proof of Proposition 2.2: Suppose that T(R1, R2) = 0. As argued in the proof of +Proposition 2.1, we have for all l1 and l2 ∈ H with ∥l1∥H ≤ R1 and ∥l2∥H ≤ R2, the real +valued random variables ⟨l1, X⟩H and ⟨l2, Y ⟩H are independent. +Let ¯l1 and ¯l2 ∈ H with ∥¯l1∥H = 1 and ∥¯l2∥H = 1. +Then, ∥R1¯l1∥H = R1 and +∥R2¯l2∥H = R2 and consequently, the independence of ⟨R1¯l1, X⟩H and ⟨R2¯l2, Y ⟩H follows +from assertion in Proposition 2.1. Then, the independence of ⟨¯l1, X⟩H and ⟨¯l2, Y ⟩H fol- +lows from Lemma 6.1. In particular, the joint probability density function f⟨¯l1 , X⟩H,⟨¯l2 , Y ⟩H +equals the product of marginal probability density functions f⟨¯l1 , X⟩H and f⟨¯l2 , Y ⟩H. Since +¯l1 and ¯l2 are arbitrary, we have T = 0. +Conversely, suppose that T = 0, which implies that the independence of ⟨l1, X⟩H +and ⟨l2, Y ⟩H with ∥l1∥H = 1 and ∥l2∥H = 1. +Hence, by Lemma 6.1, ⟨G1l1, X⟩H and +⟨G2l2, Y ⟩H are independent random variables, where G1 ≤ R1 and G2 ≤ R2. Then, for +all l1 and l2 ∈ H with ∥l1∥H ≤ R1 and ∥l2∥H ≤ R2, the joint probability density function +f⟨l1 , X⟩H,⟨l2 , Y ⟩H equals the product of marginal probability density functions f⟨l1 , X⟩H and +f⟨l2 , Y ⟩H. Hence, T(R1, R2) = 0, which completes the proof. +□ +Proof of Theorem 2.1: Note that Proposition 2.1 implies that T(R1, R1) = 0 if and +only if X and Y are independent random elements, and Proposition 2.2 implies that +T(R1, R2) = 0 ⇔ T = 0. Hence, both facts together imply that T = 0 if and only if X +and Y are independent random elements, which completes the proof. +□ +Proof of Proposition 2.3: For j = 1 and 2, we write l(M) +j += +M +� +i=1 +lj,iei. Since, lj = +∞ +� +i=1 +lj,iei +28 + +and ∥lj∥2 +H = +∞ +� +i=1 +l2 +j,i = 1, we have +lim +M→∞ +���lj − l(M) +j +��� +2 +H = lim +M→∞ +∞ +� +i=M+1 +l2 +j,i = 0, +(6.13a) +and +∥l(M) +j +∥H ↑ ∥lj∥H = 1, as M → ∞. +(6.13b) +Now +���lj − ˆlK +j +��� +H = +����� +� +lj − l(M) +j +� ++ +� +1 − +1 +∥l(M) +j +∥H +� +l(M) +j ++ +� +1 +∥l(M) +j +∥H +l(M) +j +− ˆlK +j +������ +H +≤ +���lj − l(M) +j +��� +H + +�����1 − +1 +∥l(M) +j +∥H +����� +���l(M) +j +��� +H + +����� +1 +∥l(M) +j +∥H +l(M) +j +− ˆlK +j +����� +H +≤ +���lj − l(M) +j +��� +H + +�����1 − +1 +∥l(M) +j +∥H +����� + +����� +1 +∥l(M) +j +∥H +l(M) +j +− ˆlK +j +����� +H +(6.14) +Note that +����� +1 +∥l(M) +j +∥H +l(M) +j +− ˆlK +j +����� +H += +����� +� +lj,1 +∥l(M) +j +∥H +, +lj,2 +∥l(M) +j +∥H +, · · · , +lj,M +∥l(M) +j +∥H +� +− (l(K) +j,1 , l(K) +j,2 , · · · , l(K) +j,M) +����� +RM +, +with the vector +� +lj,1 +∥l(M) +j +∥H, +lj,2 +∥l(M) +j +∥H, · · · , +lj,M +∥l(M) +j +∥H +� +in the unit sphere of RM. Therefore, by +choosing large K and appropriate ˆlK +j , the term +���� +1 +∥l(M) +j +∥Hl(M) +j +− ˆlK +j +���� +H +can be made arbitrarily +small. The proof then follows by using (6.13a) and (6.13b) in (6.14). +□ +The following Lemmas are useful in proving Theorem 2.2. +Lemma 6.2 Under (A1)–(A4) and (A6), +(I) := +� +nhn +� +1 +nh +3 +2n +� n +� +i=1 +k +�Z1,i − s +hn +, Z2,i − t +hn +�� +− fZ1,Z2(s, t) +� +converges weakly to a random variable associated with Gaussian distribution with mean +29 + += 0 and variance = fZ1,Z2(s, t) +� +k2(m1, m2)dm1dm2. +Proof of Lemma 6.2: Observe that +� +nhn +� +1 +nh +3 +2n +n +� +i=1 +k +�Z1,i − s +hn +, Z2,i − t +hn +� +− fZ1,Z2(s, t) +� += +� +nhn +� +1 +nh +3 +2n +n +� +i=1 +k +�Z1,i − s +hn +, Z2,i − t +hn +� +− E +� +1 +nh +3 +2n +n +� +i=1 +k +�Z1,i − s +hn +, Z2,i − t +hn +��� ++ +� +nhn +� +E +� +1 +nh +3 +2n +n +� +i=1 +k +�Z1,i − s +hn +, Z2,i − t +hn +�� +− fZ1,Z2(s, t) +� += +(I)A + (I)B, +where +(I)A := +� +nhn +� +1 +nh +3 +2n +n +� +i=1 +k +�Z1,i − s +hn +, Z2,i − t +hn +� +− E +� +1 +nh +3 +2n +n +� +i=1 +k +�Z1,i − s +hn +, Z2,i − t +hn +��� +, +(6.15) +and +(I)B := +� +nhn +� +E +� +1 +nh +3 +2n +n +� +i=1 +k +�Z1,i − s +hn +, Z2,i − t +hn +�� +− fZ1,Z2(s, t) +� +. +(6.16) +□ +Let us now work on (6.16). +� +nhn +� +E +� +1 +nh +3 +2n +n +� +i=1 +k +�Z1,i − s +hn +, Z2,i − t +hn +�� +− fZ1,Z2(s, t) +� += +� +nhn +� +E +� +1 +h +3 +2n +k +�Z1 − s +hn +, Z2 − t +hn +�� +− fZ1,Z2(s, t) +� += +� +nhn +1 +h +3 +2n +�� +k +�z1 − s +hn +, z2 − s +hn +� +fZ1,Z2(z1, z2)dz1dz2 − fZ1,Z2(s, t) +� += +� +nh2 +n +�� +k (m1, m2) fZ1,Z2(s + m1hn, t + m2hn)dm1dm2 − fZ1,Z2(s, t) +� +30 + += +� +nh2 +n +�� +k (m1, m2) +� +fZ1,Z2(s, t) + hn +2 +� +i=1 +�∂fZ1,Z2(ξn) +∂Zi +� +mi +� +dm1dm2 − fZ1,Z2(s, t) +� += +√nhnhn +� +2 +� +i=1 +�∂fZ1,Z2(ξn) +∂Zi +� � +mik(m1, m2)dm1dm2 +� +. +The last fact follows from (A6), i.e. +� +k(m1, m2)dm1dm2 = 1. Now, using (A1), (A2) and +(A6) (i.e. hn → 0 as n → ∞, √nhn → c as n → ∞, the partial derivatives of fZ1,Z2(., .) +are uniformly bounded and +� +mik(m1, m2)dm1dm2 < ∞), we have +(I)B → 0 as n → ∞. +(6.17) +Now, we work on (I)A (see (6.15)) and denote +Wn,i = +� +nhn +� +1 +nh +3 +2n +k +�Z1,i − s +hn +, Z2,i − t +hn +� +− E +� +1 +nh +3 +2n +k +�Z1 − s +hn +, Z2 − t +hn +��� +, i = 1, . . . , n +Note that E(Wn,i) = 0 for all i = 1, . . . , n. Observe that +W 2 +n,i = +1 +nh2 +n +� +k2 +�Z1,i − s +hn +, Z2,i − t +hn +� ++ +� +E +� +k +�Z1 − s +hn +, Z2 − t +hn +���2 +−2k +�Z1 − s +hn +, Z2 − t +hn +� +E +� +k +�Z1 − s +hn +, Z2 − t +hn +��� +⇒ E(W 2 +n,i) = +1 +nh2 +n +� +E +� +k2 +�Z1 − s +hn +, Z2 − t +hn +�� +− +� +E +� +k +�Z1 − s +hn +, Z2 − t +hn +���2� += +1 +nh2 +n +�� � +k2 +�z1 − s +hn +, z2 − t +hn +�� +fZ1,Z2(z1, z2)dz1dz2 +− +� � +k +�z1 − s +hn +, z2 − t +hn +� +fZ1,Z2(z1, z2)dz1dz2 +�2� += +1 +n +�� +k2(m1, m2)fZ1,Z2(s + m1hn, t + m2hn) +� +−h2 +n +n +� +{k(m1, m2)fZ1,Z2(s + m1hn, t + m2hn)}2 dm1dm2 +⇒ +n +� +i=1 +E(W 2 +n,i) = +�� +k2(m1, m2)fZ1,Z2(s + m1hn, t + m2hn) +� +31 + +−h2 +n +� +{k(m1, m2)fZ1,Z2(s + m1hn, t + m2hn)}2 dm1dm2. +Hence, using (A1), (A2) and (A6) (i.e., hn → 0 as n → ∞, partial derivatives of the joint +density function of (Z1, Z2) exist, and +� +k2(m1, m2)dm1dm2 < ∞, +� +m1k(m1, m2)dm1dm2 < +∞ and +� +m2k(m1, m2)dm1dm2 < ∞), we have +n +� +i=1 +E(W 2 +n,i) → fZ1,Z2(s, t) +� +k2(m1, m2)dm1dm2 := σ2 +1 as n → ∞. +(6.18) +Further, it follows from the expression of Wn,i and (I)A (see (6.15)) that (I)A = +n� +i=1 +Wn,i. +Therefore, in order to prove the asymptotic normality of (I)A, the validation of Lyapunov +condition (see Serfling (1980)) is established here. For some δ > 0, let us consider +0 ≤ +1 +σ2+δ +1 +n +� +i=1 +E|Wn,i|2+δ +≤ +1 +σ2+δ +1 +n +� +i=1 +E +���� +1 +√nhn +k +�Z1 − s +hn +, Z2 − t +hn +����� +2+δ +≤ +1 +σ2+δ +1 +× +1 +n +δ +2h2+δ +n +E +����k +�Z1 − s +hn +, Z2 − t +hn +����� +2+δ +≤ +1 +σ2+δ +1 +× +1 +n +δ +2h2+δ +n +����E +� +k +�Z1 − s +hn +, Z2 − t +hn +������ +2+δ += +1 +σ2+δ +1 +× +1 +n +δ +2h2+δ +n +���� +� +k +�z1 − s +hn +, z2 − t +hn +� +fZ1,Z2(z1, z2)(z1, z2)dz1dz2 +���� +2+δ += +1 +σ2+δ +1 +× h4+2δ +n +n +δ +2h2+δ +n +���� +� +k (m1, m2) fZ1,Z2(s + m1hn, t + m2hn)dm1dm2 +���� +2+δ += +1 +σ2+δ +1 +× h2+δ +n +n +δ +2 +���� +� +k (m1, m2) fZ1,Z2(s + m1hn, t + m2hn)dm1dm2 +���� +2+δ +→ 0 as n → ∞. +The last fact follows from the fact that hn → 0 as n → ∞, and in view of the existence of the +partial derivatives of the joint density function of (Z1, Z2), and +� +k(m1, m2)dm1dm2 < ∞, +� +m1k(m1, m2)dm1dm2 < ∞ and +� +m2k(m1, m2)dm1dm. Hence, (I)A converges weakly to +a random variable associated with Gaussian distribution with mean = 0 and variance = +fZ1,Z2(s, t) +� +k2(m1, m2)dm1dm2. Further, recall that (I) = (I)A+(I)B, and it is established +that (I)B → 0 as n → ∞ (see (6.17)), and consequently, (I) converges weakly to a +random variable associated with Gaussian distribution with mean = 0 and variance = +32 + +fZ1,Z2(s, t) +� +k2(m1, m2)dm1dm2. +□ +Lemma 6.3 Under (A1)–(A5), +(II) := +� +nhn +� +1 +nhn +n +� +i=1 +k1 +�Z1,i − s +hn +� � +1 +nhn +n +� +j=1 +k2 +�Z2,j − t +hn +� +− fZ2(t) +�� +converges weakly to a Gaussian distribution with mean = √cfZ1(s)f +′ +Z2(t) +� +mk2(m)dm and +variance = {fZ1(s)}2fZ2(t) +� +{k2(m)}2dm, where c = lim +n→∞ nh2 +n. +Proof of Lemma 6.3: Note that (II) = (II)A × (II)B, where +(II)A := +1 +nhn +n +� +i=1 +k1 +�Z1,i − s +hn +� +, +(6.19) +and +(II)B := +� +nhn +�� +1 +nhn +n +� +j=1 +k2 +�Z2,j − t +hn +� +− fZ2(t) +�� +. +(6.20) +It now follows from (6.19) that +E((II)A) += +E +� +1 +nhn +n +� +i=1 +k1 +�Z1,i − s +hn +�� += +1 +nhn +n +� +i=1 +E +� +k1 +�Z1,i − s +hn +�� += 1 +hn +E +� +k1 +�Z1 − s +hn +�� += +1 +hn +� � +k1 +�Z1 − s +hn +�� +fZ1(z1)dz1 += +� +k(m)fZ1(mhn + s)dm = +� +k(m) +� +fZ1(s) + mhnfZ′ +1(ξn) +� +dm +(f +′ +Z1 denotes the derivative of fZ1, and ξn ∈ (s, s + mhn)) += +fZ1(s) +� +k(m)dm + hnf +′ +Z1(ξn) +� +mk(m)dm. +Now, using conditions (A1), (A4) and (A5) (i.e., +� +mk(m)dm = 1, hn → 0 as n → ∞, +f +′ +Z1(.) is uniformly bounded, and +� +mk(m)dm < ∞), we have +E((II)A) → fZ1(s) as n → ∞. +(6.21) +33 + +Next, note that +0 ≤ var +� +1 +nhn +n +� +i=1 +k1 +�Z1,i − s +hn +�� += +1 +n2h2 +n +var +� n +� +i=1 +k1 +�Z1,i − s +hn +�� ++ +1 +n2h2 +n +n +� +i̸=j=1 +cov +� +k1 +�Z1,i − s +hn +� +, k1 +�Z1,j − s +hn +�� += +1 +n2h2 +n +var +� n +� +i=1 +k1 +�Z1,i − s +hn +�� +(since Z1,i ⊥ Z1,j for i ̸= j) += +1 +nh2 +n +E +� +k2 +1 +�Z1 − s +hn +�� +− +1 +nh2 +n +� +E +� +k2 +1 +�Z1 − s +hn +���2 +≤ +1 +nh2 +n +E +� +k2 +1 +�Z1 − s +hn +�� += +1 +nh2 +n +� +k2 +1 +�Z1 − s +hn +� +fZ1(z1)dz1 = +1 +nhn +� +k2 +1(m)fZ1(s + mhn)dm += +1 +nhn +� +k2 +1(m) +� +fZ1(s) + mhnf +′ +Z1(ξn) +� +dm (here ξn ∈ (s, s + mhn)) += +fZ1(s) +nhn +� +k2 +1(m)dm + f +′ +Z1(ξn) +n +� +mk2 +1(m)dm +. +Now using (A1), (A3), (A4) and (A5) (i.e., nhn → ∞ as n → ∞, fZ1(.) and f +′ +Z1(.) are +uniformly bounded, +� +k2 +1(m)dm < ∞ and +� +mk2 +1(m)dm < ∞), we have +var +� +1 +nhn +n +� +i=1 +k1 +�Z1,i − s +hn +�� +→ 0 as n → ∞. +(6.22) +Hence, using (6.21) and (6.22), we have +(II)A +p→ fZ1(s) as n → ∞. +(6.23) +Now, let us work on (II)B. (6.19) indicates that +(II)B : += +� +nhn +�� +1 +nhn +n +� +j=1 +k2 +�Z2,j − t +hn +� +− fZ2(t) +�� +34 + += +� +nhn +�� +1 +nhn +n +� +j=1 +k2 +�Z2,j − t +hn +� +− E +� +1 +nhn +n +� +j=1 +k2 +�Z2,j − t +hn +���� ++ +� +nhn +� +E +� +1 +nhn +n +� +j=1 +k2 +�Z2,j − t +hn +�� +− fZ2(t) +� += +(II)B,1 + (II)B,2, +where +(II)B,1 := +� +nhn +�� +1 +nhn +n +� +j=1 +k2 +�Z2,j − t +hn +� +− E +� +1 +nhn +n +� +j=1 +k2 +�Z2,j − t +hn +���� +(6.24) +and +(II)B,2 := +� +nhn +� +E +� +1 +nhn +n +� +j=1 +k2 +�Z2,j − t +hn +�� +− fZ2(t) +� +. +(6.25) +It now follows from (6.25) that +(II)B,2 : += +� +nhn +� +E +� +1 +nhn +n +� +j=1 +k2 +�Z2,j − t +hn +�� +− fZ2(t) +� += +� +nhn +� +E +� 1 +hn +k2 +�Z2 − t +hn +�� +− fZ2(t) +� += +� +nhn +� 1 +hn +� +k2 +�z2 − t +hn +� +fZ2(z2)dz2 − fZ2(t) +� += +� +nhn +�� +k2(m)fZ2(t + mhn)dm − fZ2(t) +� += +� +nhn +�� +k2(m) +� +fZ2(t) + mhnf +′ +Z2(t) + m2h2 +n +2 +f +′′ +Z2(ξn) +� +− fZ2(t) +� +(f +′′ +Z2 denotes the second derivative of fZ2), and ξn ∈ (t, t + mhn). +Hence, in view of (A1), (A4) and (A5) (i.e., hn → 0 as n → ∞, nh2 +n = O(1), +� +k2(m)dm = +1, +� +mk2(m)dm < ∞, f +′ +Z2 and f +′′ +Z2 are uniformly bounded), we have +(II)B,2 − √nhnf +′ +Z2(t) +� +mk2(m)dm → 0 as n → ∞. +(6.26) +35 + +Now, to study (II)B,1 (see (6.24)), let us denote +Vn,j = +1 +√nhn +� +k2 +�Z2,j − t +hn +� +− E +� +k2 +�Z2 − t +hn +��� +, j = 1, . . . , n. +Now, observe that E(Vn,j) = 0 for j = 1, . . . , n, and +V 2 +n,j = +1 +nhn +� +k2 +2 +�Z2,j − t +hn +� ++ +� +E +� +k2 +�Z2 − t +hn +���2 +− +2 +nhn +k2 +�Z2,j − t +hn +� +E +� +k2 +�Z2,j − t +hn +��� +⇒ E(V 2 +n,j) = +1 +nhn +E +� +k2 +2 +�Z2,j − t +hn +�� +− +1 +nhn +� +E +� +k2 +�Z2 − t +hn +���2 += +1 +nhn +� +k2 +2 +�z2 − t +hn +� +fZ2(z2)dz2 − +1 +nhn +�� +k2 +�z2 − t +hn +� +fZ2(z2)dz2 +�2 += +1 +nhn +��� +k2 +2(m)fZ2(t + mhn)dm +� +hn − +�� +k2(m)hnfZ2(t + mhn)dm +�2� += +� +k2 +2(m)fZ2(t + mhn)dm − hn +n +�� +k2(m)hnfZ2(t + mhn)dm +�2 +⇒ +n +� +j=1 +E(V 2 +n,j) = +� +k2 +2(m)fZ2(t + mhn)dm − hn +�� +k2(m)hnfZ2(t + mhn)dm +�2 += +� +k2 +2(m) +� +fZ2(t) + mhnf +′ +Z2(ξn) +� +dm − hn +�� +k2(m)hn +� +fZ2(t) + mhnf +′ +Z2(ξn) +� +dm +�2 +(here ξn ∈ (t, t + mhn)) +Hence, using (A1), (A3) and (A4), (i.e., hn → 0 as n → ∞, f +′ +Z2 is uniformly bounded, +� +k2(m)dm < ∞ and +� +mk2(m)dm < ∞), we have +n +� +j=1 +E(V 2 +n,j) → fZ2(t) +� +{k2(m)}2dm as n → ∞. +(6.27) +Let us now denote σ2 := fZ2(t) +� +{k2(m)}2 (see (6.27)). In order to now establish the +36 + +asymptotic normality of (II)B,1, we now consider for some δ > 0, +1 +σ2+δ +n +� +j=1 +E|Vn,j|2+δ +≤ +1 +σ2+δ E +���� +1 +√nhn +k2 +�Z2 − t +hn +����� +2+δ +≤ +1 +σ2+δ +���� +1 +√nhn +E +� +k2 +�Z2 − t +hn +������ +2+δ += +1 +σ2+δ(nhn)1+ δ +2 +���� +� +k2 +�z2 − t +hn +� +fZ2(z2)dz2 +���� +2+δ += +1 +σ2+δ(nhn)1+ δ +2 +���� +� +k2(m)fZ2(t + mhn)hndm +���� +2+δ += +h2+δ +n +σ2+δ(nhn)1+ δ +2 +���� +� +k2(m) +� +fZ2(t) + mhnf +′ +Z2(ξn) +� +dm +���� +2+δ +Hence, using (A1), (A4) and (A5) (i.e., hn → 0 as n → ∞, nhn → ∞ as n → ∞, +� +k2(m)dm < ∞ and +� +mk2(m)dm < ∞), we have +1 +σ2+δ +n +� +j=1 +E|Vn,j|2+δ → 0 as n → ∞. +(6.28) +Therefore, it follows from Lyapunov CLT (see Serfling (1980)) that (II)B,1 converges +weakly to a random variable associate with Gaussian distribution with mean = 0 and +variance = σ2 := fZ2(t) +� +{k2(m)}2dm. +As (II)B = (II)B,1 + (II)B,2, the aforementioned fact along with in view of (6.26), +one can conclude that (II)B − √nhnf +′ +Z2(t) +� +mk2(m)dm converges weakly to a random +variable associated with Gaussian distribution with mean = 0 and variance = σ2. Finally, +using (6.23) and aforesaid weak convergence of (II)B along with the fact that (II) = +(II)A × (II)B, one can conclude that (II) converges weakly to a Gaussian distribution +with mean = √cfZ1(s)f +′ +Z2(t) +� +mk2(m)dm and variance = {fZ1(s)}2fZ2(t) +� +{k2(m)}2dm. +□ +Lemma 6.4 Under (A1)–(A5), (III) := √nhn +� +fZ2(t) +� +1 +nhn +n� +i=1 +k1 +� +Z1,i−s +hn +� +− fZ1(s) +�� +converges weakly to a Gaussian distribution with mean = √cfZ2(t)f +′ +Z1(s) +� +mk1(m)dm and +variance = {fZ2(t)}2fZ1(s) +� +{k1(m)}2dm, where c = lim +n→∞ nh2 +n. +37 + +Proof of Lemma 6.4: The proof of this lemma follows from the same arguments provided +in the proof of Lemma 6.3. +□ +Lemma 6.5 Let us denote +Mn(s, t) = +1 +nh +3 +2n +� n +� +i=1 +k +�Z1,i − s +hn +, Z2,i − t +hn +�� +− +� +1 +n2h2 +n +n +� +i,j=1 +k1 +�Z1,i − s +hn +� +k2 +�Z2,j − t +hn +�� +, +and +M(s, t) = fZ1,Z2(s, t) − fZ1(s)fZ2(t), +where fZ1,Z2, fZ1 and fZ2 are probability density functions of (Z1, Z2), Z1 and Z2, re- +spectively. +Then, for a fixed (s, t), under (A1)–(A6), √nhn(Mn(s, t) − M(s, t)) con- +verges weakly to a random variable associated with Gaussian distribution with mean = +−√c +� +fZ1(s)f +′ +Z2(t) +� +mk2(m)dm + fZ2(t)f +′ +Z1(s) +� +mk1(m)dm +� +and variance = fZ1,Z2(s, t) +� +{k(m1, m2)}2dm1dm2 ++ {fZ1(s)}2fZ2(t) +� +{k2(m)}2dm + {fZ2(t)}2fZ1(s) +� +{k1(m)}2dm, where c = lim +n→∞ nh2 +n. +Proof of Lemma 6.5: Observe that +� +nhn(Mn(s, t) − M(s, t)) += +� +nhn +� +1 +nh +3 +2n +� n +� +i=1 +k +�Z1,i − s +hn +, Z2,i − t +hn +�� +− +� +1 +n2h2 +n +n +� +i,j=1 +k1 +�Z1,i − s +hn +� +k2 +�Z2,j − t +hn +��� +− +� +nhn (fZ1,Z2(s, t) − fZ1(s)fZ2(t)) += +� +nhn +� +1 +nh +3 +2n +� n +� +i=1 +k +�Z1,i − s +hn +, Z2,i − t +hn +�� +− fZ1,Z2(s, t) +� +− +� +nhn +� +1 +n2h2 +n +n +� +i,j=1 +k1 +�Z1,i − s +hn +� +k2 +�Z2,j − t +hn +� +− fZ1(s)fZ2(t) +� += +� +nhn +� +1 +nh +3 +2n +� n +� +i=1 +k +�Z1,i − s +hn +, Z2,i − t +hn +�� +− fZ1,Z2(s, t) +� +− +� +nhn +� +1 +nhn +n +� +i=1 +k1 +�Z1,i − s +hn +� � +1 +nhn +n +� +j=1 +k2 +�Z2,j − t +hn +� +− fZ2(t) +�� +38 + +− +� +nhn +� +fZ2(t) +� +1 +nhn +n +� +i=1 +k2 +�Z1,i − s +hn +� +− fZ1(s) +�� += +(I) − (II) − (III), +where +(I) := +� +nhn +� +1 +nh +3 +2n +� n +� +i=1 +k +�Z1,i − s +hn +, Z2,i − t +hn +�� +− fZ1,Z2(s, t) +� +, +(6.29) +(II) := +� +nhn +� +1 +nhn +n +� +i=1 +k1 +�Z1,i − s +hn +� � +1 +nhn +n +� +j=1 +k2 +�Z2,j − t +hn +� +− fZ2(t) +�� +, +(6.30) +and +(III) := +� +nhn +� +fZ2(t) +� +1 +nhn +n +� +i=1 +k2 +�Z1,i − s +hn +� +− fZ1(s) +�� +. +(6.31) +It follows from the assertions in Lemma 6.2, 6.3 and 6.4 that (I), (II) and (III) con- +verge weakly to a certain Gaussian random variables. In order to establish the asymptotic +normality of (I) − (II) − (III), let us derive the asymptotic distribution of a1 × (I) + a2 × +(II) + a3 × (III), where ai ∈ R (i = 1, 2 and 3) are arbitrary constants. +Consider +a1 × (I) + a2 × (II) + a3 × (III) += +a1 × {(I)A + (I)B} + a2[(II)A × {(II)B,1 + (II)B,2}] + a3 × {(III)A + (III)B} += +{a1 × (I)A + a2 × (II)A × (II)B,1 + a3 × (III)A} ++ +{a1 × (I)B + a2 × (II)A × (II)B,2 + a3 × (III)B} +:= +C + D, +where C := {a1 × (I)A + a2 × (II)A × (II)B,1 + a3 × (III)A}, D := {a1 × (I)B + a2 × +(II)A × (II)B,2 + a3 × (III)B}, (I)A is defined in (6.15), (I)B is defined as (6.16), (II)A is +defined as (6.19), (II)B,1 is defined as (6.24), (II)B,2 is defined as (6.25), +(III)A = +� +nhn +� +fZ2(t) +� +1 +nhn +n +� +i=1 +k1 +�Z1,i − s +hn +� +− E +� +1 +nhn +n +� +i=1 +k1 +�Z1,i − s +hn +���� +, +39 + +and +(III)B = +� +nhn +� +fZ2(t) +� +E +� +1 +nhn +n +� +i=1 +k2 +�Z1,i − s +hn +�� +− fZ1(s) +�� +. +Let us first consider D. As arguing for (II)B in the proof of Lemma 6.3, (III)B → +√cfZ2(t)f +′ +Z1(s) +� +mk1(m)dm as n → ∞. Along with the facts in (6.17), (6.23) , (6.26), one +can conclude that +D +p→ √c +� +a2fZ1(s)f +′ +Z2(t) +� +mk2(m)dm + a3fZ2(t)f +′ +Z1(s) +� +mk1(m)dm +� +as n → ∞. +(6.32) +Now, observe that +C = a1 +� +nhn +� +1 +nh +3 +2n +n +� +i=1 +k +�Z1,i − s +hn +, Z2,i − t +hn +� +− E +� +1 +nh +3 +2n +n +� +i=1 +k +�Z1,i − s +hn +, Z2,i − t +hn +��� ++ +a2 +� +nhn +� +1 +nhn +n +� +i=1 +k1 +�Z1,i − s +hn +�� �� +1 +nhn +n +� +j=1 +k2 +�Z2,j − t +hn +� +− E +� +1 +nhn +n +� +j=1 +k2 +�Z2,j − t +hn +���� ++ +a3 +� +nhn +� +fZ2(t) +� +1 +nhn +n +� +i=1 +k1 +�Z1,i − s +hn +� +− E +� +1 +nhn +n +� +i=1 +k1 +�Z1,i − s +hn +���� +. +Let us now denote +Ln,i = a1 +� +nhn +� +1 +nh +3 +2n +k +�Z1,i − s +hn +, Z2,i − t +hn +� +− E +� +1 +nh +3 +2n +k +�Z1,i − s +hn +, Z2,i − t +hn +��� ++ +a2 +� +nhn +� 1 +nhn +k1 +�Z1,i − s +hn +�� �� 1 +nhn +k2 +�Z2,j − t +hn +� +− E +� 1 +nhn +k2 +�Z2,j − t +hn +���� ++ +a3 +� +nhn +� +fZ2(t) +� 1 +nhn +k1 +�Z1,i − s +hn +� +− E +� 1 +nhn +k1 +�Z1,i − s +hn +���� +. +Note that C = +n� +i=1 +Ln,i, and E(Ln,i) = 0 for all i = 1, . . . , n. Further note that +n +� +i=1 +E(L2 +n,i) → a2 +1fZ1,Z2(s, t) +� +k2(m1, m2)dm1dm2 ++ +a2 +2{fZ1(s)}2fZ2(t) +� +{k2(m)}2dm + a2 +3{fZ2(t)}2fZ1(s) +� +{k1(m)}2dm := σ2 +2 +40 + +as n → ∞. +(6.33) +The last fact follows from the assertions in Lemmas 6.2, 6.3 and 6.4, and in view of the +independence between the random variables Z1 and Z2, and Covariance ((I)A, (II)A × +(II)B,2) → 0 as n → ∞ and Covariance ((I)A, (II)A × (III)A) → 0 as n → ∞. +Now, in order to show the asymptotic normality of C = +n� +i=1 +Ln,i, for some δ > 0, we +consider +1 +σ2+δ +2 +n +� +i=1 +E|Ln,i|2+δ += +1 +σ2+δ +2 +n +� +i=1 +E|a1Wn,i + a2 +1 +nhn +k1 +�Z1,i − s +hn +� +Vn,i + a3Bn,i|2+δ +≤ +1 +σ2+δ +2 +� +|a1|2+δ +n +� +i=1 +E|Wn,i|2+δ + |a2|2+δ +n +� +i=1 +fZ1(s)E|Vn,i|2+δ + |a3|2+δ +n +� +i=1 +E|Bn,i|2+δ +� +→ 0 as n → ∞, +where Wn,i is defined in the proof of Lemma 6.2, Vn,i is defined in the proof of Lemma 6.3, +and +Bn,i = +� +nhn +� +fZ2(t) +� 1 +nhn +k1 +�Z1,i − s +hn +� +− E +� 1 +nhn +k1 +�Z1,i − s +hn +���� +. +The last limiting fact follows from the fact that +n� +i=1 +E|Wn,i|2+δ → 0 as n → ∞ (see the +proof of Lemma 6.2), +n� +i=1 +E|Vn,i|2+δ → 0 as n → ∞ (see the proof of Lemma 6.3), and +n� +i=1 +E|Bn,i|2+δ → 0 as n → ∞ (exactly the same proof as it is for Vn,i in Lemma 6.3). +Therefore, C converges weakly to a random variable associated with Gaussian dis- +tribution with mean = 0 and variance = σ2 +2 (see (6.33) for the explicit expression). +Next, since a1 × (I) + a2 × (II) + a3 × (III) = C + D, a1 × (I) + a2 × (II) + a3 × +(III) converges weakly to a random variable associate with Gaussian distribution with +mean = √c +� +a2fZ1(s)f +′ +Z2(t) +� +mk2(m)dm + a3fZ2(t)f +′ +Z1(s) +� +mk1(m)dm +� +and variance = σ2 +2 +in view of (6.32) using an application of Slutsky’s theorem (see Serfling (1980)). +Fi- +nally, as a1, a2 and a3 are arbitrary constants, using Cramer-Wold device (see Serfling +(1980)), one can conclude that ((I), (II), (III)) converges weakly to a certain trivari- +41 + +ate Gaussian distribution, and hence, √nhn(Mn(s, t) − M(s, t)) = (I) − (II) − (III) +converges weakly to a random variable associated with Gaussian distribution with mean += −√c +� +fZ1(s)f +′ +Z2(t) +� +mk2(m)dm + fZ2(t)f +′ +Z1(s) +� +mk1(m)dm +� +and +variance = fZ1,Z2(s, t) +� +{k(m1, m2)}2dm1dm2 ++ {fZ1(s)}2fZ2(t) +� +{k2(m)}2dm + {fZ2(t)}2fZ1(s) +� +{k1(m)}2dm. +□ +Lemma 6.6 Under (A1), (A5) and (A6), √nhn +� +lim +G→∞ lim +L→∞(Tn − T G,L +n +) +� +p→ 0 as n → ∞. +Proof of Lemma 6.6: Recall Tn from (2.7) : +Tn += +sup +θj,i∈(− π +2 , π +2 ) +j=1,2;i=1,...,M−2 +θj,M−1∈(−π,π) +j=1,2 +s∈R,t∈R +����� +1 +nh +3 +2n +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +� +− +1 +n2h2 +n +n +� +i,j=1 +k1 +� +⟨ˆlK +1 , Xi⟩H − s +hn +� +k2 +� +⟨ˆlK +2 , Yj⟩H − t +hn +������ , +and recall from (3.12) that +T G,L +n += +sup +θj,i∈(− π +2 , π +2 ) +j=1,2;i=1,...,M−1 +θj,M−2∈(−π,π) +j=1,2 +s∈{s1,...,sL} +t∈{t1,...,tL} +����� +1 +nh +3 +2n +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +� +− +1 +n2h2 +n +n +� +i,j=1 +k1 +� +⟨ˆlK +1 , Xi⟩H − s +hn +� +k2 +� +⟨ˆlK +2 , Yj⟩H − t +hn +������ . +Note that it follows from the assertion in Proposition 3.1 that +sup +s∈{s1,...,sL} +t∈{t1,...,tL} +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +� +−sup +s∈R +t∈R +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +� +→ 0 +almost surely as G → ∞ and L → ∞ (iteratively) for any n ≥ 1, where si ∈ [−G, G] and +42 + +ti ∈ [−G, G] for i = 1, . . . , L. This implies that +E +� +�� +sup +s∈{s1,...,sL} +t∈{t1,...,tL} +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +� +− sup +s∈R +t∈R +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +�� +�� → 0 +(6.34) +for any n ≥ 1 as G → ∞ and L → ∞ (iteratively), which follows from dominated +convergence theorem (see, e.g., Serfling (1980)) because +� +�� +sup +s∈{s1,...,sL} +t∈{t1,...,tL} +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +� +− sup +s∈R +t∈R +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +�� +�� +is bounded for all G and L and any n ≥ 1. Hence, +E +� +�� +sup +s∈{s1,...,sL} +t∈{t1,...,tL} +√nhn +nh +3 +2n +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +� +− sup +s∈R +t∈R +√nhn +nh +3 +2n +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +�� +� → 0 +(6.35) +as n → ∞ (along with G → ∞ and L → ∞, iteratively) in view of (A1) (i.e., √nhn → √c +as n → ∞). +Arguing exactly in a similar way, one can conclude that +E +� +�� +sup +s∈{s1,...,sL} +t∈{t1,...,tL} +√nhn +nh +3 +2n +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +� +− sup +s∈R +t∈R +√nhn +nh +3 +2n +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +�� +� +2 +→ 0 +(6.36) +as n → ∞ (along with G → ∞ and L → ∞ (iteratively)). +43 + +Hence, using (6.35) and (6.36), we have +� +� +� +� +� +sup +s∈{s1,...,sL} +t∈{t1,...,tL} +√nhn +nh +3 +2n +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +� +− sup +s∈R +t∈R +√nhn +nh +3 +2n +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +�� +� +� +p→ 0 +(6.37) +as n → ∞ (along with G → ∞ and L → ∞ (iteratively)). +Similarly, one can show that +√nhn +n2h2 +n +� +� +� +� +� +sup +s∈{s1,...,sL} +t∈{t1,...,tL} +n +� +i=1 +k1 +� +⟨ˆlK +1 , Xi⟩H − s +hn +� +k2 +� +⟨ˆlK +2 , Yi⟩H − t +hn +� +− sup +s∈R +t∈R +n +� +i=1 +k1 +� +⟨ˆlK +1 , Xi⟩H − s +hn +� +k2 +� +⟨ˆlK +2 , Yi⟩H − t +hn +�� +� +� +p→ 0. +(6.38) +In view of (6.37) and (6.38), we have √nhn +� +lim +G→∞ lim +L→∞(Tn − T G,L +n +) +� +p→ 0 as n → ∞, which +completes the proof. +□ +Lemma 6.7 For distinct pairs (s1, t1), · · · , (sm, tm), consider the random vector +� +nhn +� +� +� +� +� +� +Mn(s1, t1) − M(s1, t1) +Mn(s2, t2) − M(s2, t2) +· · · +Mn(sm, tm) − M(sm, tm) +� +� +� +� +� +� +where Mn(·, ·) and M(·, ·) are the same as in Lemma 6.5. Then, under (A1)–(A6), the +sequence of random vectors converge in distribution to an m-dimensional multivariate Nor- +mal distribution with independent components, where the l-th component follows a univari- +ate Normal distribution with mean += −√c +� +fZ1(sl)f +′ +Z2(tl) +� +mk2(m)dm + fZ2(tl)f +′ +Z1(sl) +� +mk1(m)dm +� +44 + +and variance += fZ1,Z2(sl, tl) +� +k2(m1, m2)dm1dm2+{fZ1(sl)}2fZ2(tl) +� +{k2(m)}2+fZ1(sl){fZ2(tl)}2 +� +{k1(m)}2dm +for l = 1, 2, · · · , m, where c = lim +n→∞ nh2 +n. +Proof of Lemma 6.7: We proceed as in Lemma 6.5. Here, we fix α1, · · · .αm ∈ R and +look at the convergence in distribution of √nhn +m +� +l=1 +αl(Mn(sl, tl) − M(sl, tl)) as n goes to +∞. Observe that +� +nhn +m +� +l=1 +αl(Mn(sl, tl) − M(sl, tl)) = +m +� +l=1 +αl.(I)sl,tl − +m +� +l=1 +αl.(II)sl,tl − +m +� +l=1 +αl.(III)sl,tl, +(6.39) +where +(I)sl,tl = +� +nhn +� +1 +nh +3 +2n +� n +� +i=1 +k +�Z1,i − sl +hn +, Z2,i − tl +hn +�� +− fZ1,Z2(sl, tl) +� +, +(II)sl,tl = +� +nhn +� +1 +nhn +n +� +i=1 +k1 +�Z1,i − sl +hn +� � +1 +nhn +n +� +j=1 +k2 +�Z2,j − tl +hn +� +− fZ2(tl) +�� +and +(III)sl,tl = +� +nhn +� +fZ2(tl) +� +1 +nhn +n +� +i=1 +k2 +�Z1,i − sl +hn +� +− fZ1(sl) +�� +for l = 1, · · · , m. +We look at the joint distribution of (�m +l=1 αl.(I)sl,tl, �m +l=1 αl.(II)sl,tl, �m +l=1 αl.(III)sl,tl). +Fix real scalars a1, a2 and a3 and, consider +a1 +m +� +l=1 +αl.(I)sl,tl + a2 +m +� +l=1 +αl.(II)sl,tl + a3 +m +� +l=1 +αl.(III)sl,tl = ˜Cn + ˜Dn +45 + +with +˜Cn := a1 +n +� +i=1 +� m +� +l=1 +αlWn,i,sl,tl +� ++ a2 +n +� +i=1 +� m +� +l=1 +αl.(II)sl,tl,A Vn,i,sl,tl +� ++ a3 +n +� +i=1 +� m +� +l=1 +αlBn,i,sl,tl +� +(6.40) +and +˜Dn := a1 +m +� +l=1 +αl.(I)sl,tl,B + a2 +m +� +l=1 +αl.(II)sl,tl,A × (II)sl,tl,B,2 + a3 +m +� +l=1 +αl.(III)sl,tl,B, (6.41) +where +(I)sl,tl,B = +� +nhn +� +E +� +1 +nh +3 +2n +n +� +i=1 +k +�Z1,i − sl +hn +, Z2,i − tl +hn +�� +− fZ1,Z2(sl, tl) +� +, +(II)sl,tl,A = +1 +nhn +n +� +i=1 +k1 +�Z1,i − sl +hn +� +, +(II)sl,tl,B,2 = +� +nhn +� +E +� +1 +nhn +n +� +j=1 +k2 +�Z2,j − tl +hn +�� +− fZ2(tl) +� +, +(III)sl,tl,B = +� +nhn +� +fZ2(tl) +� +E +� +1 +nhn +n +� +i=1 +k2 +�Z1,i − sl +hn +�� +− fZ1(sl) +�� +, +Wn,i,sl,tl = +� +nhn +� +1 +nh +3 +2n +k +�Z1,i − sl +hn +, Z2,i − tl +hn +� +− E +� +1 +nh +3 +2n +k +�Z1 − sl +hn +, Z2 − tl +hn +��� +, +Vn,i,sl,tl = +1 +√nhn +� +k2 +�Z2,i − tl +hn +� +− E +� +k2 +�Z2 − tl +hn +��� +Bn,i,sl,tl = +� +nhn +� +fZ2(tl) +� 1 +nhn +k1 +�Z1,i − sl +hn +� +− E +� 1 +nhn +k1 +�Z1,i − sl +hn +���� +for l = 1, · · · , m and i = 1, · · · , n. +As argued in Lemma 6.5, we have +˜Dn +p +−−−→ +n→∞ +√c +� +a2 +m +� +l=1 +αlfZ1(sl)f +′ +Z2(tl) +� +mk2(m)dm + a3 +m +� +l=1 +αlfZ2(tl)f +′ +Z1(sl) +� +mk1(m)dm +� +. +46 + +Now, we consider the term ˜Cn. We have E ˜Cn = 0 and +lim +n E +� +˜Cn +�2 += lim +n V ar +� +˜Cn +� += a2 +1 +� m +� +l=1 +α2 +l fZ1,Z2(sl, tl) +� � +k2(m1, m2)dm1dm2 ++ a2 +2 +� m +� +l=1 +α2 +l {fZ1(sl)}2fZ2(tl) +� � +{k2(m)}2dm ++ a2 +3 +� m +� +l=1 +α2 +l fZ1(sl){fZ2(tl)}2 +� � +{k1(m)}2dm +We establish the asymptotic normality of ˜Cn. For some δ > 0, repeatedly applying +Jensen’s inequality, we observe that +n +� +i=1 +E +����� +� +a1 +m +� +l=1 +αlWn,i,sl,tl +� ++ +� +a2 +m +� +l=1 +αl.(II)sl,tl,A Vn,i,sl,tl +� ++ +� +a3 +m +� +l=1 +αlBn,i,sl,tl +������ +2+δ +≤31+δ +n +� +i=1 +|a1|2+δE +����� +m +� +l=1 +αlWn,i,sl,tl +����� +2+δ ++31+δ +n +� +i=1 +|a2|2+δE +����� +m +� +l=1 +αl.(II)sl,tl,A Vn,i,sl,tl +����� +2+δ ++31+δ +n +� +i=1 +|a3|2+δE +����� +m +� +l=1 +αlBn,i,sl,tl +����� +2+δ +≤31+δm1+δ|a1|2+δ +m +� +l=1 +|αl|2+δ +n +� +i=1 +E |Wn,i,sl,tl|2+δ ++31+δm1+δ|a2|2+δ +m +� +l=1 +|αl|2+δE |(II)sl,tl,A|2+δ +n +� +i=1 +E |Vn,i,sl,tl|2+δ ++31+δm1+δ|a3|2+δ +m +� +l=1 +|αl|2+δ +n +� +i=1 +E |Bn,i,sl,tl|2+δ +n→∞ +−−−→ 0. +In the final step above, we argue as in Lemma 6.5. By Slutky’s Theorem, we have the +47 + +aymptotic normality of ˜Cn + ˜Dn and consequently, the result follows. +□ +Proof of Theorem 2.2: The proof follows from the proof of Lemma 6.6 and the proof +of Lemma 6.7 along with an application of continuous mapping theorem. +□ +Proof of Proposition 2.4: Observe that the asymptotic power of the test under H1 is +PH1[ +� +nhnTn ≥ ˆcα] (since under H0, T = 0) += P[ +� +nhn(Tn − t0) ≥ ˆcα − +� +nhnt0] (assume that T = t0 > 0 under H1) += P[Z ≥ ˆcα − +� +nhnt0], +where Z follows certain non-degenate distribution. Since √nhn → ∞ as n → ∞ (see +condition (A1)) and t0 > 0, we have +lim +n→∞ P[Z ≥ ˆcα − +� +nhnt0] = 1. +This completes that proof. +□ +The following lemma is used in the proof of Proposition 3.1. +Lemma 6.8 Let f : R2 → R be any continuous function. Let ∥f∥∞ denote +sup +(t,s)∈R2 |f(t, s)| +and ∥f∥∞,G denote +sup +(t,s)∈[−G,G]×[−G,G] +|f(t, s)|, for any G > 0. Given any G > 0 and any +positive integer L, consider the set +SG,L := {−G + i 2G +L : i = 0, 1, · · · , L} +of equally spaced points in the interval [−G, G]. Then, we have the following limits. +(i) For any G > 0, +lim +L→∞ +max +(t,s)∈SG,L×SG,L |f(t, s)| = ∥f∥∞,G. +(ii) We have +lim +G→∞ lim +L→∞ +max +(t,s)∈SG,L×SG,L |f(t, s)| = lim +G→∞ ∥f∥∞,G = ∥f∥∞. +48 + +Proof of Lemma 6.8: Fix G > 0 and note that the continuous function f is bounded +on [−G, G] × [−G, G] (see (Rudin, 1976, Theorem 4.15)). Further, since the continuous +function |f| on [−G, G] × [−G, G] achieves its supremum within the set (see (Rudin, 1976, +Theorem 4.16)), there exists (t′, s′) ∈ [−G, G] × [−G, G] such that |f(t′, s′)| = ∥f∥∞,G. +Since f is continuous at (t′, s′), corresponding to the above ϵ > 0, we have δ > 0 such that +|f(t, s) − f(t′, s′)| < ϵ +2 +whenever (t, s) ∈ [−G, G] × [−G, G] with +� +(t − t′)2 + (s − s′)2 < δ. +Given a positive integer L, we can cover the set [−G, G]×[−G, G] by squares with corners +from SG,L × SG,L and with lengths of a side +2G +L . +Here, the diagonals in these squares +are of length +2 +√ +2G +L , and as such, any point in such a square is within a distance +√ +2G +L +from a corner. +In particular, for (t′, s′), there exists (t′′, s′′) ∈ SG,L × SG,L such that +� +(t′′ − t′)2 + (s′′ − s′)2 ≤ +√ +2G +L . We can now choose large L such that +√ +2G +L +< δ and hence, +|f(t′′, s′′) − f(t′, s′)| < ϵ +2 and in particular, +|f(t′, s′)| < |f(t′′, s′′)| + ϵ +2. +Then, +|f(t′′, s′′)| ≤ ∥f∥∞,G < |f(t′′, s′′)| + ϵ, +with (t′′, s′′) ∈ SG,L × SG,L for sufficiently large L. +Since ϵ > 0 is arbitrary, for any fixed G > 0, we have proved the first statement, i.e., +lim +L→∞ +max +(t,s)∈SG,L×SG,L |f(t, s)| = ∥f∥∞,G. +To prove the second statement, it is enough to show that limG→∞ ∥f∥∞,G = ∥f∥∞. +First, consider the case when ∥f∥∞ < ∞. Then, given ϵ > 0, there exists (t′, s′) ∈ R2 +such that +|f(t′, s′)| ≤ ∥f∥∞ < |f(t′, s′)| + ϵ. +49 + +Then, there exists G > 0 large such that (t′, s′) ∈ [−G, G] × [−G, G] and hence +|f(t′, s′)| ≤ ∥f∥∞,G ≤ ∥f∥∞ < |f(t′, s′)| + ϵ ≤ ∥f∥∞,G + ϵ. +Hence, we have limG→∞ ∥f∥∞,G = ∥f∥∞ when ∥f∥∞ < ∞. +When ∥f∥∞ = ∞, for any R > 0, there exists (t′, s′) ∈ R2 such that +|f(t′, s′)| ≥ R. +Then, there exists G > 0 large such that (t′, s′) ∈ [−G, G] × [−G, G] and hence +R ≤ |f(t′, s′)| ≤ ∥f∥∞,G. +Hence, we have limG→∞ ∥f∥∞,G = ∞ = ∥f∥∞. This completes the proof. +□ +Proof of Proposition 3.1: First, we observe that given finitely many real-valued continu- +ous functions f1, f2, · · · , fm on R2, the function (t, s) �→ max{f1(t, s), f2(t, s), · · · , fm(t, s)} +is also continuous. To see this, note that (t, s) �→ max{f1(t, s), f2(t, s)} = +1 +2|f1(t, s) + +f2(t, s)| − 1 +2(f1(t, s) − f2(t, s)) is a continuous function on R2. +Consequently, (t, s) �→ +max{f1(t, s), f2(t, s), f3(t, s)} = max{max{f1(t, s), f2(t, s)}, f3(t, s)} is also continuous. It- +erating this way, we have the above observation. +Now, for every fixed n ≥ 1, consider the real valued continuous function +(t, s) �→ +max +θj,i∈(− π +2 , π +2 ) +j=1,2;i=1,...,M−1 +θj,M−2∈(−π,π) +j=1,2 +����� +1 +nh +3 +2n +n +� +i=1 +k +� +⟨ˆlK +1 , Xi⟩H − s +hn +, ⟨ˆlK +2 , Yi⟩H − t +hn +� +− +1 +n2h2 +n +n +� +i,j=1 +k1 +� +⟨ˆlK +1 , Xi⟩H − s +hn +� +k2 +� +⟨ˆlK +2 , Yj⟩H − t +hn +������ +on R2. Under (A6), the required almost sure convergence follows from Lemma 6.8. +□ +As a consequence of assumption (A3) that the random elements X and Y are norm +bounded and the joint probability density function of the random variables ⟨l1, X⟩H and +⟨l2, Y ⟩H is uniformly bounded, we show that the marginal probability density functions of +50 + +the random variables ⟨l1, X⟩H and ⟨l2, Y ⟩H are also bounded. +Proposition 6.1 Suppose that the Hilbert valued random elements X and Y are essen- +tially bounded, i.e., there exist M1 > 0 and M2 > 0 such that P(∥X∥H ≤ M1) = 1 and +P(∥Y ∥H ≤ M2) = 1. Then, for all l1 and l2 ∈ H, the random variables ⟨l1 , X⟩H and +⟨l2 , Y ⟩H are also essentially bounded. Furthermore, in this case, if the joint probability +density function f⟨l1 , X⟩H,⟨l2 , Y ⟩H is bounded, then so are the marginal probability density +functions f⟨l1 , X⟩H and f⟨l2 , Y ⟩H. +Proof: For x, y ∈ H, we have | ⟨x , y⟩H | ≤ ∥x∥H∥y∥H. Then, for all l1, l2 ∈ H, P(| ⟨l1 , X⟩H | ≤ +∥l1∥HM1) = 1 and P(| ⟨l2 , Y ⟩H | ≤ ∥l2∥HM2) = 1. This proves that the random variables +⟨l1 , X⟩H and ⟨l2 , Y ⟩H are essentially bounded. +Now, if f⟨l1 , X⟩H,⟨l2 , Y ⟩H(x, y) ≤ R for all x ∈ [−M1, M1], y ∈ [−M2, M2], then +f⟨l1 , X⟩H(x) = +� M2 +−M2 +f⟨l1 , X⟩H,⟨l2 , Y ⟩H(x, y) dy ≤ 2RM2, ∀x +and +f⟨l2 , Y ⟩H(y) = +� M1 +−M1 +f⟨l1 , X⟩H,⟨l2 , Y ⟩H(x, y) dx ≤ 2RM1, ∀y +This completes the proof. +□ +7 +Appendix B : Additional Real Data Analysis and +Simulation Studies +7.1 +Real Data Analysis +This section consists of two more real data analyses. Those data sets are well-known the +Coffee data and the Berkeley growth data. The coffee data is available at https://www.cs. +ucr.edu/~eamonn/time_series_data_2018/, and it contains spectroscopy readings taken +at 286 wavelength values for 14 samples of each of the two varieties of coffee, namely, +Arabica and Robusta. +The readings are illustrated in the diagrams in Figure 3, and +each observation is viewed as an element in the separable Hilbert space L2([0, 1]) after +51 + +Figure 3: +Spectroscopy readings of mean curve of Arabica coffee and the Robusta coffee. +Source : https: // www. mdpi. com/ 2304-8158/ 9/ 6/ 788/ htm +normalization by an appropriate scaling factor as we did in the analysis of Canadian +weather data in Section 4. Following the same methodology described in Section 4, we +compute the p-value, and the p-value is obtained as 0.069, which indicates that the data +does not favour the null hypothesis at 7% level of significance, i.e., spectroscopy readings +of the Arabica coffee and the Robusta coffee have strong enough statistical dependence. +Note that this result is not unexpected as it seems from the diagrams in Figure 3 that the +curves associated with spectroscopy readings of the Arabica coffee and the Robusta coffee +are not independent. Now, as this data is not favouring the null hypothesis, here also, we +estimate the size and power of test based on Tn for this data, and in order to carry out +this study, we follow the same methodology as we did for the Canadian weather data in +Section 4. Using that methodology, at 5% level of significance, the estimated size is 0.058, +and the estimated power is 0.587. It further indicates the proposed test based on Tn can +achieve the nominal level and perform good in terms of power. +The Berkeley growth data is available at https://rdrr.io/cran/fda/man/growth. +html. It contains the heights of 39 boys and 54 girls measured at 31 time points between +52 + +Arabica +Robusta +1.8 +(log 1/R) +1.6 +.4 +Diffusereflectance +1.2 +0.8 +999 +1063 11411229 1333 1458 1607 1791 2023 2323 +Wavelength (nm)Figure 4: +The growth of heights : +male and female. +Source : +https: // fda. +readthedocs. io/ en/ latest/ auto_ examples +the ages 1 and 18 years, and the curves are recorded at 101 equispaced ages in the interval +[1, 18]. The heights are illustrated in the diagrams in Figure 4, and each observation is +viewed as an element in the separable Hilbert space L2([0, 1]) after normalization by an +appropriate scaling factor. For this data set, we obtain p-value as 0.074, which indicates +that the data does not favour the null hypothesis at 8% level of significance, i.e., in other +words, the heights of boys and girls have strong enough statistical dependence. +Note +that this result is not unexpected as it seems from the diagrams in Figure 4 that the +height curves associated with boys and girls are not independent. Now, as this data is not +favouring the null hypothesis, here also, we estimate the size and power of the test based +on Tn. At 5% level of significance, the estimated size is 0.057, and the estimated power is +0.788, which further indicates that the proposed test is capable of detecting dependence +structure of the random elements in real data also. +53 + +Berkeley Growth Study +male +female +200 +180 +160 +height +140 +120 +100 +80 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +age7.2 +Simulation Studies +Here we study the performance of the proposed test for a few more cases when the sample +size is finite. The finite sample power of the test is estimated for the following examples. +Example 7 : Let (X, Y ) be L2([0, 1]) × L2([0, 1]) valued random element. Suppose that +X(t) (t ∈ [0, 1]) follows a Gaussian process B(t) with E(B(t)) = 0 and E(B(t)B(s)) = +min(t, s) for all t ∈ [0, 1] and s ∈ [0, 1], and Y (t) +d= {X(t)}2. +Example 8 : Let (X, Y ) be L2([0, 1]) × L2([0, 1]) valued random element. Suppose that +X(t) (t ∈ [0, 1]) follows a Gaussian process B(t) with E(B(t)) = 0 and E(B(t)B(s)) = +min(t, s) for all t ∈ [0, 1] and s ∈ [0, 1], and Y (t) +d= eX(t). +Example 9 : Let (X, Y ) be L2([0, 1]) × L2([0, 1]) valued random element. Suppose that +X(t) (t ∈ [0, 1]) follows a t-process with 3 degrees of freedom, where t-process with k(k ≥ 1) +degrees of freedom is defined as +B(t) +√ +N +k +, where B(t) is a Gaussian process with E(B(t)) = 0 +and E(B(t)B(s)) = min(t, s) for all t ∈ [0, 1] and s ∈ [0, 1] and independent of N, which +follows Chi squared distribution with k degrees of freedom. Here, Y (t) +d= {X(t)}2. +Example 10 : Let (X, Y ) be L2([0, 1]) × L2([0, 1]) valued random element. Suppose that +X(t) (t ∈ [0, 1]) follows a t-process with 3 degrees of freedom. Here, Y (t) +d= eX(t). +For all these aforementioned examples, in order to generate data from the Brownian +motion, the data are generated from associated certain multivariate normal distributions. +We here study the performance of the proposed test for the sample sizes n = 20, 50, 100 +and 500, and the estimated powers of the proposed test are reported in Table 3. +The results reported in Table 3 indicate that the test is more powerful when X follows +t-process with 3 degrees of freedom than when X follows Brownian motion. It may be due +to the fact that for a fixed t, X(t) distributed with a t-process with 3 degrees of freedom +has a heavier tail than X(t) distributed with a certain Gaussian process. In terms of the +relationship between Y and X, it is observed that the proposed test is more powerful when +Y +d= eX than when Y +d= X2. It is also expected as the exponential function has a faster +growth rate than polynomial function. +54 + +model +n = 20 +n = 50 +n = 100 +n = 500 +Example 7 (α = 5%) +0.441 +0.553 +0.685 +0.832 +Example 7 (α = 10%) +0.483 +0.605 +0.733 +0.886 +Example 8 (α = 5%) +0.459 +0.601 +0.824 +0.954 +Example 8 (α = 10%) +0.499 +0.643 +0.877 +0.965 +Example 9 (α = 5%) +0.484 +0.655 +0.872 +0.963 +Example 9 (α = 10%) +0.511 +0.688 +0.901 +0.983 +Example 10 (α = 5%) +0.502 +0.713 +0.907 +0.986 +Example 10 (α = 10%) +0.532 +0.741 +0.933 +0.992 +Table 3: The estimated power of the proposed test for different sample sizes n. The level +of significance, i.e., α are 5% and 10%. +References +Bergsma, W. and Dassios, A. (2014). A consistent test of independence based on a sign +covariance related to kendall’s tau. Bernoulli, 20(2):1006–1028. +Berrett, T. B., Kontoyiannis, I., and Samworth, R. J. (2021). Optimal rates for indepen- +dence testing via u-statistic permutation tests. 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Cambridge University Press. +57 + diff --git a/UNAyT4oBgHgl3EQfhPjd/content/tmp_files/load_file.txt b/UNAyT4oBgHgl3EQfhPjd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..218a80195441a6375fad2d2b109f1b5287c43b3a --- /dev/null +++ b/UNAyT4oBgHgl3EQfhPjd/content/tmp_files/load_file.txt @@ -0,0 +1,1673 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf,len=1672 +page_content='Testing Independence of Infinite Dimensional Random Elements: A Sup-norm Approach Suprio Bhar IIT Kanpur Department of Mathematics and Statistics Kanpur 208016, India email: suprio@iitk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='in Subhra Sankar Dhar IIT Kanpur Department of Mathematics and Statistics Kanpur 208106, India email: subhra@iitk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='in Abstract In this article, we study the test for independence of two random elements X and Y lying in an infinite dimensional space H (specifically, a real separable Hilbert space equipped with the inner product ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='⟩H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In the course of this study, a measure of as- sociation is proposed based on the sup-norm difference between the joint probability density function of the bivariate random vector (⟨l1, X⟩H, ⟨l2, Y ⟩H) and the prod- uct of marginal probability density functions of the random variables ⟨l1, X⟩H and ⟨l2, Y ⟩H, where l1 ∈ H and l2 ∈ H are two arbitrary elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' It is established that the proposed measure of association equals zero if and only if the random elements are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In order to carry out the test whether X and Y are independent or not, the sample version of the proposed measure of association is considered as the test statistic after appropriate normalization, and the asymptotic distributions of the test statistic under the null and the local alternatives are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The performance of the new test is investigated for simulated data sets and the practicability of the 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='00375v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='ST] 1 Jan 2023 test is shown for three real data sets related to climatology, biological science and chemical science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Keywords: Climate, Measure of Association, Projection, Separable Hilbert Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 Key Ideas and Literature Review For univariate and multivariate data, there have been several attempts to test whether two or more random variables or vectors are independent or not in various situations (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', Blum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (1961), Szekely et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (2007), Genest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (1961), Einmahl and Van Keilegom (2008), Dette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (2013), Bergsma and Dassios (2014), Dhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (2016), Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (2017), Dhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (2018), Drton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (2020), Chatterjee (2021), Berrett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (2021), She et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (2022b), She et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (2022a) and a few references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' To the best of our knowledge, all these tests are based on a certain measure of association, which equals zero if and only if the finite dimensional random variables or vectors are independent, and consequently, the tests based on those measures of associations lead to consistent tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Therefore, one first needs to develop a measure of association having such necessary and sufficient relation with independence in the infinite dimensional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Once such a relation is found, in principle, it allows us to formulate a reasonable test statistic for checking whether two random elements are independent or not in infinite dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' This article addresses this issue in the subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Let us first discuss the utility of the framework in terms of infinite dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In many interdisciplinary subjects like Biology, Economics, Climatology or Finance, we recently come across many data sets, where the dimension of the data is much larger than the sample size of the data set, and in majority of the cases, the standard multivariate techniques cannot be implemented because of their high dimensionalities relative to the sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In order to overcome this problem, one can embed such data into a suit- able infinite dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' For instance, functional data (see Ramsay and Silverman (2002), Ferraty and Vieu (2006)) is an infinite dimensional data, and one can analyse the 2 functional data using the techniques adopted for infinite dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In this work, we consider random elements lying in a real separable Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The separability of the space allows us to write the random elements in the countably infinite orthonormal expansion of a suitable basis, making it easier to handle the theoretical issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In the con- text of measure of association or the test for independence of infinite dimensional random elements, we would like to mention that only a few limited numbers of articles are available in the literature on this topic, and the contributions of the major ones are described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Almost a decade ago, Lyons (2013) (see also Lyons (2018) and Lyons (2021)) studied the distance covariance, which is proposed by Szekely et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (2007) for finite dimensional random vectors, in a certain metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' However, Lyons (2013) did not study the corresponding testing of hypothesis problems based on the proposed measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Before the aforesaid work, Cuesta-Albertos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (2006) studied the random projection based goodness of fit test and two sample tests for the infinite dimensional element, which can be applied on the test for independence of the infinite dimensional random elements as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' During the almost same period, Gretton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (2007) proposed a new methodology based on the Hilbert-Schmidt independence criterion (HSIC) for testing independence of two finite dimensional random vectors, and it can too be applied for the Hilbert space valued random elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' However, none of these articles used the density functions of the random variables obtained by the certain projections of the random elements lying in infinite dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' This article proposes a methodology for testing the independence of random elements defined in infinite dimensional space based on the probability density functions of projections of said random elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 Contributions We consider two random elements X and Y lying in a real separable Hilbert space H (equipped with the inner product ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='⟩H) and propose a methodology of testing indepen- dence between X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' This is the first major contribution of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The method- ology is developed based on sup-norm distance between the joint probability density of the bivariate random vector (⟨l1, X⟩H, ⟨l2, Y ⟩H) and the product of marginal probability density functions of the random variables ⟨l1, X⟩H and ⟨l2, Y ⟩H, where l1 ∈ H and l2 ∈ H 3 are two arbitrary elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' This measure of association equals with zero for all l1 ∈ H and l2 ∈ H if and only if X and Y are independent random elements, and it is non-negative as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The second major contribution is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' For a given data, a sample version of the measure with appropriate normalization is proposed, which is considered as the test statistic for testing independence of infinite dimensional random elements, and the asymptotic distribution of the sample version under null and local/contiguous alternatives is derived after appropriate normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' These theoretical results enable us to study the consistency and the asymptotic power of the corresponding test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Unlike other tests, since the test statistic used in this methodology depends on the choice of the kernel and the associated bandwidth, the performance of the proposed test can be enhanced by suitable choices of the kernel and the bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The third major contribution of the work is the implementation of the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' As the test statistic is based on the supremum over the infinite choices of infinite dimensional random elements l1 and l2, the exact computation of the test statistic is intractable for a given data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' To overcome this problem, l1 and l2 are chosen over certain finite collection of possibilities of the choices, and it is shown that this method approximates the actual statistic when the number of possible choices of l1 and l2 are sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Overall, it is established that the approximated test is easily implementable, and it gives satisfactory results in analysing real data as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3 Challenges In the course of this work, we overcome a few mathematical challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The first challenge involves the interpretation of independence between two infinite dimensional random el- ements, and it is resolved using the concept of projection towards all possible directions (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2, we further reduce the problem by relying on vectors of unit length, rather that those with arbitrary lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' This procedure reduces the complexity of the optimization problem associated with infinite dimensional projection vector to a large extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The second challenge concerns the optimization problem asso- ciated with time parameters involved with the test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' To overcome this issue, we 4 approximate the test statistic in two steps: first, we use a finite dimensional approximation for any random vector on an orthonormal basis and then, we compute the test statistic on a sufficiently large number of time points chosen over a sufficiently large interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Us- ing advanced techniques of analysis, it is shown that the later version of the test statistic approximates arbitrarily well the original test statistic (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The third challenge is related to asymptotic distribution of the test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' As the key term of the test statistic involves a triangular array and a handful number of complex terms, the asymptotic distribution of the key term follows after careful use of CLT associ- ated with triangular array (see the proofs of Lemmas 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Finally, the asymptotic distribution of the test statistic follows from Cram´er–Wold theorem and continuous mapping theorem (see van der Vaart (1998)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4 Applications We analyse well known climate data, which consists of daily temperature and precipitation at thirty five locations in Canada averaged over 1960 to 1994 (see Ramsay and Silverman (2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Note that the dimension of the data equals 365, which is much larger than the sample size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', 35) of the data, and embedding such a high dimensional data into a specific Hilbert space is legitimate enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' From the point of view of climate science, it is of interest to know how far the temperature depends on the precipitation at a given time point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The analysis of this real data using the proposed methodology gives some insight about the long standing issue in climatology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Along with it, two more data are also analysed to show the practicability of the proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' One data set is well-known Berkeley growth data, which consists the heights of 39 boys and 54 girls measured at 31 time points between the ages 1 and 18 years, and from medical science’s point of view, it is of interest to know whether the growth (in terms of height) of the boys and the girls are independent or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' We address it using our proposed test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Another data set is Coffee data, which contains spectroscopy readings taken at 286 wavelength values for 14 samples of each of the two varieties of coffee, namely, Arabica and Robusta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' As coffee is one of the most popular beverages, one may be interested to know whether the chemical features of the Arabica coffee and the Robusta coffee are 5 independent or not, and this issue is also addressed using our test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5 Organization of the Article The article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Section 2 introduces the basic concepts of separable Hilbert space valued random elements, and Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 proposes the criterion for checking independence between two random elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Afterwards, for a given data, the estimation of the criterion is studied in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2, and Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 formulates the test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The asymptotic properties of the test statistic and associated test are investigated in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3, and Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 studies the performance of the proposed test in terms of the asymptotic power for various examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The performance of the proposed test for finite sample sizes is studied in Section 3, and real data analysis is carried out in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Section 5 contains some concluding remarks, and finally, Sections 6 and 7 contain all technical details and additional numerical results, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 2 Methodology We first formulate the statement of the problem and subsequently construct an appropriate test statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Let H denote a real separable Hilbert space equipped with the inner product ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='⟩H and the norm ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='||H = � ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='⟩H, and suppose that (Ω, A, P) is a probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Let us now consider an H × H valued bivariate random element (X, Y ), which is a measurable mapping from (Ω, A, P) into H×H equipped with its Borel σ-algebra B(H×H) generated by the open sets of H × H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In other words, for any Borel set U1 ∈ H and U2 ∈ H, (X−1(U1), Y −1(U2)) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' We now want to check whether the random elements X and Y are independent or not, and in this regard, a new criterion for checking independence is proposed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 Proposed Criterion for Independence We here propose a criteria for checking independence based on certain one-dimensional projections of the random elements X and Y lying on a separable Hilbert space H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In 6 order to formulate the criterion, a useful lemma is stated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 Let X and Y be two H valued random elements on a probability space (Ω, A, P), where H is a real separable Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then H-valued X and H-valued Y are two in- dependent random elements if and only if the real valued random variable ⟨l1, X⟩H and the real valued random variable ⟨l2, Y ⟩H are independent for every l1 ∈ H and l2 ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In other words, P(⟨l1 , X⟩H ∈ A, ⟨l2 , Y ⟩H ∈ B) = P(⟨l1 , X⟩H ∈ A)P(⟨l2 , Y ⟩H ∈ B) for all Borel subsets A and B of R if and only if P(X ∈ C, Y ∈ D) = P(X ∈ C)P(Y ∈ D) for all Borel subsets C and D of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The assertion in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 indicates that P(A∩B) = PX(A)PY (B) for all A ∈ A and B ∈ A, where PX and PY are marginal measures associated with X and Y , respectively of the product measure P if and only if Q(C ∩ D) = QX(C)QY (D), where Q is the product measure of (⟨l1, X⟩H, ⟨l2, Y ⟩H), QX is the measure associated with ⟨l1, X⟩H, and QY is the measure associated with ⟨l2, Y ⟩H, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Here C ∈ B(R) and D ∈ B(R) are two arbitrary Borel sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' This fact indicates that if ⟨l1, X⟩H and ⟨l2, Y ⟩H are absolutely continuous random variables, then the joint probability density function of ⟨l1, X⟩H and ⟨l2, Y ⟩H equals with the product of marginal probability density functions of ⟨l1, X⟩H and ⟨l2, Y ⟩H for all l1 and l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Writing with notation, suppose that f(⟨l1,X⟩H,⟨l2,Y ⟩H)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' ), f⟨l1,X⟩H(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=') and f⟨l2,Y ⟩H(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=') are the joint probability density, marginal probability density functions of (⟨l1, X⟩H, ⟨l2, Y ⟩H), ⟨l1, X⟩H and ⟨l2, Y ⟩H, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then X and Y are independent random elements if and only if f(⟨l1,X⟩H,⟨l2,Y ⟩H)(s, t) − f⟨l1,X⟩H(s)f⟨l2,Y ⟩H(t) = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1) for all s ∈ R, t ∈ R, l1 ∈ H and l2 ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The identity in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1) motivates us to formulate a 7 criterion for checking independence between X and Y is the following : T(R1, R2) := sup ||l1||H≤R1,||l2||H≤R2 sup s∈R,t∈R |f(⟨l1,X⟩H,⟨l2,Y ⟩H)(s, t) − f⟨l1,X⟩H(s)f⟨l2,Y ⟩H(t)|, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2) where R1 and R2 are two arbitrary large positive real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1), we have the following: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 T(R1, R2) = 0 if and only if X and Y are H-valued independent random elements, where T(R1, R2) is as defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In other words, T(R1, R2) is degenerate at 0 if and only if X and Y are H-valued independent random elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' However, note that finding the supremum over {l1 : ||l1||H ≤ R1} and {l2 : ||l2||H ≤ R2} is not easily tractable in practice, and to overcome it, one can find the supremum over the boundary of the unit sphere, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', {l1 : ||l1||H = 1} and {l2 : ||l2||H = 1} defined in H, and this equivalence is asserted in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 Let T := sup ||l1||H=1,||l2||H=1 sup s∈R,t∈R |f(⟨l1,X⟩H,⟨l2,Y ⟩H)(s, t) − f⟨l1,X⟩H(s)f⟨l2,Y ⟩H(t)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3) Then T(R1, R2) = 0 ⇔ T = 0, where T(R1, R2) is the same as defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Finally, the following theorem characterizes the independence of X and Y based on equality of T with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 Let (X, Y ) be a bivariate random element taking values on H × H, where H is a real separable Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then, X and Y are independent if and only if T = 0, where T is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The assertion in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 implies that the testing independence between the random elements X and Y defined in H is equivalent to testing T = 0, and hence, in order to test 8 whether T = 0 or not (OR X and Y are independent random elements or not) for a given data, one needs to have an appropriate estimator of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 Estimation of T Let (X1, Y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , (Xn, Yn) be an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='d sequence of random elements, and they are identically distributed with (X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In order to estimate T, one first needs to find l1 and l2 from the unit sphere in H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' However, in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3), l1 and l2 appear in the joint and marginal probability density functions and as such, we end up with a two fold issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' First one involves an infinite-dimensional optimization in l1 and l2 and the second one is the estimation of the corresponding joint and marginal probability density functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The following may be one of the procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 Approximation of l1 and l2 In view of the fact that H is a real separable Hilbert space, and since l1 ∈ H and l2 ∈ H, we have l1 = ∞ � i=1 l1,iei and l2 = ∞ � i=1 l2,iei, where for a fixed i, ei = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , 0 � �� � i−1 , 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=') is an infinite dimensional basis in H, and l1,i and l2,i (i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=') are the coefficients of the orthonormal expansion of l1 and l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Note that ∞ � i=1 l2 1,i = 1, and ∞ � i=1 l2 2,i = 1, and ����� �����l1 − M � i=1 l1,iei ����� ����� 2 H → 0 and ����� �����l2 − M � i=1 l2,iei ����� ����� 2 H → 0 as M → ∞ as H is a real separable Hilbert space (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', Rudin (1991)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Using this fact, we propose an approximate choices of lj (j = 1 and 2) as ˆlM j = M � i=1 lj,iei, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4) 9 where M is a sufficiently large positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Afterwards, to compute the criterion T or estimate T, we compute the supremum over (lj,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , lj,M) for j = 1 and 2, where M � i=1 l2 j,i = 1, and this optimization problem can approximately be solved using polar transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' For j = 1 and 2, let us consider the following transformation into the spherical coordi- nate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' lj,1 = cos θj,1, lj,2 = sin θj,1 cos θj,2, lj,3 = sin θj,1 sin θj,2 cos θj,3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' lj,M−1 = sin θj,1 sin θj,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' sin θj,M−2 cos θj,M−1, lj,M = sin θj,1 sin θj,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' sin θj,M−2 sin θj,M−1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5) where for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , M − 2, θj,i ∈ (− π 2, π 2) and θj,M−1 ∈ (−π, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Observe that for j = 1 and 2, M � i=1 l2 j,i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In practice, in the course of implementing the test, we consider K equally spaced choices of θj,i from (− π 2, π 2) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , M − 2, and similarly, K equally spaced θj,M−1 are chosen from (−π, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Using this grid searching of θj,i (j = 1 and 2, and i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , M − 1), suppose that the supremum associated with l1 and l2 involved in T (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3)) attains at some θ(K) j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i for j = 1 and 2, and i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , M, and corresponding associated lj,i are denoted by l(K) j,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Finally, the approximated choices of lj is considered as ˆlK j = M � i=1 l(K) j,i ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='6) Observe that M � i=1 (l(K) j,i )2 = 1 for j = 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The following proposition guarantees that ˆlK j is approximate lj (j = 1 and 2) arbitrarily well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Note that ˆlK j (j = 1 and 2) depend on M as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' However, for notational convenience, we do not explicitly write M in the expression of ˆlK j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3 For j = 1 and 2, ��� ���ˆlK j − lj ��� ��� H → 0 almost surely as M → ∞ and K → ∞, where ˆlK j is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 10 Observe that ˆlK j is also unknown in practice as the joint and marginal probability density functions involved in T (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3)) are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Therefore, as indicated from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3 that ˆlK j is a good approximation of lj (j = 1 and 2), and since computing ˆlK j is much less complex relative to computing lj (j = 1 and 2), we will work on ˆlK j in formulating the test statistic and studying its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 Formulation of Test Statistic As we said earlier, formally speaking we want to test H0 : X ⊥ Y against H1 : X ̸⊥ Y , where X and Y are two random elements lying in a real separable Hilbert space H, and it follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 that it is equivalent to test H∗ 0 : T = 0 against H∗ 1 : T ̸= 0, where T is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In order to test H∗ 0 against H∗ 1, one needs to have an appropriate estimator of T, and as an initial step in the course of this work, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 studied how to approximate l1 and l2 involved in T (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3)), and the approximated lj (j = 1 and 2) are denoted by ˆlK j defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Here one first needs to project the infinite dimensional data into one dimensional space through ˆlK 1 and ˆlK 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Let us fist consider the data (X1, Y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , (Xn, Yn), which be- long to the same probability space of (X, Y ), and suppose that the transformed data are (⟨ˆlK 1 , X1⟩H, ⟨ˆlK 2 , Y1⟩H), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , (⟨ˆlK 1 , Xn⟩H, ⟨ˆlK 2 , Yn⟩H), where the projection vectors ˆlK 1 and ˆlK 2 are unknown in practice as we discussed at the end of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In order to identify the optimum ˆlK 1 and ˆlK 2 for a given data, we need to estimate the joint probability density function f(⟨l1,X⟩H,⟨l2,Y ⟩H)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=') and the marginal probability density functions f⟨l1,X⟩H(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=') and f⟨l2,Y ⟩H(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' We describe the estimation of the probability density functions in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Let k : R2 → (0, ∞) be a bivariate function such that � k(x, y)dxdy = 1, k1 : R → (0, ∞) and k2 : R → (0, ∞) are such that � k1(x)dx = 1 and � k2(x)dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Note that k, k1 (and k2) can be considered as the kernels involved in the estimator of the bivariate and univariate probability density functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' For details on kernel density estimation, the readers may refer to Silverman (1998), and a few more technical conditions on the kernels will be described at the appropriate places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Now, suppose that Tn is an appropriate estimator T based on the kernel density estimators k, k1 and k2, which can be formulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 11 Tn = sup θj,i∈(− π 2 , π 2 ) j=1,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',M−1 θj,M−2∈(−π,π) j=1,2 s∈R,t∈R ����� 1 nh 3 2n n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn � − 1 n2h2 n n � i,j=1 k1 � ⟨ˆlK 1 , Xi⟩H − s hn � k2 � ⟨ˆlK 2 , Yj⟩H − t hn ������ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='7) where hn is a sequence of bandwidth such that hn → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Note that ˆlK j involves θj,i (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='6)), and for this reason, supremum taken over ˆlK j eventually boils down to the supremum taken over θj,i (j = 1 and 2, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , M) in the expression of Tn (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The asymptotic properties of Tn are studied in the subsequent section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3 Asymptotic Properties of Tn Recall that the testing of hypothesis problem H0 : X ⊥ Y against H1 : X ̸⊥ Y is equivalent to testing of hypothesis problem H∗ 0 : T = 0 against H∗ 1 : T ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In order to check whether for a given data T = 0 or not, the test statistic Tn (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='7)) is formulated in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2, and in order to carry out the test based on Tn, the distributional feature of Tn is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' However, due to complex formulation of Tn, the derivation of exact distribution of Tn may not be tractable, and to overcome it, we here derive the asymptotic distribution of Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' We first assume the following technical conditions and then state the asymptotic distribution of Tn under local alternatives, and the asymptotic distribution under null hypothesis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', H0 or H∗ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (A1) The sequence of bandwidth {hn}n≥1 is such that hn → 0 as n → ∞, nh2 n → c (for some constant c ∈ R) as n → ∞, and nhn → ∞ as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (A2) The partial derivatives of the joint probability density function f of the bivariate random vector (⟨l1, X⟩H, ⟨l2, Y ⟩H) exist, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', ∂f(⟨l1,X⟩H,⟨l2,Y ⟩H)(z1,z2) ∂z1 and ∂f(⟨l1,X⟩H,⟨l2,Y ⟩H)(z1,z2) ∂z2 exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (A3) The random elements X and Y are norm bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', ||X||H and ||Y ||H are bounded random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Moreover, the joint probability density function of ⟨l1, X⟩H and ⟨l2, Y ⟩H is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 12 (A4) The first and the second derivatives of the probability density functions of the random variables ⟨l1, X⟩H and ⟨l2, Y ⟩H are uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (A5) For i = 1 and 2, the kernels ki : R → A ⊂ R+ are such that � ki(m)dm = 1, � mki(m)dm < ∞, � k2 i (m)dm < ∞, � m2ki(m)dm < ∞ and � m{ki(m)}2dm < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Here A is a bounded set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (A6) The bivariate kernel k : R2 → A ⊂ R+ is such that � m1k(m1, m2)dm1dm2 < ∞, � m2k(m1, m2)dm1dm2 < ∞, � m1{k(m1, m2)}2dm1dm2 < ∞ and � m2{k(m1, m2)}2dm1dm2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Here A is a bounded set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 Condition (A1) indicates that for various choices of hn, the asymptotic re- sults described in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 holds, and such conditions are common across the literature in kernel density estimation (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', Silverman (1998)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Condition (A2) is satisfied when the directional derivatives of the joint probability density function of ||X||H and ||Y ||H exist, and these are indeed realistic assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Further, when the infinite dimensional random elements X and Y are norm bounded, and the joint probability density function of the random variables ⟨l1, X⟩H and ⟨l2, Y ⟩H is uniformly bounded (see Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1), then the marginal probability density functions of the random variables ⟨l1, X⟩H and ⟨l2, Y ⟩H are also bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' For instance, for well-known Gaussian process, the conditions (A3) and (A4) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Condition (A5) implies certain mild restriction on the tail behaviour of the chosen kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' However, for compact support of the kernel, there is no need to put restric- tions on the tail behaviour of the chosen kernels as the integrations involving kernels are finite as long as the the kernels are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Condition (A6) asserted some Integrability assumptions on the chosen bivariate kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In this case also, if the support of the chosen bivariate kernel is compact, the continuity of the chosen bivariate kernel is good enough to satisfy the integrability assumptions in condition (A6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 13 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 Let us denote T G,L n = sup θj,i∈(− π 2 , π 2 ) j=1,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',M−1 θj,M−2∈(−π,π) j=1,2 s∈{s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',sL} t∈{t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',tL} ����� 1 nh 3 2n n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn � − 1 n2h2 n n � i,j=1 k1 � ⟨ˆlK 1 , Xi⟩H − s hn � k2 � ⟨ˆlK 2 , Yj⟩H − t hn ������ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='8) where si = −G + i 2G L and ti = −G + i 2G L for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , L are equi-distributied points in [−G, G], and Tn is the same as defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then, under (A1), (A5) and (A6), � nhn � lim G→∞ lim L→∞(Tn − T G,L n ) � p→ 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Moreover, under conditions (A1)–(A6) and when T = λ √nhn for some λ > 0, √nhnT G,L n − λ converges weakly to the distribution of sup s,t∈{−G+i 2G L |i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',L} {|Zs,t|} as n → ∞, K → ∞ and M → ∞ (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='6) for the definitions of K and M), where Zs,t is a random variable associated with normal distribution with mean = −√c � fZ1(s)f ′ Z2(t) � mk2(m)dm + fZ2(t)f ′ Z1(s) � mk1(m)dm � and variance = fZ1,Z2(s, t) � k2(m1, m2)dm1dm2+ {fZ1(s)}2fZ2(t) � {k2(m)}2+fZ1(s){fZ2(t)}2 � {k1(m)}2dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Here c = lim n→∞ nh2 n and (Z1, Z2) = (⟨l1, X⟩H, ⟨l2, Y ⟩H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 The first part in the statement of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 indicates that T G,L n is an appropriate approximation of Tn for sufficiently large G and L after normalized by √nhn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Here G controls the length of the interval of the time parameters, and L controls the number of time points after discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The motivation of constructing T G,L n will be discussed in details in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Moreover, with the same normalization factor √nhn, T G,L n − λ 14 converges weakly to a certain random variable associated with the functional of multivariate normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' All together, one can approximately compute the asymptotic power of the test based on Tn using the assertion in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3 Observe that Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 is stated when T = λ √nhn, and since √nhn → ∞ as n → ∞ (see Condition (A1)), it indicates that the statement of this theorem is valid for local alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' As a result, the assertion of this theorem enables us to compute the approximated asymptotic power of the test based on Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Further, as a special case, when λ = 0, the asymptotic distribution of Tn under null hypothesis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', T = 0) follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2, and it is formally stated in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4 Under conditions (A1)-(A6) and when T = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', under null hypothesis), √nhnT G,L n converges weakly to the distribution of sup s,t∈{−G+i 2G L |i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',L} {|Zs,t|} as n → ∞, K → ∞ and M → ∞, where Zs,t denotes a random variable associated with normal distribution with mean and variance, which are the same as described in the statement of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4 asserts the asymptotic distribution T G,L n under the null hypothesis, and therefore, one can approximately estimate the critical value of the test based on Tn using the asymptotic distribution of T G,L n stated in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4 as Tn and T G,L n can be made arbitrary close to each other for sufficiently large G and L (see the first part of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' All in all, one can explicitly express the power of the test based on Tn when T = t0 using Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In notation, if ˆcα is such that PH0[√nhnTn > ˆcα] → α as n → ∞, the the power of the test based on Tn under H1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', when T = t0) is given by PH1 �� nhnTn > ˆcα � = P �� nhn(Tn − t0) > ˆcα − � nhnt0 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='9) and the asymptotic power of the test based on Tn under local alternatives (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', T = λ √nhn as mentioned in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2) is given by P� T= λ √nhn � �� nhnTn > ˆcα � = P �� nhnTn − λ > ˆcα − λ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='10) 15 Moreover, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='9), one can establish the consistency of the test based on Tn, which is stated in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4 Let ˆcα is such that PH0[√nhnTn > ˆcα] → α as n → ∞, where α ∈ (0, 1) is a preassigned constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then, the test based on Tn, which rejects H0 if √nhnTn ≥ ˆcα, is a consistent test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In other words, PH1[√nhnTn ≥ ˆcα] → 1 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4 asserts that the test based on Tn is expected to perform well in terms of power for a sufficiently large sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Next, we study the performance of the test based on Tn through the asymptotic power study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 Asymptotic Power Study Recall that Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 states the asymptotic distribution of T G,L n under local alternatives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', when T = λ √nhn (λ ≥ 0), and one can compute the asymptotic power of the test based on Tn for various choices of the random processes in separable Hilbert space and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In the course of study, the critical value with α% level of significance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', ˆcα is computed using the result described in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4, and using the obtained ˆcα, one can approximate the asymptotic power of the test based on Tn using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' We consider following three examples : Example 1 : Let (X, Y ) be L2([0, 1]) × L2([0, 1]) valued random element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Suppose that X(t) (t ∈ [0, 1]) has the same distribution as U exp(t), where U follows uniform distribution over [0, 1], and Y (t) = ||X||L2([0,1]) × B(t), where B(t) (t ∈ [0, 1]) is a Gaussian process with E(B(t)) = 0 and E(B(t)B(s)) = min(t, s) for all t ∈ [0, 1] and s ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Example 2 : Let (X, Y ) be L2([0, 1]) × L2([0, 1]) valued random element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Suppose that X(t) (t ∈ [0, 1]) has the same distribution as U exp(t), where U follows uniform distribution over [0, 1], and Y (t) = ||X||L2([0,1]) × B1(t), where B1(t) (t ∈ [0, 1]) is fractional Brownian motion with Hurst index = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='25, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', E(B1(t)) = 0 and E(B1(t)B1(s)) = 1 2(|t|0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5 + |s|0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5 − |t − s|0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5) for all t ∈ [0, 1] and s ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Example 3 : Let (X, Y ) be L2([0, 1]) × L2([0, 1]) valued random element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Suppose that X(t) (t ∈ [0, 1]) has the same distribution as U exp(t), where U follows uniform distribution over [0, 1], and Y (t) = ||X||L2([0,1]) × B2(t), where B2(t) (t ∈ [0, 1]) is fractional Brownian 16 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4 Example 1 λ Asymptotic Power 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='7 Example 2 λ Asymptotic Power 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='7 Example 3 λ Asymptotic Power Figure 1: Asymptotic power of the proposed test for Examples 1, 2 and 3 at 5% level of significance when λ varies from 0 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' motion with Hurst index = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='75, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', E(B2(t)) = 0 and E(B2(t)B2(s)) = 1 2(|t|1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5 + |s|1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5 − |t − s|1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5) for all t ∈ [0, 1] and s ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Note that in order to compute the asymptotic power of the test, one first needs to compute the critical value, which can be estimated using the result stated in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', when T = 0, and it is equivalent to the fact that X and Y are indepen- dent random elements in H (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Now, in order to satisfy this statisti- cal independence between X and Y , we first consider X(t) and Y (t) are two indepen- dent random elements in L2([0, 1]) associated with standard Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' After- wards, the terms involving the joint and the marginal probability density functions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', f(⟨l1,X⟩H,⟨l2,Y ⟩H)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' ), f⟨l1,X⟩H(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' ), and f⟨l2,Y ⟩H(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=')) appeared in the mean and the variance of the asymptotic distribution of Tn under H0 (see Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4) are estimated by bivariate and marginal kernel density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' For the bivariate kernel density function, we consider the two fold product of the marginal kernel density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In this study, for estimating the marginal probability density function, we use the well-known Epanechnikov Kernel because of its optimal properties (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', Silverman (1998)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Finally, α% (denoted as ˆcα) critical value is computed as the (1 − α)-th quantile of the normal distribution described in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Precisely speaking, ˆcα = G−1 H0(α), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='11) 17 where GH0 is the cumulative distribution function of √nhnT G,L n (see Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4 for the explicit form of GH0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In the study, we consider G = 20 and L = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Now, for Examples 1, 2 and 3, the approximated asymptotic powers of the test based on Tn are illustrated in the diagrams in Figure 1 for various choices of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In general, it is observed from all three diagrams that the asymptotic power of the test based on Tn increases as λ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The increasing feature of asymptotic power with respect to λ for all three examples has been seen in view of the fact that X and Y are not independent random elements and as the value of T is deviating more from the null hypothesis as λ is deviating from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Moreover, among Examples 1, 2 and 3, observe that the asymptotic powers for Examples 2 and 3 are more than that for Example 1 since in Examples 2 and 3, Y (t) at various time points t ∈ [0, 1] are correlated whereas in Example 1, Y (t) at various time points t ∈ [0, 1] are uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 3 Finite Sample Level and Power Study The asymptotic power study in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 indicates that the power of the test based on Tn is fairly good when the sample size is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Besides, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4 asserts the consistency of the test, which also indicates that the proposed test will perform well in terms of power for a sufficiently large sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Overall, these facts only guarantee the expected good performance of the proposed test when the sample size is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Here, we now want to see the performance of the proposed test when the sample size is small or moderately small/large, and in order to carry out this study, one needs to compute Tn for a given data, which is described in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 Computation of Tn Observe that the exact distribution of Tn is intractable because of its complex structure, and for that reason, one cannot compute the critical value from the inverse of the exact dis- tribution of Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In order to overcome this issue, one requires to approximate the sampling distribution of Tn using Monte-Carlo simulation, and in this procedure, the computation of Tn for a given sample is requisite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Further, for the same reason provided above, one 18 needs to compute Tn for a given sample for computing the power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Recall Tn from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='7) : Tn = sup θj,i∈(− π 2 , π 2 ) j=1,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',M−2 θj,M−1∈(−π,π) j=1,2 s∈R,t∈R ����� 1 nh 3 2n n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn � − 1 n2h2n n� i,j=1 k1 � ⟨ˆlK 1 ,Xi⟩H−s hn � k2 � ⟨ˆlK 2 ,Yj⟩H−t hn ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Note that in the expression in Tn, the supremum is taken over finitely many choices of θj,i (j = 1 and 2, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , M −1), which are involved in ˆlK 1 and ˆlK 2 and over s ∈ R and t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' As R in an interval with infinite length, the computation of supremum on s and t over R becomes unmanageable, and in order to overcome this issue, we take a sufficiently large interval [−G, G] and do the grid search on {s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , sL} and {t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , tL}, where si ∈ [−G, G] and ti ∈ [−G, G] (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , L) are equally spaced points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' This approximation procedure makes the computation doable in practice since domain of both variables s and t become finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Therefore, in practice, we will compute the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' T G,L n = sup θj,i∈(− π 2 , π 2 ) j=1,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',M−1 θj,M−2∈(−π,π) j=1,2 s∈{s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',sL} t∈{t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',tL} ����� 1 nh 3 2n n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn � − 1 n2h2 n n � i,j=1 k1 � ⟨ˆlK 1 , Xi⟩H − s hn � k2 � ⟨ˆlK 2 , Yj⟩H − t hn ������ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='12) where G ∈ R and L ∈ Z+ are sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 along with the first part of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 guide our choice of T G,L n as an approximation of Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 For all n ≥ 1, under (A5) and (A6), lim G→∞ lim L→∞(Tn − T G,L n ) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' = 0, where Tn is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='7), and T G,L n is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 Estimated Level and Power Here, we discuss how one can estimate the level and the power of the test based on Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In this study, we choose M = 10 and K = 20 as we have observed that any larger values than the aforementioned chosen values does not much change the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' For details about the variables M and K, recall Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3 and the discussion before the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Besides, as Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 indicates, the choices of G and L are also an issue of concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' After having a thorough investigation, it is found that G = 20 and L = 10 are reasonably good choices in terms of computational complexity and the precision of this finite approx- imation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Throughout the study, we consider k1 and k2 in the expression in Tn (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='7)) as Epanechnikov kernel (see Silverman (1998)) for its optimal property, and the bivariate kernel k is considered as the two folds product of Epanechnikov kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' As the bandwidth, we choose hn = cn− 1 6, where c is estimated by cross validation technique (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', Duong and Hazelton (2005) and Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (1992)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In the course of this study, the experiments are carried out for n = 20, 50 and 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In order to carry out the critical value of the test, we generate two independent data sets X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , Xn) and Y = (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , Yn) from standard Brownian motion, and replicate this experiment 1000 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Note that as X and Y are independent, the null hypothesis H0 : X ⊥ Y (⇔ T = 0) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Now, let Tn,i be the value of Tn for the i-th replicate (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , 1000), and these 1000 values of Tn,i provide the estimated distribution of Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Therefore, (1 − α)-th (α ∈ (0, 1)) quantile of Tn,i (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , 1000), denoted as ˆcα can be considered as the estimated critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Now, in order to estimate the level of significance, we generate two independent data sets, namely, X ′ = (X ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , X ′ n) and Y ′ = (Y ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , Y ′ n) and replicate this experiment M times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Suppose that Tn,i is the value of Tn for the i-th replication, and the estimated level is defined as 1 M M � i=1 1(Tn,i>ˆcα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Next, in order to estimate the power of the test, we generate two dependent data sets, namely, X ′′ = (X ′′ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , X ′′ n) and Y ′′ = (Y ′′ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , Y ′′ n ) and replicate this experiment M ′ times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Suppose that Tn,j is the value of Tn for the j-th replication, and the estimated power is defined as 1 M′ M ′ � j=1 1(Tn,j>ˆcα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' To estimate the level, we consider the following three examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 20 Example 4 : Two independent data sets X = {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , Xn} and Y = {Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , Yn} are generated from standard Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Example 5 : Two independent data sets X = {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , Xn} and Y = {Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , Yn} are generated from fractional Brownian motion with Hurst index = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Example 6 : Two independent data sets X = {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , Xn} and Y = {Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , Yn} are generated from fractional Brownian motion with Hurst index = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' model n = 20 n = 50 n = 100 n = 500 Example 4 (α = 5%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='051 Example 4 (α = 10%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='107 Example 5 (α = 5%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='052 Example 5 (α = 10%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='106 Example 6 (α = 5%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='064 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='056 Example 6 (α = 10%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='111 Table 1: The estimated size of the proposed test for different sample sizes n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The levels of significance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', α are 5% and 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In order to generate the data from the fractional/standard Brownian motion, we generate the data corresponding to a certain large dimensional multivariate normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The estimated levels for Examples 4, 5 and 6 are reported in Table 1 for the sample sizes n = 20, 50, 100 and 500 at 5% and 10% level of significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The reported values indicate that the estimated level never deviated more than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5% from desired level of significance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', 5% and 10%) when the sample sizes are 20 and 50 and not deviated more than 1% when the sample sizes are 100 and 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' To estimate the power, we generate the data (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , Xn) and (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , Yn) from the distributions as described in Examples 1, 2 and 3 in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' At 5% and 10% level of significance, the estimated powers for those examples are reported in Table 2 when the sample sizes are 20, 50 and 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In all these cases, it indicates the power increases as the sample size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Besides, as we observed in the asymptotic power study (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1), the proposed test becomes more powerful when the data is generated from the distribution described in Examples 2 and 3 than that of Example 1, and this fact is expected since the correlation structure involved in the fractional Brownian motion having Hurst parameter ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 21 model n = 20 n = 50 n = 100 n = 500 Example 1 (α = 5%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='506 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='678 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='791 Example 1 (α = 10%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='399 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='601 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='882 Example 2 (α = 5%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='445 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='624 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='785 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='911 Example 2 (α = 10%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='503 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='632 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='867 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='949 Example 3 (α = 5%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='641 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='805 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='902 Example 3 (α = 10%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='517 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='666 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='841 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='956 Table 2: The estimated power of the proposed test for different sample sizes n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The level of significance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', α are 5% and 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' A few more simulation studies are carried out in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 4 Real Data Analysis Here, we implement our proposed test on a real data, which consists of daily temperature and precipitation at n = 35 locations in Canada averaged over 1960 to 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Ramsay and Silverman (2005) studied this data set in the context of functional principal component analysis, and the soft version of the data may be found in https://climate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='gc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' ca/historical_data/search_historic_data_e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The dimension of the data equals 365, which is much larger than the sample size = 35 of the data, and therefore, embedding such a high dimensional data into a specific Hilbert space is legitimate enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' For this data set, suppose that Xi (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , 35) is the average daily temperature for each day of the year at the i-th location, and Yi is the base 10 logarithm of the corresponding average precipitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Here we consider X and Y as L2([0, 1])-valued random elements, where 365 days are considered as the equally spaced 365 time points over [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The curves associated with X and Y are demonstrated in the diagrams in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In order to check whether X and Y are independent random elements or not, we compute the p-value of the test based on Tn using Bootstrap resampling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' At the beginning, to compute Tn (precisely speaking, T G,L n ), we choose tuning parameters M = 10, K = 20, G = 20 and L = 10 as this issue has been discussed at the beginning of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Similarly, we choose the bandwidth hn = cn− 1 6, where c is estimated by well-known cross validation technique (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', Duong and Hazelton (2005)), and as said before, k1 22 Figure 2: The left diagram plots the average daily temperature and the right diagram plots the average daily precipitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Source : https: // fda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' io/ en/ latest/ auto_ examples 23 Canadian Weather Arctic Atlantic Continental Pacific 20 10 temperature (C) 0 10 20 30 0 50 100 150 200 250 300 350 dayCanadian Weather 16 14 12 precipitation (mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=') 10 00 6 4 0 50 100 150 200 250 300 350 dayand k2 are chosen as Epanechnikov kernel, and the bivariate kernel k is considered as the two folds product of Epanechnikov kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Based on these aforesaid choices, we generate 500 Bootstrap resamples with size = 35 from the original data (X1, Y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , (X35, Y35), and suppose that t0 is the value of Tn for the original data (X1, Y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , (X35, Y35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Let tj (j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , 500) be the value of Tn for the j-th resample, and the p-value is defined as 1 500 500 � j=1 1(tj>t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' This methodology gives us the p-value as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='078, which indicates the data does not favour the null hypothesis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', the temperature curve and the precipitation curve in those 35 locations in Canada over the period from 1960 to 1994 are not statistically independent at 8% level of significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Even from the climate science point of view, it is expected that the temperature and the amount of precipitation are supposed to have some dependence structure, and hence, the result shown by the proposed test is reasonable enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' For this data, as the temperature and the precipitation are dependent according to our proposed test, one may be interested to know the estimated size and power of the test for this data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In order to compute estimated size and power, we combine (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , X35) and (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , Y35) and mix these 70 observations randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Let us denote the new sample as Z = (Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , Z70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Now, partition Z into two parts, namely, Z1 = (Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , Z35) and Z2 = (Z36, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , Z70), and note that Z1 and Z2 are two independent data sets as these are constructed by random mixing of the combined sample of X and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Next, from Z1 and Z2, by Bootstrap methodology, we generate M many resamples Z1,i and Z2,i (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , M) and compute the test statistic Tn for each resample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Suppose that ti is the value of Tn for the i-th resample, and then, (1 − α)-th quantile of (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , tM) is considered as the estimated critical value (denoted as ˆcα) at α% level of significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In order to estimate the size of the test, we generate M1 many resamples Z1,i and Z2,i (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , M1), and the estimated size can be defined as 1 M1 M1 � j=1 1(tj>ˆcα), where tj is the value of Tn for j-th resample (j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , M1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Next, in order to estimate the power, using Bootstrap methodology, we generate M2 many resamples from the original data (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , X35) and (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , Y35), and note that in each resample, X-data and Y -data are statistically dependent since original (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , X35) and (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , Y35) are statistically dependent data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Suppose that tk (k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , M2) is the value of Tn for the k-th resample, and the estimated power can 24 be defined as 1 M2 M1 � k=1 1(tk>ˆcα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In this study, we consider M1 = M2 = 500 and estimate the power and the size of the test when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Using all these choices, we obtain the estimated size = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='058 and the estimated power = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Overall, these facts indicate that the proposed test can achieve the nominal level of the test and poses good power when the random elements are statistically dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Two more real data are analysed using the proposed methodology in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 5 Concluding Remarks This article studies the test for independence of two random elements taking values on an infinite dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In this test, the test statistic is formulated based on the sup norm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', L∞ norm) distance between the joint probability density function of two dimen- sional certain projection of bivariate random element and product of marginal probability density functions of corresponding marginal random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The asymptotic distribu- tion of the test statistic under certain local alternatives have been derived, and it has been observed that the proposed test performs well for various choices of local alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Be- sides, the usefulness of the test is shown on well-known data sets, and simulation studies also indicate that the proposed test performs well under different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' It follows from the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 that the main crux of the proof is the derivation of the pointwise asymptotic properties of ����� 1 nh 3 2n n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn � − 1 n2h2 n n � i,j=1 k1 � ⟨ˆlK 1 , Xi⟩H − s hn � k2 � ⟨ˆlK 2 , Yj⟩H − t hn ������ , and afterwards, the asymptotic distribution of T G,L n follows from some arguments related continuous mapping theorem (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', Serfling (1980)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In this context, we would like to point out that one may consider any other appropriate norm like Lp norm, when p ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The study of performance of the test based on Lp norm may be of interest for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Another issue may need to be discussed, which is related to proposed criterion T (see 25 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' One may argue whether T can be thought of as a measure of association or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Note that under some technical conditions, one can argue that T ≤ M, where M > 0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Hence, it may be appropriate to consider T ′ = T M as a measure of association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Observe that T ′ ∈ [0, 1], and for any two random elements X and Y , T ′(X, Y ) = 0 if and only if X and Y are independent random elements, which follows from the assertion in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In order to establish T ′ as a measure of association, one needs to characterize the case when T ′(X, Y ) = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', in other words, the readers may be interested to know the perfect dependence structure between the random elements X and Y , which is not done in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The characterization of perfect dependence between X and Y when T ′(X, Y ) = 1 may be of another interest for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Acknowldgement: The first author acknowledges the research grant DST/INSPIRE/04/2017/002835, Government of India, and the second author is thankful to the research grants MTR/2019/000039 and CRG/2022/001489, Government of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 6 Appendix A : Technical Details Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1: Let X and Y be independent, and fix l1 and l2 ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Observe that the mappings ⟨l1 , ·⟩H : H → R and ⟨l2 , ·⟩H : H → R defined by x �→ ⟨l1 , x⟩H and x �→ ⟨l2 , x⟩H are measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Now, for all Borel subsets A and B of R, we have P(⟨l1 , X⟩H ∈ A, ⟨l2 , Y ⟩H ∈ B) = P(X ∈ ⟨l1 , ·⟩−1 H (A), Y ∈ ⟨l1 , ·⟩−1 H (B)) = P(X ∈ ⟨l1 , ·⟩−1 H (A))P(Y ∈ ⟨l1 , ·⟩−1 H (B)) (since X and Y are independent) = P(⟨l1 , X⟩H ∈ A)P(⟨l2 , Y ⟩H ∈ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Therefore, the two random variables ⟨l1 , X⟩H and ⟨l2 , Y ⟩H are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Conversely, suppose that ⟨l1 , X⟩H and ⟨l2 , Y ⟩H are independent for all l1 and l2 ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 26 Since, H is separable, consider an orthonormal basis {en : n = 1, 2, · · · }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Observe that X H= ∞ � n=1 ⟨X , en⟩H en, Y H= ∞ � n=1 ⟨Y , en⟩H en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Here A H= B denotes ||A − B||H = 0 for any A ∈ H and B ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Since the two families of real valued random variables {⟨X , en⟩H : n = 1, 2, · · · } and {⟨Y , en⟩H : n = 1, 2, · · · } are independent, so are X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' □ The following lemma is useful in proving Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 Let (Z, W) denote an R2 valued random vector, and fix non-zero real num- bers α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then, (Z, W) is absolutely continuous if and only if (αZ, βW) is absolutely continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Moreover, Z and W are independent if and only if αZ and βW are indepen- dent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1: Consider the transformation (x, y)t ∈ R2 �→ (αx, βy)t ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Observe that the transformation is one-to-one and onto, and the Jacobian of the transfor- mation is a non-zero constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Hence, the first statement related to absolute continuity follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' To prove the second part, let A and B be Borel subsets of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then, 1 αA (elementwise product) and 1 βB (elementwise product) are also Borel subsets of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Now, αZ and βW are independent ⇐⇒ P(αZ ∈ A, βW ∈ B) = P(αZ ∈ A) P(βW ∈ B), for all Borel subsets A, B ⇐⇒ P(Z ∈ 1 αA, W ∈ 1 βB) = P(Z ∈ 1 αA) P(W ∈ 1 βB), for all Borel subsets A, B ⇐⇒ P(Z ∈ A′, W ∈ B′) = P(Z ∈ A′) P(W ∈ B′), for all Borel subsets A′, B′ ⇐⇒ Z and W are independent This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' □ Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1: Let X and Y be independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1, ⟨l1, X⟩H and ⟨l2, Y ⟩H are also independent, for all l1 and l2 ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In particular, the joint proba- bility density function f⟨l1 , X⟩H,⟨l2 , Y ⟩H equals the product of marginal probability density 27 functions f⟨l1 , X⟩H and f⟨l2 , Y ⟩H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Hence, T(R1, R2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Conversely, suppose that T(R1, R2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then, for all l1 and l2 ∈ H with ∥l1∥H ≤ R1 and ∥l2∥H ≤ R2, the joint probability density function f⟨l1 , X⟩H,⟨l2 , Y ⟩H equals the product of marginal probability density functions f⟨l1 , X⟩H and f⟨l2 , Y ⟩H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Therefore, ⟨l1, X⟩H and ⟨l2, Y ⟩H are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Now, consider ¯l1, ¯l2 ∈ H with ∥¯l1∥H > R1 and ∥¯l2∥H > R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Since ��� R1 ∥¯l1∥H¯l1 ��� H = R1 and ��� R2 ∥¯l2∥H¯l2 ��� H = R2, independence of ⟨ R1 ∥¯l1∥H¯l1, X⟩H and ⟨ R2 ∥¯l2∥H¯l2, Y ⟩H, which follows from the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Finally, the independence of ⟨¯l1, X⟩H and ⟨¯l2, Y ⟩H follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The argument is completed by applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' □ Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2: Suppose that T(R1, R2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' As argued in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1, we have for all l1 and l2 ∈ H with ∥l1∥H ≤ R1 and ∥l2∥H ≤ R2, the real valued random variables ⟨l1, X⟩H and ⟨l2, Y ⟩H are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Let ¯l1 and ¯l2 ∈ H with ∥¯l1∥H = 1 and ∥¯l2∥H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then, ∥R1¯l1∥H = R1 and ∥R2¯l2∥H = R2 and consequently, the independence of ⟨R1¯l1, X⟩H and ⟨R2¯l2, Y ⟩H follows from assertion in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then, the independence of ⟨¯l1, X⟩H and ⟨¯l2, Y ⟩H fol- lows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In particular, the joint probability density function f⟨¯l1 , X⟩H,⟨¯l2 , Y ⟩H equals the product of marginal probability density functions f⟨¯l1 , X⟩H and f⟨¯l2 , Y ⟩H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Since ¯l1 and ¯l2 are arbitrary, we have T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Conversely, suppose that T = 0, which implies that the independence of ⟨l1, X⟩H and ⟨l2, Y ⟩H with ∥l1∥H = 1 and ∥l2∥H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Hence, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1, ⟨G1l1, X⟩H and ⟨G2l2, Y ⟩H are independent random variables, where G1 ≤ R1 and G2 ≤ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then, for all l1 and l2 ∈ H with ∥l1∥H ≤ R1 and ∥l2∥H ≤ R2, the joint probability density function f⟨l1 , X⟩H,⟨l2 , Y ⟩H equals the product of marginal probability density functions f⟨l1 , X⟩H and f⟨l2 , Y ⟩H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Hence, T(R1, R2) = 0, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' □ Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1: Note that Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 implies that T(R1, R1) = 0 if and only if X and Y are independent random elements, and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 implies that T(R1, R2) = 0 ⇔ T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Hence, both facts together imply that T = 0 if and only if X and Y are independent random elements, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' □ Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3: For j = 1 and 2, we write l(M) j = M � i=1 lj,iei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Since, lj = ∞ � i=1 lj,iei 28 and ∥lj∥2 H = ∞ � i=1 l2 j,i = 1, we have lim M→∞ ���lj − l(M) j ��� 2 H = lim M→∞ ∞ � i=M+1 l2 j,i = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='13a) and ∥l(M) j ∥H ↑ ∥lj∥H = 1, as M → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='13b) Now ���lj − ˆlK j ��� H = ����� � lj − l(M) j � + � 1 − 1 ∥l(M) j ∥H � l(M) j + � 1 ∥l(M) j ∥H l(M) j − ˆlK j ������ H ≤ ���lj − l(M) j ��� H + �����1 − 1 ∥l(M) j ∥H ����� ���l(M) j ��� H + ����� 1 ∥l(M) j ∥H l(M) j − ˆlK j ����� H ≤ ���lj − l(M) j ��� H + �����1 − 1 ∥l(M) j ∥H ����� + ����� 1 ∥l(M) j ∥H l(M) j − ˆlK j ����� H (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='14) Note that ����� 1 ∥l(M) j ∥H l(M) j − ˆlK j ����� H = ����� � lj,1 ∥l(M) j ∥H , lj,2 ∥l(M) j ∥H , · · · , lj,M ∥l(M) j ∥H � − (l(K) j,1 , l(K) j,2 , · · · , l(K) j,M) ����� RM , with the vector � lj,1 ∥l(M) j ∥H, lj,2 ∥l(M) j ∥H, · · · , lj,M ∥l(M) j ∥H � in the unit sphere of RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Therefore, by choosing large K and appropriate ˆlK j , the term ���� 1 ∥l(M) j ∥Hl(M) j − ˆlK j ���� H can be made arbitrarily small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The proof then follows by using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='13a) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='13b) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' □ The following Lemmas are useful in proving Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 Under (A1)–(A4) and (A6), (I) := � nhn � 1 nh 3 2n � n � i=1 k �Z1,i − s hn , Z2,i − t hn �� − fZ1,Z2(s, t) � converges weakly to a random variable associated with Gaussian distribution with mean 29 = 0 and variance = fZ1,Z2(s, t) � k2(m1, m2)dm1dm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2: Observe that � nhn � 1 nh 3 2n n � i=1 k �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − t hn � − fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t) � = � nhn � 1 nh 3 2n n � i=1 k �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − t hn � − E � 1 nh 3 2n n � i=1 k �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − t hn ��� + � nhn � E � 1 nh 3 2n n � i=1 k �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − t hn �� − fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t) � = (I)A + (I)B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' where (I)A := � nhn � 1 nh 3 2n n � i=1 k �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − t hn � − E � 1 nh 3 2n n � i=1 k �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − t hn ��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='15) and (I)B := � nhn � E � 1 nh 3 2n n � i=1 k �Z1,i − s hn , Z2,i − t hn �� − fZ1,Z2(s, t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='16) □ Let us now work on (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' � nhn � E � 1 nh 3 2n n � i=1 k �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − t hn �� − fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t) � = � nhn � E � 1 h 3 2n k �Z1 − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2 − t hn �� − fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t) � = � nhn 1 h 3 2n �� k �z1 − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' z2 − s hn � fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' z2)dz1dz2 − fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t) � = � nh2 n �� k (m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' m2) fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s + m1hn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t + m2hn)dm1dm2 − fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t) � 30 = � nh2 n �� k (m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' m2) � fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t) + hn 2 � i=1 �∂fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(ξn) ∂Zi � mi � dm1dm2 − fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t) � = √nhnhn � 2 � i=1 �∂fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(ξn) ∂Zi � � mik(m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' m2)dm1dm2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The last fact follows from (A6), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' � k(m1, m2)dm1dm2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Now, using (A1), (A2) and (A6) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' hn → 0 as n → ∞, √nhn → c as n → ∞, the partial derivatives of fZ1,Z2(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=') are uniformly bounded and � mik(m1, m2)dm1dm2 < ∞), we have (I)B → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='17) Now, we work on (I)A (see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='15)) and denote Wn,i = � nhn � 1 nh 3 2n k �Z1,i − s hn , Z2,i − t hn � − E � 1 nh 3 2n k �Z1 − s hn , Z2 − t hn ��� , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , n Note that E(Wn,i) = 0 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Observe that W 2 n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i = 1 nh2 n � k2 �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − t hn � + � E � k �Z1 − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2 − t hn ���2 −2k �Z1 − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2 − t hn � E � k �Z1 − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2 − t hn ��� ⇒ E(W 2 n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i) = 1 nh2 n � E � k2 �Z1 − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2 − t hn �� − � E � k �Z1 − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2 − t hn ���2� = 1 nh2 n �� � k2 �z1 − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' z2 − t hn �� fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' z2)dz1dz2 − � � k �z1 − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' z2 − t hn � fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' z2)dz1dz2 �2� = 1 n �� k2(m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' m2)fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s + m1hn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t + m2hn) � −h2 n n � {k(m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' m2)fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s + m1hn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t + m2hn)}2 dm1dm2 ⇒ n � i=1 E(W 2 n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i) = �� k2(m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' m2)fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s + m1hn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t + m2hn) � 31 −h2 n � {k(m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' m2)fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s + m1hn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t + m2hn)}2 dm1dm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Hence, using (A1), (A2) and (A6) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', hn → 0 as n → ∞, partial derivatives of the joint density function of (Z1, Z2) exist, and � k2(m1, m2)dm1dm2 < ∞, � m1k(m1, m2)dm1dm2 < ∞ and � m2k(m1, m2)dm1dm2 < ∞), we have n � i=1 E(W 2 n,i) → fZ1,Z2(s, t) � k2(m1, m2)dm1dm2 := σ2 1 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='18) Further, it follows from the expression of Wn,i and (I)A (see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='15)) that (I)A = n� i=1 Wn,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Therefore, in order to prove the asymptotic normality of (I)A, the validation of Lyapunov condition (see Serfling (1980)) is established here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' For some δ > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' let us consider 0 ≤ 1 σ2+δ 1 n � i=1 E|Wn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i|2+δ ≤ 1 σ2+δ 1 n � i=1 E ���� 1 √nhn k �Z1 − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2 − t hn ����� 2+δ ≤ 1 σ2+δ 1 × 1 n δ 2h2+δ n E ����k �Z1 − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2 − t hn ����� 2+δ ≤ 1 σ2+δ 1 × 1 n δ 2h2+δ n ����E � k �Z1 − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2 − t hn ������ 2+δ = 1 σ2+δ 1 × 1 n δ 2h2+δ n ���� � k �z1 − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' z2 − t hn � fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' z2)(z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' z2)dz1dz2 ���� 2+δ = 1 σ2+δ 1 × h4+2δ n n δ 2h2+δ n ���� � k (m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' m2) fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s + m1hn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t + m2hn)dm1dm2 ���� 2+δ = 1 σ2+δ 1 × h2+δ n n δ 2 ���� � k (m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' m2) fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s + m1hn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t + m2hn)dm1dm2 ���� 2+δ → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The last fact follows from the fact that hn → 0 as n → ∞, and in view of the existence of the partial derivatives of the joint density function of (Z1, Z2), and � k(m1, m2)dm1dm2 < ∞, � m1k(m1, m2)dm1dm2 < ∞ and � m2k(m1, m2)dm1dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Hence, (I)A converges weakly to a random variable associated with Gaussian distribution with mean = 0 and variance = fZ1,Z2(s, t) � k2(m1, m2)dm1dm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Further, recall that (I) = (I)A+(I)B, and it is established that (I)B → 0 as n → ∞ (see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='17)), and consequently, (I) converges weakly to a random variable associated with Gaussian distribution with mean = 0 and variance = 32 fZ1,Z2(s, t) � k2(m1, m2)dm1dm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3 Under (A1)–(A5), (II) := � nhn � 1 nhn n � i=1 k1 �Z1,i − s hn � � 1 nhn n � j=1 k2 �Z2,j − t hn � − fZ2(t) �� converges weakly to a Gaussian distribution with mean = √cfZ1(s)f ′ Z2(t) � mk2(m)dm and variance = {fZ1(s)}2fZ2(t) � {k2(m)}2dm, where c = lim n→∞ nh2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3: Note that (II) = (II)A × (II)B, where (II)A := 1 nhn n � i=1 k1 �Z1,i − s hn � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='19) and (II)B := � nhn �� 1 nhn n � j=1 k2 �Z2,j − t hn � − fZ2(t) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='20) It now follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='19) that E((II)A) = E � 1 nhn n � i=1 k1 �Z1,i − s hn �� = 1 nhn n � i=1 E � k1 �Z1,i − s hn �� = 1 hn E � k1 �Z1 − s hn �� = 1 hn � � k1 �Z1 − s hn �� fZ1(z1)dz1 = � k(m)fZ1(mhn + s)dm = � k(m) � fZ1(s) + mhnfZ′ 1(ξn) � dm (f ′ Z1 denotes the derivative of fZ1, and ξn ∈ (s, s + mhn)) = fZ1(s) � k(m)dm + hnf ′ Z1(ξn) � mk(m)dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Now, using conditions (A1), (A4) and (A5) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', � mk(m)dm = 1, hn → 0 as n → ∞, f ′ Z1(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=') is uniformly bounded, and � mk(m)dm < ∞), we have E((II)A) → fZ1(s) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='21) 33 Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' note that 0 ≤ var � 1 nhn n � i=1 k1 �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn �� = 1 n2h2 n var � n � i=1 k1 �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn �� + 1 n2h2 n n � i̸=j=1 cov � k1 �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' k1 �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='j − s hn �� = 1 n2h2 n var � n � i=1 k1 �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn �� (since Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i ⊥ Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='j for i ̸= j) = 1 nh2 n E � k2 1 �Z1 − s hn �� − 1 nh2 n � E � k2 1 �Z1 − s hn ���2 ≤ 1 nh2 n E � k2 1 �Z1 − s hn �� = 1 nh2 n � k2 1 �Z1 − s hn � fZ1(z1)dz1 = 1 nhn � k2 1(m)fZ1(s + mhn)dm = 1 nhn � k2 1(m) � fZ1(s) + mhnf ′ Z1(ξn) � dm (here ξn ∈ (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' s + mhn)) = fZ1(s) nhn � k2 1(m)dm + f ′ Z1(ξn) n � mk2 1(m)dm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Now using (A1), (A3), (A4) and (A5) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', nhn → ∞ as n → ∞, fZ1(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=') and f ′ Z1(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=') are uniformly bounded, � k2 1(m)dm < ∞ and � mk2 1(m)dm < ∞), we have var � 1 nhn n � i=1 k1 �Z1,i − s hn �� → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='22) Hence, using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='21) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='22), we have (II)A p→ fZ1(s) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='23) Now, let us work on (II)B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='19) indicates that (II)B : = � nhn �� 1 nhn n � j=1 k2 �Z2,j − t hn � − fZ2(t) �� 34 = � nhn �� 1 nhn n � j=1 k2 �Z2,j − t hn � − E � 1 nhn n � j=1 k2 �Z2,j − t hn ���� + � nhn � E � 1 nhn n � j=1 k2 �Z2,j − t hn �� − fZ2(t) � = (II)B,1 + (II)B,2, where (II)B,1 := � nhn �� 1 nhn n � j=1 k2 �Z2,j − t hn � − E � 1 nhn n � j=1 k2 �Z2,j − t hn ���� (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='24) and (II)B,2 := � nhn � E � 1 nhn n � j=1 k2 �Z2,j − t hn �� − fZ2(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='25) It now follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='25) that (II)B,2 : = � nhn � E � 1 nhn n � j=1 k2 �Z2,j − t hn �� − fZ2(t) � = � nhn � E � 1 hn k2 �Z2 − t hn �� − fZ2(t) � = � nhn � 1 hn � k2 �z2 − t hn � fZ2(z2)dz2 − fZ2(t) � = � nhn �� k2(m)fZ2(t + mhn)dm − fZ2(t) � = � nhn �� k2(m) � fZ2(t) + mhnf ′ Z2(t) + m2h2 n 2 f ′′ Z2(ξn) � − fZ2(t) � (f ′′ Z2 denotes the second derivative of fZ2), and ξn ∈ (t, t + mhn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Hence, in view of (A1), (A4) and (A5) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', hn → 0 as n → ∞, nh2 n = O(1), � k2(m)dm = 1, � mk2(m)dm < ∞, f ′ Z2 and f ′′ Z2 are uniformly bounded), we have (II)B,2 − √nhnf ′ Z2(t) � mk2(m)dm → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='26) 35 Now, to study (II)B,1 (see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='24)), let us denote Vn,j = 1 √nhn � k2 �Z2,j − t hn � − E � k2 �Z2 − t hn ��� , j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Now, observe that E(Vn,j) = 0 for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' and V 2 n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='j = 1 nhn � k2 2 �Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='j − t hn � + � E � k2 �Z2 − t hn ���2 − 2 nhn k2 �Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='j − t hn � E � k2 �Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='j − t hn ��� ⇒ E(V 2 n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='j) = 1 nhn E � k2 2 �Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='j − t hn �� − 1 nhn � E � k2 �Z2 − t hn ���2 = 1 nhn � k2 2 �z2 − t hn � fZ2(z2)dz2 − 1 nhn �� k2 �z2 − t hn � fZ2(z2)dz2 �2 = 1 nhn ��� k2 2(m)fZ2(t + mhn)dm � hn − �� k2(m)hnfZ2(t + mhn)dm �2� = � k2 2(m)fZ2(t + mhn)dm − hn n �� k2(m)hnfZ2(t + mhn)dm �2 ⇒ n � j=1 E(V 2 n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='j) = � k2 2(m)fZ2(t + mhn)dm − hn �� k2(m)hnfZ2(t + mhn)dm �2 = � k2 2(m) � fZ2(t) + mhnf ′ Z2(ξn) � dm − hn �� k2(m)hn � fZ2(t) + mhnf ′ Z2(ξn) � dm �2 (here ξn ∈ (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t + mhn)) Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' using (A1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (A3) and (A4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', hn → 0 as n → ∞, f ′ Z2 is uniformly bounded, � k2(m)dm < ∞ and � mk2(m)dm < ∞), we have n � j=1 E(V 2 n,j) → fZ2(t) � {k2(m)}2dm as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='27) Let us now denote σ2 := fZ2(t) � {k2(m)}2 (see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='27)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In order to now establish the 36 asymptotic normality of (II)B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' we now consider for some δ > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 1 σ2+δ n � j=1 E|Vn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='j|2+δ ≤ 1 σ2+δ E ���� 1 √nhn k2 �Z2 − t hn ����� 2+δ ≤ 1 σ2+δ ���� 1 √nhn E � k2 �Z2 − t hn ������ 2+δ = 1 σ2+δ(nhn)1+ δ 2 ���� � k2 �z2 − t hn � fZ2(z2)dz2 ���� 2+δ = 1 σ2+δ(nhn)1+ δ 2 ���� � k2(m)fZ2(t + mhn)hndm ���� 2+δ = h2+δ n σ2+δ(nhn)1+ δ 2 ���� � k2(m) � fZ2(t) + mhnf ′ Z2(ξn) � dm ���� 2+δ Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' using (A1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (A4) and (A5) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', hn → 0 as n → ∞, nhn → ∞ as n → ∞, � k2(m)dm < ∞ and � mk2(m)dm < ∞), we have 1 σ2+δ n � j=1 E|Vn,j|2+δ → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='28) Therefore, it follows from Lyapunov CLT (see Serfling (1980)) that (II)B,1 converges weakly to a random variable associate with Gaussian distribution with mean = 0 and variance = σ2 := fZ2(t) � {k2(m)}2dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' As (II)B = (II)B,1 + (II)B,2, the aforementioned fact along with in view of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='26), one can conclude that (II)B − √nhnf ′ Z2(t) � mk2(m)dm converges weakly to a random variable associated with Gaussian distribution with mean = 0 and variance = σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Finally, using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='23) and aforesaid weak convergence of (II)B along with the fact that (II) = (II)A × (II)B, one can conclude that (II) converges weakly to a Gaussian distribution with mean = √cfZ1(s)f ′ Z2(t) � mk2(m)dm and variance = {fZ1(s)}2fZ2(t) � {k2(m)}2dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4 Under (A1)–(A5), (III) := √nhn � fZ2(t) � 1 nhn n� i=1 k1 � Z1,i−s hn � − fZ1(s) �� converges weakly to a Gaussian distribution with mean = √cfZ2(t)f ′ Z1(s) � mk1(m)dm and variance = {fZ2(t)}2fZ1(s) � {k1(m)}2dm, where c = lim n→∞ nh2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 37 Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4: The proof of this lemma follows from the same arguments provided in the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5 Let us denote Mn(s, t) = 1 nh 3 2n � n � i=1 k �Z1,i − s hn , Z2,i − t hn �� − � 1 n2h2 n n � i,j=1 k1 �Z1,i − s hn � k2 �Z2,j − t hn �� , and M(s, t) = fZ1,Z2(s, t) − fZ1(s)fZ2(t), where fZ1,Z2, fZ1 and fZ2 are probability density functions of (Z1, Z2), Z1 and Z2, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then, for a fixed (s, t), under (A1)–(A6), √nhn(Mn(s, t) − M(s, t)) con- verges weakly to a random variable associated with Gaussian distribution with mean = −√c � fZ1(s)f ′ Z2(t) � mk2(m)dm + fZ2(t)f ′ Z1(s) � mk1(m)dm � and variance = fZ1,Z2(s, t) � {k(m1, m2)}2dm1dm2 + {fZ1(s)}2fZ2(t) � {k2(m)}2dm + {fZ2(t)}2fZ1(s) � {k1(m)}2dm, where c = lim n→∞ nh2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5: Observe that � nhn(Mn(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t) − M(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t)) = � nhn � 1 nh 3 2n � n � i=1 k �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − t hn �� − � 1 n2h2 n n � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='j=1 k1 �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn � k2 �Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='j − t hn ��� − � nhn (fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t) − fZ1(s)fZ2(t)) = � nhn � 1 nh 3 2n � n � i=1 k �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − t hn �� − fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t) � − � nhn � 1 n2h2 n n � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='j=1 k1 �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn � k2 �Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='j − t hn � − fZ1(s)fZ2(t) � = � nhn � 1 nh 3 2n � n � i=1 k �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − t hn �� − fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t) � − � nhn � 1 nhn n � i=1 k1 �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn � � 1 nhn n � j=1 k2 �Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='j − t hn � − fZ2(t) �� 38 − � nhn � fZ2(t) � 1 nhn n � i=1 k2 �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn � − fZ1(s) �� = (I) − (II) − (III),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' where (I) := � nhn � 1 nh 3 2n � n � i=1 k �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − s hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − t hn �� − fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='29) (II) := � nhn � 1 nhn n � i=1 k1 �Z1,i − s hn � � 1 nhn n � j=1 k2 �Z2,j − t hn � − fZ2(t) �� , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='30) and (III) := � nhn � fZ2(t) � 1 nhn n � i=1 k2 �Z1,i − s hn � − fZ1(s) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='31) It follows from the assertions in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4 that (I), (II) and (III) con- verge weakly to a certain Gaussian random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In order to establish the asymptotic normality of (I) − (II) − (III), let us derive the asymptotic distribution of a1 × (I) + a2 × (II) + a3 × (III), where ai ∈ R (i = 1, 2 and 3) are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Consider a1 × (I) + a2 × (II) + a3 × (III) = a1 × {(I)A + (I)B} + a2[(II)A × {(II)B,1 + (II)B,2}] + a3 × {(III)A + (III)B} = {a1 × (I)A + a2 × (II)A × (II)B,1 + a3 × (III)A} + {a1 × (I)B + a2 × (II)A × (II)B,2 + a3 × (III)B} := C + D, where C := {a1 × (I)A + a2 × (II)A × (II)B,1 + a3 × (III)A}, D := {a1 × (I)B + a2 × (II)A × (II)B,2 + a3 × (III)B}, (I)A is defined in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='15), (I)B is defined as (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='16), (II)A is defined as (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='19), (II)B,1 is defined as (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='24), (II)B,2 is defined as (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='25), (III)A = � nhn � fZ2(t) � 1 nhn n � i=1 k1 �Z1,i − s hn � − E � 1 nhn n � i=1 k1 �Z1,i − s hn ���� , 39 and (III)B = � nhn � fZ2(t) � E � 1 nhn n � i=1 k2 �Z1,i − s hn �� − fZ1(s) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Let us first consider D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' As arguing for (II)B in the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3, (III)B → √cfZ2(t)f ′ Z1(s) � mk1(m)dm as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Along with the facts in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='17), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='23) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='26), one can conclude that D p→ √c � a2fZ1(s)f ′ Z2(t) � mk2(m)dm + a3fZ2(t)f ′ Z1(s) � mk1(m)dm � as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='32) Now, observe that C = a1 � nhn � 1 nh 3 2n n � i=1 k �Z1,i − s hn , Z2,i − t hn � − E � 1 nh 3 2n n � i=1 k �Z1,i − s hn , Z2,i − t hn ��� + a2 � nhn � 1 nhn n � i=1 k1 �Z1,i − s hn �� �� 1 nhn n � j=1 k2 �Z2,j − t hn � − E � 1 nhn n � j=1 k2 �Z2,j − t hn ���� + a3 � nhn � fZ2(t) � 1 nhn n � i=1 k1 �Z1,i − s hn � − E � 1 nhn n � i=1 k1 �Z1,i − s hn ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Let us now denote Ln,i = a1 � nhn � 1 nh 3 2n k �Z1,i − s hn , Z2,i − t hn � − E � 1 nh 3 2n k �Z1,i − s hn , Z2,i − t hn ��� + a2 � nhn � 1 nhn k1 �Z1,i − s hn �� �� 1 nhn k2 �Z2,j − t hn � − E � 1 nhn k2 �Z2,j − t hn ���� + a3 � nhn � fZ2(t) � 1 nhn k1 �Z1,i − s hn � − E � 1 nhn k1 �Z1,i − s hn ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Note that C = n� i=1 Ln,i, and E(Ln,i) = 0 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Further note that n � i=1 E(L2 n,i) → a2 1fZ1,Z2(s, t) � k2(m1, m2)dm1dm2 + a2 2{fZ1(s)}2fZ2(t) � {k2(m)}2dm + a2 3{fZ2(t)}2fZ1(s) � {k1(m)}2dm := σ2 2 40 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='33) The last fact follows from the assertions in Lemmas 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4, and in view of the independence between the random variables Z1 and Z2, and Covariance ((I)A, (II)A × (II)B,2) → 0 as n → ∞ and Covariance ((I)A, (II)A × (III)A) → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Now, in order to show the asymptotic normality of C = n� i=1 Ln,i, for some δ > 0, we consider 1 σ2+δ 2 n � i=1 E|Ln,i|2+δ = 1 σ2+δ 2 n � i=1 E|a1Wn,i + a2 1 nhn k1 �Z1,i − s hn � Vn,i + a3Bn,i|2+δ ≤ 1 σ2+δ 2 � |a1|2+δ n � i=1 E|Wn,i|2+δ + |a2|2+δ n � i=1 fZ1(s)E|Vn,i|2+δ + |a3|2+δ n � i=1 E|Bn,i|2+δ � → 0 as n → ∞, where Wn,i is defined in the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2, Vn,i is defined in the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3, and Bn,i = � nhn � fZ2(t) � 1 nhn k1 �Z1,i − s hn � − E � 1 nhn k1 �Z1,i − s hn ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The last limiting fact follows from the fact that n� i=1 E|Wn,i|2+δ → 0 as n → ∞ (see the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2), n� i=1 E|Vn,i|2+δ → 0 as n → ∞ (see the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3), and n� i=1 E|Bn,i|2+δ → 0 as n → ∞ (exactly the same proof as it is for Vn,i in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Therefore, C converges weakly to a random variable associated with Gaussian dis- tribution with mean = 0 and variance = σ2 2 (see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='33) for the explicit expression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Next, since a1 × (I) + a2 × (II) + a3 × (III) = C + D, a1 × (I) + a2 × (II) + a3 × (III) converges weakly to a random variable associate with Gaussian distribution with mean = √c � a2fZ1(s)f ′ Z2(t) � mk2(m)dm + a3fZ2(t)f ′ Z1(s) � mk1(m)dm � and variance = σ2 2 in view of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='32) using an application of Slutsky’s theorem (see Serfling (1980)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Fi- nally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' as a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' a2 and a3 are arbitrary constants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' using Cramer-Wold device (see Serfling (1980)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' one can conclude that ((I),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (II),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (III)) converges weakly to a certain trivari- 41 ate Gaussian distribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' and hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' √nhn(Mn(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t) − M(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t)) = (I) − (II) − (III) converges weakly to a random variable associated with Gaussian distribution with mean = −√c � fZ1(s)f ′ Z2(t) � mk2(m)dm + fZ2(t)f ′ Z1(s) � mk1(m)dm � and variance = fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' t) � {k(m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' m2)}2dm1dm2 + {fZ1(s)}2fZ2(t) � {k2(m)}2dm + {fZ2(t)}2fZ1(s) � {k1(m)}2dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='6 Under (A1), (A5) and (A6), √nhn � lim G→∞ lim L→∞(Tn − T G,L n ) � p→ 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='6: Recall Tn from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='7) : Tn = sup θj,i∈(− π 2 , π 2 ) j=1,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',M−2 θj,M−1∈(−π,π) j=1,2 s∈R,t∈R ����� 1 nh 3 2n n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn � − 1 n2h2 n n � i,j=1 k1 � ⟨ˆlK 1 , Xi⟩H − s hn � k2 � ⟨ˆlK 2 , Yj⟩H − t hn ������ , and recall from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='12) that T G,L n = sup θj,i∈(− π 2 , π 2 ) j=1,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',M−1 θj,M−2∈(−π,π) j=1,2 s∈{s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',sL} t∈{t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',tL} ����� 1 nh 3 2n n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn � − 1 n2h2 n n � i,j=1 k1 � ⟨ˆlK 1 , Xi⟩H − s hn � k2 � ⟨ˆlK 2 , Yj⟩H − t hn ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Note that it follows from the assertion in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 that sup s∈{s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',sL} t∈{t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',tL} n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn � −sup s∈R t∈R n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn � → 0 almost surely as G → ∞ and L → ∞ (iteratively) for any n ≥ 1, where si ∈ [−G, G] and 42 ti ∈ [−G, G] for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' , L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' This implies that E � �� sup s∈{s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',sL} t∈{t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',tL} n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn � − sup s∈R t∈R n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn �� �� → 0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='34) for any n ≥ 1 as G → ∞ and L → ∞ (iteratively), which follows from dominated convergence theorem (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', Serfling (1980)) because � �� sup s∈{s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',sL} t∈{t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',tL} n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn � − sup s∈R t∈R n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn �� �� is bounded for all G and L and any n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Hence, E � �� sup s∈{s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',sL} t∈{t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',tL} √nhn nh 3 2n n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn � − sup s∈R t∈R √nhn nh 3 2n n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn �� � → 0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='35) as n → ∞ (along with G → ∞ and L → ∞, iteratively) in view of (A1) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', √nhn → √c as n → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Arguing exactly in a similar way, one can conclude that E � �� sup s∈{s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',sL} t∈{t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',tL} √nhn nh 3 2n n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn � − sup s∈R t∈R √nhn nh 3 2n n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn �� � 2 → 0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='36) as n → ∞ (along with G → ∞ and L → ∞ (iteratively)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 43 Hence, using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='35) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='36), we have � � � � � sup s∈{s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',sL} t∈{t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',tL} √nhn nh 3 2n n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn � − sup s∈R t∈R √nhn nh 3 2n n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn �� � � p→ 0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='37) as n → ∞ (along with G → ∞ and L → ∞ (iteratively)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Similarly, one can show that √nhn n2h2 n � � � � � sup s∈{s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',sL} t∈{t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',tL} n � i=1 k1 � ⟨ˆlK 1 , Xi⟩H − s hn � k2 � ⟨ˆlK 2 , Yi⟩H − t hn � − sup s∈R t∈R n � i=1 k1 � ⟨ˆlK 1 , Xi⟩H − s hn � k2 � ⟨ˆlK 2 , Yi⟩H − t hn �� � � p→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='38) In view of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='37) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='38), we have √nhn � lim G→∞ lim L→∞(Tn − T G,L n ) � p→ 0 as n → ∞, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='7 For distinct pairs (s1, t1), · · · , (sm, tm), consider the random vector � nhn � � � � � � Mn(s1, t1) − M(s1, t1) Mn(s2, t2) − M(s2, t2) · · Mn(sm, tm) − M(sm, tm) � � � � � � where Mn(·, ·) and M(·, ·) are the same as in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then, under (A1)–(A6), the sequence of random vectors converge in distribution to an m-dimensional multivariate Nor- mal distribution with independent components, where the l-th component follows a univari- ate Normal distribution with mean = −√c � fZ1(sl)f ′ Z2(tl) � mk2(m)dm + fZ2(tl)f ′ Z1(sl) � mk1(m)dm � 44 and variance = fZ1,Z2(sl, tl) � k2(m1, m2)dm1dm2+{fZ1(sl)}2fZ2(tl) � {k2(m)}2+fZ1(sl){fZ2(tl)}2 � {k1(m)}2dm for l = 1, 2, · · · , m, where c = lim n→∞ nh2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='7: We proceed as in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Here, we fix α1, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='αm ∈ R and look at the convergence in distribution of √nhn m � l=1 αl(Mn(sl, tl) − M(sl, tl)) as n goes to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Observe that � nhn m � l=1 αl(Mn(sl, tl) − M(sl, tl)) = m � l=1 αl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (I)sl,tl − m � l=1 αl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (II)sl,tl − m � l=1 αl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (III)sl,tl, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='39) where (I)sl,tl = � nhn � 1 nh 3 2n � n � i=1 k �Z1,i − sl hn , Z2,i − tl hn �� − fZ1,Z2(sl, tl) � , (II)sl,tl = � nhn � 1 nhn n � i=1 k1 �Z1,i − sl hn � � 1 nhn n � j=1 k2 �Z2,j − tl hn � − fZ2(tl) �� and (III)sl,tl = � nhn � fZ2(tl) � 1 nhn n � i=1 k2 �Z1,i − sl hn � − fZ1(sl) �� for l = 1, · · · , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' We look at the joint distribution of (�m l=1 αl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (I)sl,tl, �m l=1 αl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (II)sl,tl, �m l=1 αl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (III)sl,tl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Fix real scalars a1, a2 and a3 and, consider a1 m � l=1 αl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (I)sl,tl + a2 m � l=1 αl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (II)sl,tl + a3 m � l=1 αl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (III)sl,tl = ˜Cn + ˜Dn 45 with ˜Cn := a1 n � i=1 � m � l=1 αlWn,i,sl,tl � + a2 n � i=1 � m � l=1 αl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (II)sl,tl,A Vn,i,sl,tl � + a3 n � i=1 � m � l=1 αlBn,i,sl,tl � (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='40) and ˜Dn := a1 m � l=1 αl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (I)sl,tl,B + a2 m � l=1 αl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (II)sl,tl,A × (II)sl,tl,B,2 + a3 m � l=1 αl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (III)sl,tl,B, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='41) where (I)sl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='tl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='B = � nhn � E � 1 nh 3 2n n � i=1 k �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − sl hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − tl hn �� − fZ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='Z2(sl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' tl) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (II)sl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='tl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='A = 1 nhn n � i=1 k1 �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − sl hn � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (II)sl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='tl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 = � nhn � E � 1 nhn n � j=1 k2 �Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='j − tl hn �� − fZ2(tl) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (III)sl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='tl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='B = � nhn � fZ2(tl) � E � 1 nhn n � i=1 k2 �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − sl hn �� − fZ1(sl) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Wn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='sl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='tl = � nhn � 1 nh 3 2n k �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − sl hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − tl hn � − E � 1 nh 3 2n k �Z1 − sl hn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Z2 − tl hn ��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Vn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='sl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='tl = 1 √nhn � k2 �Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − tl hn � − E � k2 �Z2 − tl hn ��� Bn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='sl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='tl = � nhn � fZ2(tl) � 1 nhn k1 �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − sl hn � − E � 1 nhn k1 �Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i − sl hn ���� for l = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' m and i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' As argued in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5, we have ˜Dn p −−−→ n→∞ √c � a2 m � l=1 αlfZ1(sl)f ′ Z2(tl) � mk2(m)dm + a3 m � l=1 αlfZ2(tl)f ′ Z1(sl) � mk1(m)dm � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 46 Now, we consider the term ˜Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' We have E ˜Cn = 0 and lim n E � ˜Cn �2 = lim n V ar � ˜Cn � = a2 1 � m � l=1 α2 l fZ1,Z2(sl, tl) � � k2(m1, m2)dm1dm2 + a2 2 � m � l=1 α2 l {fZ1(sl)}2fZ2(tl) � � {k2(m)}2dm + a2 3 � m � l=1 α2 l fZ1(sl){fZ2(tl)}2 � � {k1(m)}2dm We establish the asymptotic normality of ˜Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' For some δ > 0, repeatedly applying Jensen’s inequality, we observe that n � i=1 E ����� � a1 m � l=1 αlWn,i,sl,tl � + � a2 m � l=1 αl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (II)sl,tl,A Vn,i,sl,tl � + � a3 m � l=1 αlBn,i,sl,tl ������ 2+δ ≤31+δ n � i=1 |a1|2+δE ����� m � l=1 αlWn,i,sl,tl ����� 2+δ +31+δ n � i=1 |a2|2+δE ����� m � l=1 αl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (II)sl,tl,A Vn,i,sl,tl ����� 2+δ +31+δ n � i=1 |a3|2+δE ����� m � l=1 αlBn,i,sl,tl ����� 2+δ ≤31+δm1+δ|a1|2+δ m � l=1 |αl|2+δ n � i=1 E |Wn,i,sl,tl|2+δ +31+δm1+δ|a2|2+δ m � l=1 |αl|2+δE |(II)sl,tl,A|2+δ n � i=1 E |Vn,i,sl,tl|2+δ +31+δm1+δ|a3|2+δ m � l=1 |αl|2+δ n � i=1 E |Bn,i,sl,tl|2+δ n→∞ −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In the final step above, we argue as in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' By Slutky’s Theorem, we have the 47 aymptotic normality of ˜Cn + ˜Dn and consequently, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' □ Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2: The proof follows from the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='6 and the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='7 along with an application of continuous mapping theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' □ Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4: Observe that the asymptotic power of the test under H1 is PH1[ � nhnTn ≥ ˆcα] (since under H0, T = 0) = P[ � nhn(Tn − t0) ≥ ˆcα − � nhnt0] (assume that T = t0 > 0 under H1) = P[Z ≥ ˆcα − � nhnt0], where Z follows certain non-degenate distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Since √nhn → ∞ as n → ∞ (see condition (A1)) and t0 > 0, we have lim n→∞ P[Z ≥ ˆcα − � nhnt0] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' This completes that proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' □ The following lemma is used in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='8 Let f : R2 → R be any continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Let ∥f∥∞ denote sup (t,s)∈R2 |f(t, s)| and ∥f∥∞,G denote sup (t,s)∈[−G,G]×[−G,G] |f(t, s)|, for any G > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Given any G > 0 and any positive integer L, consider the set SG,L := {−G + i 2G L : i = 0, 1, · · · , L} of equally spaced points in the interval [−G, G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then, we have the following limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (i) For any G > 0, lim L→∞ max (t,s)∈SG,L×SG,L |f(t, s)| = ∥f∥∞,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (ii) We have lim G→∞ lim L→∞ max (t,s)∈SG,L×SG,L |f(t, s)| = lim G→∞ ∥f∥∞,G = ∥f∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 48 Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='8: Fix G > 0 and note that the continuous function f is bounded on [−G, G] × [−G, G] (see (Rudin, 1976, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='15)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Further, since the continuous function |f| on [−G, G] × [−G, G] achieves its supremum within the set (see (Rudin, 1976, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='16)), there exists (t′, s′) ∈ [−G, G] × [−G, G] such that |f(t′, s′)| = ∥f∥∞,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Since f is continuous at (t′, s′), corresponding to the above ϵ > 0, we have δ > 0 such that |f(t, s) − f(t′, s′)| < ϵ 2 whenever (t, s) ∈ [−G, G] × [−G, G] with � (t − t′)2 + (s − s′)2 < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Given a positive integer L, we can cover the set [−G, G]×[−G, G] by squares with corners from SG,L × SG,L and with lengths of a side 2G L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Here, the diagonals in these squares are of length 2 √ 2G L , and as such, any point in such a square is within a distance √ 2G L from a corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In particular, for (t′, s′), there exists (t′′, s′′) ∈ SG,L × SG,L such that � (t′′ − t′)2 + (s′′ − s′)2 ≤ √ 2G L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' We can now choose large L such that √ 2G L < δ and hence, |f(t′′, s′′) − f(t′, s′)| < ϵ 2 and in particular, |f(t′, s′)| < |f(t′′, s′′)| + ϵ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then, |f(t′′, s′′)| ≤ ∥f∥∞,G < |f(t′′, s′′)| + ϵ, with (t′′, s′′) ∈ SG,L × SG,L for sufficiently large L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Since ϵ > 0 is arbitrary, for any fixed G > 0, we have proved the first statement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', lim L→∞ max (t,s)∈SG,L×SG,L |f(t, s)| = ∥f∥∞,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' To prove the second statement, it is enough to show that limG→∞ ∥f∥∞,G = ∥f∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' First, consider the case when ∥f∥∞ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then, given ϵ > 0, there exists (t′, s′) ∈ R2 such that |f(t′, s′)| ≤ ∥f∥∞ < |f(t′, s′)| + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 49 Then, there exists G > 0 large such that (t′, s′) ∈ [−G, G] × [−G, G] and hence |f(t′, s′)| ≤ ∥f∥∞,G ≤ ∥f∥∞ < |f(t′, s′)| + ϵ ≤ ∥f∥∞,G + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Hence, we have limG→∞ ∥f∥∞,G = ∥f∥∞ when ∥f∥∞ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' When ∥f∥∞ = ∞, for any R > 0, there exists (t′, s′) ∈ R2 such that |f(t′, s′)| ≥ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then, there exists G > 0 large such that (t′, s′) ∈ [−G, G] × [−G, G] and hence R ≤ |f(t′, s′)| ≤ ∥f∥∞,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Hence, we have limG→∞ ∥f∥∞,G = ∞ = ∥f∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' □ Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1: First, we observe that given finitely many real-valued continu- ous functions f1, f2, · · · , fm on R2, the function (t, s) �→ max{f1(t, s), f2(t, s), · · · , fm(t, s)} is also continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' To see this, note that (t, s) �→ max{f1(t, s), f2(t, s)} = 1 2|f1(t, s) + f2(t, s)| − 1 2(f1(t, s) − f2(t, s)) is a continuous function on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Consequently, (t, s) �→ max{f1(t, s), f2(t, s), f3(t, s)} = max{max{f1(t, s), f2(t, s)}, f3(t, s)} is also continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' It- erating this way, we have the above observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Now, for every fixed n ≥ 1, consider the real valued continuous function (t, s) �→ max θj,i∈(− π 2 , π 2 ) j=1,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=',M−1 θj,M−2∈(−π,π) j=1,2 ����� 1 nh 3 2n n � i=1 k � ⟨ˆlK 1 , Xi⟩H − s hn , ⟨ˆlK 2 , Yi⟩H − t hn � − 1 n2h2 n n � i,j=1 k1 � ⟨ˆlK 1 , Xi⟩H − s hn � k2 � ⟨ˆlK 2 , Yj⟩H − t hn ������ on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Under (A6), the required almost sure convergence follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' □ As a consequence of assumption (A3) that the random elements X and Y are norm bounded and the joint probability density function of the random variables ⟨l1, X⟩H and ⟨l2, Y ⟩H is uniformly bounded, we show that the marginal probability density functions of 50 the random variables ⟨l1, X⟩H and ⟨l2, Y ⟩H are also bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 Suppose that the Hilbert valued random elements X and Y are essen- tially bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', there exist M1 > 0 and M2 > 0 such that P(∥X∥H ≤ M1) = 1 and P(∥Y ∥H ≤ M2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then, for all l1 and l2 ∈ H, the random variables ⟨l1 , X⟩H and ⟨l2 , Y ⟩H are also essentially bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Furthermore, in this case, if the joint probability density function f⟨l1 , X⟩H,⟨l2 , Y ⟩H is bounded, then so are the marginal probability density functions f⟨l1 , X⟩H and f⟨l2 , Y ⟩H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Proof: For x, y ∈ H, we have | ⟨x , y⟩H | ≤ ∥x∥H∥y∥H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Then, for all l1, l2 ∈ H, P(| ⟨l1 , X⟩H | ≤ ∥l1∥HM1) = 1 and P(| ⟨l2 , Y ⟩H | ≤ ∥l2∥HM2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' This proves that the random variables ⟨l1 , X⟩H and ⟨l2 , Y ⟩H are essentially bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Now, if f⟨l1 , X⟩H,⟨l2 , Y ⟩H(x, y) ≤ R for all x ∈ [−M1, M1], y ∈ [−M2, M2], then f⟨l1 , X⟩H(x) = � M2 −M2 f⟨l1 , X⟩H,⟨l2 , Y ⟩H(x, y) dy ≤ 2RM2, ∀x and f⟨l2 , Y ⟩H(y) = � M1 −M1 f⟨l1 , X⟩H,⟨l2 , Y ⟩H(x, y) dx ≤ 2RM1, ∀y This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' □ 7 Appendix B : Additional Real Data Analysis and Simulation Studies 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='1 Real Data Analysis This section consists of two more real data analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Those data sets are well-known the Coffee data and the Berkeley growth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The coffee data is available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' ucr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='edu/~eamonn/time_series_data_2018/, and it contains spectroscopy readings taken at 286 wavelength values for 14 samples of each of the two varieties of coffee, namely, Arabica and Robusta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The readings are illustrated in the diagrams in Figure 3, and each observation is viewed as an element in the separable Hilbert space L2([0, 1]) after 51 Figure 3: Spectroscopy readings of mean curve of Arabica coffee and the Robusta coffee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Source : https: // www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' mdpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' com/ 2304-8158/ 9/ 6/ 788/ htm normalization by an appropriate scaling factor as we did in the analysis of Canadian weather data in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Following the same methodology described in Section 4, we compute the p-value, and the p-value is obtained as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='069, which indicates that the data does not favour the null hypothesis at 7% level of significance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', spectroscopy readings of the Arabica coffee and the Robusta coffee have strong enough statistical dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Note that this result is not unexpected as it seems from the diagrams in Figure 3 that the curves associated with spectroscopy readings of the Arabica coffee and the Robusta coffee are not independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Now, as this data is not favouring the null hypothesis, here also, we estimate the size and power of test based on Tn for this data, and in order to carry out this study, we follow the same methodology as we did for the Canadian weather data in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Using that methodology, at 5% level of significance, the estimated size is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='058, and the estimated power is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' It further indicates the proposed test based on Tn can achieve the nominal level and perform good in terms of power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The Berkeley growth data is available at https://rdrr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='io/cran/fda/man/growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' It contains the heights of 39 boys and 54 girls measured at 31 time points between 52 Arabica Robusta 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='8 (log 1/R) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='4 Diffusereflectance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='8 999 1063 11411229 1333 1458 1607 1791 2023 2323 Wavelength (nm)Figure 4: The growth of heights : male and female.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Source : https: // fda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' io/ en/ latest/ auto_ examples the ages 1 and 18 years, and the curves are recorded at 101 equispaced ages in the interval [1, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The heights are illustrated in the diagrams in Figure 4, and each observation is viewed as an element in the separable Hilbert space L2([0, 1]) after normalization by an appropriate scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' For this data set, we obtain p-value as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='074, which indicates that the data does not favour the null hypothesis at 8% level of significance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', in other words, the heights of boys and girls have strong enough statistical dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Note that this result is not unexpected as it seems from the diagrams in Figure 4 that the height curves associated with boys and girls are not independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Now, as this data is not favouring the null hypothesis, here also, we estimate the size and power of the test based on Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' At 5% level of significance, the estimated size is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='057, and the estimated power is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='788, which further indicates that the proposed test is capable of detecting dependence structure of the random elements in real data also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 53 Berkeley Growth Study male female 200 180 160 height 140 120 100 80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='5 age7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='2 Simulation Studies Here we study the performance of the proposed test for a few more cases when the sample size is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The finite sample power of the test is estimated for the following examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Example 7 : Let (X, Y ) be L2([0, 1]) × L2([0, 1]) valued random element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Suppose that X(t) (t ∈ [0, 1]) follows a Gaussian process B(t) with E(B(t)) = 0 and E(B(t)B(s)) = min(t, s) for all t ∈ [0, 1] and s ∈ [0, 1], and Y (t) d= {X(t)}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Example 8 : Let (X, Y ) be L2([0, 1]) × L2([0, 1]) valued random element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Suppose that X(t) (t ∈ [0, 1]) follows a Gaussian process B(t) with E(B(t)) = 0 and E(B(t)B(s)) = min(t, s) for all t ∈ [0, 1] and s ∈ [0, 1], and Y (t) d= eX(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Example 9 : Let (X, Y ) be L2([0, 1]) × L2([0, 1]) valued random element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Suppose that X(t) (t ∈ [0, 1]) follows a t-process with 3 degrees of freedom, where t-process with k(k ≥ 1) degrees of freedom is defined as B(t) √ N k , where B(t) is a Gaussian process with E(B(t)) = 0 and E(B(t)B(s)) = min(t, s) for all t ∈ [0, 1] and s ∈ [0, 1] and independent of N, which follows Chi squared distribution with k degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Here, Y (t) d= {X(t)}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Example 10 : Let (X, Y ) be L2([0, 1]) × L2([0, 1]) valued random element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Suppose that X(t) (t ∈ [0, 1]) follows a t-process with 3 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Here, Y (t) d= eX(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' For all these aforementioned examples, in order to generate data from the Brownian motion, the data are generated from associated certain multivariate normal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' We here study the performance of the proposed test for the sample sizes n = 20, 50, 100 and 500, and the estimated powers of the proposed test are reported in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The results reported in Table 3 indicate that the test is more powerful when X follows t-process with 3 degrees of freedom than when X follows Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' It may be due to the fact that for a fixed t, X(t) distributed with a t-process with 3 degrees of freedom has a heavier tail than X(t) distributed with a certain Gaussian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' In terms of the relationship between Y and X, it is observed that the proposed test is more powerful when Y d= eX than when Y d= X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' It is also expected as the exponential function has a faster growth rate than polynomial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 54 model n = 20 n = 50 n = 100 n = 500 Example 7 (α = 5%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='441 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='553 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='685 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='832 Example 7 (α = 10%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='483 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='605 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='733 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='886 Example 8 (α = 5%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='459 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='601 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='824 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='954 Example 8 (α = 10%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='643 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='877 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='965 Example 9 (α = 5%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='484 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='655 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='872 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='963 Example 9 (α = 10%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='511 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='901 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='983 Example 10 (α = 5%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='502 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='713 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='907 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='986 Example 10 (α = 10%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='532 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='741 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='933 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='992 Table 3: The estimated power of the proposed test for different sample sizes n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' The level of significance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', α are 5% and 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' References Bergsma, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' and Dassios, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' A consistent test of independence based on a sign covariance related to kendall’s tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Bernoulli, 20(2):1006–1028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Berrett, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', Kontoyiannis, I.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Scandinavian Journal of Statistics, 40(1):21–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Dhar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', Bergsma, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=', and Dassios, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Asymptotic Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} +page_content=' 57' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAyT4oBgHgl3EQfhPjd/content/2301.00375v1.pdf'} diff --git a/UdAzT4oBgHgl3EQfJvt-/content/tmp_files/2301.01085v1.pdf.txt b/UdAzT4oBgHgl3EQfJvt-/content/tmp_files/2301.01085v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..98eda2bfb94ad9488d6d25c8a9886b9815179498 --- /dev/null +++ b/UdAzT4oBgHgl3EQfJvt-/content/tmp_files/2301.01085v1.pdf.txt @@ -0,0 +1,4034 @@ +arXiv:2301.01085v1 [econ.EM] 3 Jan 2023 +The Chained Difference-in-Differences +Christophe Bell´ego∗ +David Benatia† +Vincent Dortet-Bernardet‡ +January 4, 2023 +Abstract +This paper studies the identification, estimation, and inference of long-term +(binary) treatment effect parameters when balanced panel data is not avail- +able, or consists of only a subset of the available data. We develop a new +estimator: the chained difference-in-differences, which leverages the overlap- +ping structure of many unbalanced panel data sets. This approach consists in +efficiently aggregating a collection of short-term treatment effects estimated on +multiple incomplete panels. Our estimator accommodates (1) multiple time +periods, (2) variation in treatment timing, (3) treatment effect heterogeneity, +and (4) general missing data patterns. We establish the asymptotic proper- +ties of the proposed estimator and discuss identification and efficiency gains +in comparison to existing methods. Finally, we illustrate its relevance through +(i) numerical simulations, and (ii) an application about the effects of an inno- +vation policy in France. +Keywords: Difference-in-Differences, Dynamic treatment effects, Event study, +Unbalanced panel, Attrition, Treatment effect heterogeneity, GMM. +JEL Codes: C14 C2 C23 O3 J38. +∗CREST (UMR 9194), ENSAE, Institut Polytechnique de Paris, 5 Avenue Henry Le Chatelier, +91120 Palaiseau, France (e-mail: christophe.bellego@ensae.fr). +†HEC Montr´eal, D´epartement d’´Economie Appliqu´ee, 3000 Chemin de la Cˆote-Sainte-Catherine, +Montr´eal, QC H3T 2A7, Canada (Corresponding author, e-mail: david.benatia@hec.ca). +‡Direction G´en´erale des Entreprises (DGE), Minist`ere Fran¸cais de l’´Economie et des Finances +(e-mail: vincent.dortet-bernadet@finances.gouv.fr). +The authors thank Laurent Davezies, Cl´ement de Chaisemartin, Xavier D’Haultefœuille, Yan- +nick Guyonvarch, Xavier Jaravel, and all participants to seminars and conferences for insightful +discussions and comments. R packages are available upon request. + +1 +Introduction +Many public policies take years before having effects on their targeted outcomes. +For instance, most innovation policies have long run objectives such as making new +discoveries, enhancing knowledge, and increasing technology development, but in- +duce only little innovation in the short-run (Bakker, 2013; O’Connor and Rice, 2013; +Gross et al., 2018). Measuring these long-term effects is however difficult for two +principal reasons. First, identification of treatment effects from observational data +raises significant challenges, whereas randomized controlled experiments are often +too costly, or raise ethical concerns. Second, treatment effects are frequently esti- +mated from panel survey data, where the subjects of interest (e.g. individuals or +firms) are not consistently observed over the entire time frame. This problem can be +caused by attrition (Hausman and Wise, 1979), if individuals drop out of the sample, +or because of the survey design itself, if individuals are frequently replaced to prevent +attrition (Nijman et al., 1991). +In this paper, we study the identification, estimation, and inference of long-term +treatment effects in settings where balanced panel data is not available, or con- +sists of only a subset of the available data. We develop a new estimator, the chained +difference-in-differences (DiD), which leverages the overlapping structure of many un- +balanced panel data sets (Baltagi and Song, 2006). The intuition behind the chained +DiD estimator is simple. Letting Yt be an outcome of interest observed in three dis- +tinct time periods t0 < t1 < t2, the long difference Yt2−Yt0 can always be decomposed +into a sum of short differences (Yt2 − Yt1) + (Yt1 − Yt0). Our estimator generalizes +this simple idea by optimally aggregating short-term treatment effect parameters, +or “chain links”, obtained from (possibly many) overlapping incomplete panels. For +instance, one subsample of individuals might be used to estimate the DiD from t0 to +t1 while another would be used to estimate the DiD from t1 to t2. The sum of both +effects identifies the long-term DiD from t0 to t2, which may not be feasible or may +suffer from efficiency losses if a large part of the sample is discarded by using only +a balanced subsample. Our estimator is designed for multiple time periods, it ac- +commodates variations in treatment timing, treatment effect heterogeneity, general +missing data patterns, and it may deliver substantial efficiency gains compared to +2 + +estimators using only subsets of the data. +Building upon the influential work of Callaway and Sant’Anna (2021), we show that +our estimator is consistent, asymptotically normal, and computationally simple. +Their multiplier bootstrap remains asymptotically valid in our setting. We identify +three main advantages of our approach: (1) it does not require having a balanced +panel subsample as it is the case with a standard DiD; (2) identification rests on +a weaker assumption about missing data compared to the cross-section DiD, which +treats the sample as repeated cross-sectional data; and (3) it may also deliver effi- +ciency gains compared to other approaches using only subsets of the available data. +The proposed approach is especially relevant with regard to the significant interest +for long-term evaluations of interventions, with many applications related to educa- +tion and labor economics (Angrist et al., 2006; Kahn, 2010; Oreopoulos et al., 2012; +Autor and Houseman, 2010; Garca-Prez et al., 2020; Lechner et al., 2011; Mroz and Savage, +2006; Stevens, 1997). The estimation of treatment effects generally consists in per- +forming a DiD focusing only on units that are observed over the entire time frame: +the balanced subsample. This approach, hereafter referred to as the long DiD, allows +getting rid of both time- and individual-specific unobservable heterogeneity, but may +also involve discarding many individuals with missing observations when estimating +long-term effects. The evaluation of long-term effects is sometimes difficult, if not +infeasible, because of such missing data problems. Missing observations in panel data +sets may exist for two principal reasons: (1) by design or (2) because of attrition. +First, the design of rotating panel surveys attempts to alleviate the burden of admin- +istering and responding to statistical surveys, and to prevent attrition, by replacing +subjects regularly (Heshmati, 1998). The survey is administered to a cohort of sub- +jects in a limited number of periods. The cohort is then replaced by another cohort +randomly drawn from the population of interest. The resulting data set is hence +composed of a collection of incomplete panels. The data is said to have an over- +lapping structure if there is at least one period in which two separate cohorts are +administered the survey. +The most famous example of rotating panel survey is the Current Population Survey +(CPS), where each cohort is interviewed for a total of 8 months (over a period of 16 +3 + +months), and part of the sample is replaced each month by a new subsample.1 This +survey is one of the most widely used data sources in economic and social research. +It has been used or cited in 1000+ articles between 2000 and 2022, including publi- +cations in top journals such as the American Economic Review, Journal of Political +Economy, Quarterly Journal of Economics, American Sociological Review, and De- +mography.2 Other rotating panel surveys, without claiming to be exhaustive, include +the Medical Expenditure Panel Survey, the General Social Survey since 2006, and the +Consumer Expenditure Survey (Blundell et al., 2008) in the U.S., the Labour Force +Survey in the U.K., and the Enquˆete Emploi (Labour Force Survey) and Enquˆete +sur les Moyens Consacr´es `a la R&D (R&D Survey) in France. +Despite their prevalence, the econometrics literature on rotating panels is almost +non-existent (Baltagi and Song, 2006). +Heshmati (1998) uses rotating panels for +production function estimation. Nijman et al. (1991) study the optimal choice of the +rotation period for estimating a linear combination of period means. Likewise, our +estimator consists in a linear combination of parameters corresponding to the long- +term average treatment effect parameter. To the best of our knowledge, our paper +is the first to study identification, estimation, and inference of treatment effects in +this context. +Second, individuals may also drop out of the surveys and cause attrition, which +can be particularly severe for long panels. This attrition raises some concerns be- +cause it is associated with selection. +Attrition can be due to either “ignorable” +or “non-ignorable” selection rules (Verbeek and Nijman, 1996). Ignorable attrition +implies that missing data occurs completely at random, and as such focusing on +a balanced panel subsample does not threaten identification. Verbeek and Nijman +(1992), among others, propose a test of the ignorability assumption. Non-ignorable +attrition means that missing data is related to either observable or unobservable +factors. Hirano et al. (2001) provide important identification results for the addi- +tive non-ignorable class of attrition models, which nest several well-known meth- +ods (Hausman and Wise, 1979; Little and Rubin, 1989). These results have been +extended to multi-periods panels by Hoonhout and Ridder (2019) and apply to our +1The design is detailed at https://www.bls.gov/opub/hom/cps/design.htm. +2This figure is based on a Google Scholar search conducted on December 5, 2022. +4 + +setting. Bhattacharya (2008) studies the properties of a sieves-based semi-parametric +estimator of the attrition function initially proposed by Hirano et al. (2001). These +methods are often referred to as models of selection on unobservables because the +probability of attrition is allowed to depend on variables that are not observed when +an individual drops out, but not on unobservable error terms (Moffit et al., 1999). In- +verse propensity weighting is the most popular method for addressing non-ignorable +attrition caused by observable factors, even beyond the estimation of average treat- +ment effects (Chaudhuri et al., 2018). There also exist methods using instrumental +variables to address attrition due to latent factors (Fr¨olich and Huber, 2014). Other +approaches focus on particular structures of attrition, like monotonically missing +data (Chaudhuri, 2020; Barnwell and Chaudhuri, 2021). +Attrition is generally addressed before estimating treatment effects with the long +DiD, either by reweighting the observations in the balanced subsample with in- +verse propensity scores (Hirano et al., 2003), or by imputing the missing observa- +tions (Hirano et al., 1998). However, discarding a possibly large proportion of the +data by focusing on the balanced subsample may lead to significant efficiency losses +(Baltagi and Song, 2006; Chaudhuri, 2020; Barnwell and Chaudhuri, 2021), or may +lead to discarding the complete dataset, as illustrated in our application using the +French R&D survey. If there are too few individuals observed over the entire time +horizon, the data is typically treated as repeated cross-sections and treatment effects +are estimated with the cross-section DiD (Abadie, 2005; Callaway and Sant’Anna, +2021).3 +The identification of the average treatment effect using the cross-section DiD requires +not only a parallel trends assumption on the population, but also that the sampling +process is (conditionally) independent of the levels of idiosyncratic shocks to the +outcome variable. This assumption is violated if, for instance, units with larger un- +observed individual shocks are relatively more likely to be sampled in the treatment +group in later periods. In such a case, the treatment groups observed early on will +differ in unobservable ways from those observed later on. An identification problem +arises as soon as the sampling process does not affect the observed control groups +3This approach consists in taking the difference of differences of averages, where different sets +of units are used to compute each of the four averages. +5 + +in the exact same way. As an illustration, the survey sampling in our application +depends on the level of a firm’s internal R&D expenditure by design, which may lead +to biases. Instead, our approach only requires that the sampling process be (condi- +tionally) independent of the trend of those idiosyncratic shocks, as is the case in our +application. This weaker assumption is sufficient because the chained DiD allows +eliminating individual heterogeneity like the long DiD before taking expectations. It +is similar to a parallel trends assumption but conditional on being sampled. There- +fore, the chained DiD is robust to some forms of attrition caused by unobservable +heterogeneity, unlike the cross-section DiD. Note that the attrition models discussed +earlier can also be used as a first-step in our framework. Although we do not correct +for the efficiency losses of using first-step plug-in estimates in our paper, we identify +some suitable options to do so (Frazier and Renault, 2017; Chaudhuri et al., 2019; +Sant’Anna and Zhao, 2020). +Recent developments in the literature about treatments effects with multiple periods +and treatment heterogeneity have revealed that two-way fixed-effects estimators fail +to identify the treatment effect parameters of interest (de Chaisemartin and D’Haultfœuille, +2020; Athey and Imbens, 2022; Goodman-Bacon, 2021; Borusyak et al., 2021; Sun and Abraham, +2021). Our paper builds upon Callaway and Sant’Anna (2021) which address this is- +sue by generalizing the approach developed in Abadie (2005) to multiple periods and +varying treatment timing. We contribute further to this literature by extending this +framework to settings with incomplete panel data. An alternative approach would +have been to adapt the general framework developed by de Chaisemartin and D’Haultfœuille +(2020) to our setting, or the other related papers focused on staggered adoption de- +signs and event studies with multiple periods. However, the chained DiD fits very +well within the framework developed by Callaway and Sant’Anna (2021) which fo- +cuses on estimating all group-time average treatment effects before aggregating all +those parameters into summary parameters of interest. Likewise, our approach focus +on (even smaller) building blocks: one-period-difference group-time average treat- +ment effects, which measure the increase of average treatment effect of group g from +period t−1 to period t. In the general case, these blocks naturally become k-period- +difference group-time average treatment effects. +We illustrate the performance of the chained DiD in two ways. First, we use sim- +6 + +ulations to compare the long DiD, chained DiD and cross-section DiD in terms of +bias and variance under several data generating processes (DGP). Our simulations +include a stratified panel data set, composed of a balanced panel and a rotating +panel. Second, we study the long-term employment effects of a large-scale innova- +tion policy in France giving grants to collaborative R&D projects. Technical progress +and innovation, stimulated by R&D activities, are key to economic growth (Scherer, +1982; Howitt and Aghion, 1998; Griffith et al., 2004) but firms tend to invest too +little because of the public good nature of innovations (Arrow, 1962; Nelson, 1959). +Therefore, measuring the long term effects of subsidized R&D is important to inform +policymakers and improve future policies. +This application is especially relevant because the French R&D survey, which con- +tains firm-level data on R&D activities, consists of rotating panels but does not in- +clude a balanced subsample except for the largest firms. In addition, it is possible to +fully observe a limited number of variables for all firms close to those provided by the +R&D survey by using administrative data. We are thus able to compare the results of +each of the three estimators by focusing on two of those variables: total employment +and highly qualified workforce. The long DiD is applied to the complete data and +serves as the benchmark estimates. The chained DiD and cross-section DiD estima- +tors are applied to both data sets, and to an “artificial” unbalanced panel which is +generated by discarding all observations from the complete administrative data set +which are missing in the R&D survey. This application is somehow comparable to a +simulation exercise but uses real data. +Our results show that the policy had a positive effect on employment for firms that +received a grant to participate to a collaborative R&D project. We find that the +estimates as well as their standard errors obtained with the chained DiD estimator +are close to those obtained with the long DiD. In contrast, the cross-section estimator +delivers biased estimates which also lack sufficient precision to detect any statistically +significant effect associated with the policy. +The remainder of the paper is organized as follows. Section 2 presents our method- +ology and asymptotic results. Numerical simulations are in Section 3. Section 4 +contains the application to R&D policy. Section 5 concludes the paper. +7 + +2 +Identification, Estimation and Inference +2.1 +Basic Framework +We first present the main insights in a simple framework. The main notation is +as follows. +There are T periods and each particular time period is denoted by +t = 1, ..., T . In a standard DiD setup, T = 2, no one is treated in t = 1, and all +treatments take place in t = 2. To gain intuition, we first focus on the case where +all treatments take place in t = 2 but assume T > 2. +Define G to be a binary variable equal to one if an individual is in the treatment +group, and C = 1 −G as a binary variable equal to one for individuals in the control +group. Also, define Dt to be a binary variable equal to one if an individual is treated +in t, and equal to zero otherwise. Let Yt(0) denote the potential outcome at time +t without the treatment and Yt(1) denote its counterpart with the treatment. The +observed outcome in each period can be written as Yt = DtYt(1) + (1 − Dt)Yt(0). +We focus on the long-term average treatment effect on the treated which corresponds +to the average treatment effect in period t > 2 on individuals in the treatment group, +hence first treated in period 2. It is formally defined by +ATT(t) = E [Yt(1) − Yt(0)|G = 1] . +(1) +The identification of ATT(t) with panel data has attracted much research attention. +In this paper, we are interested in a case where balanced panel data is not available. +The missing data pattern. +We assume that a new random sample of nt individ- +uals is drawn at each period t to replace a subsample of previously observed individ- +uals. This replacement can occur due to attrition or by design. For the moment, we +assume that subsamples may differ in size nt but individuals are only observed for +two consecutive periods as in many rotating panel design.4 Therefore, one cannot +observe the entire path {Y1, Y2, ..., YT } for any individual. The key feature of our +4In practice, rotating panels may involve subgroups sampled over a different number of con- +secutive periods. The result of this paper continues to apply, so the same estimator and inference +procedure can be used even with a more sophisticated sampling process as presented in Section 2.3. +8 + +approach is that there is some overlap across subsamples, that is there are at least +two different subsamples observed in each period 1 < t < T . +This structure is in stark contrast with the literature about attrition in panel data, +which typically assumes that there always exists a balanced subsample where individ- +uals are observed throughout the entire time frame (Hirano et al., 2001; Hoonhout and Ridder, +2019). Although there is no need for such a balanced subsample here, we extend our +method to more general settings in Section 2.3. +This extension is illustrated in +Sections 3 and 4. +To characterize the sampling process, we define St to be a binary variable equal to +one for individuals observed at t and zero otherwise, and use St,t+1 to denote StSt+1 +which indicates if an individual is observed at both t and t + 1. The missing data +pattern is summarized in Table 1 for T = 3. In the general framework, we will +assume that treatments can also vary with some observable covariates X and that +individuals can be observed in non-consecutive periods. +Table 1: Missing data pattern in a three-period panel data set +Obs. Indicators +Variables +Sub-population +S1 +S2 +S3 +Y1 +Y2 +Y3 +X +Incomplete Panel 1 +1 +1 +0 +× +× +· +× +Incomplete Panel 2 +0 +1 +1 +· +× +× +× +The long DiD. +The common approach to identify treatment effects is to consider +the long difference-in-differences defined by +ATT(t) =E[Yt(1) − Y1(0)|G = 1] − E[Yt(0) − Y1(0)|C = 1], +(2) +under the standard parallel trend assumption E[Yt(0) − Y1(0)|G = 1] = E[Yt(0) − +Y1(0)|C = 1]. ATT(t) corresponds to the long-term effect for t > 2. Unfortunately, +individuals are never observed more than two consecutive periods in this framework. +Calculating the averages of Yit − Yi1 for t > 2 for the treatment and control groups +is hence infeasible. +9 + +The cross-section DiD. +If the panel consists of incomplete panel data, the iden- +tification of the parameter of interest can be achieved by assuming that the sampling +process St is independent of (Yt, D1, D2, ..., DT ) in addition to the parallel trend as- +sumption (Abadie, 2005). In this case, ATT(t) is identified by the “cross-section +DiD” given by +ATTCS = (E[Yt|StG = 1] − E[Y1|S1G = 1]) − (E[Yt|StC = 1] − E[Y1|S1C = 1]) . +The sampling assumption allows replacing the averages of difference, E[Yt(0) − +Y1(0)|a = 1] for a ∈ {G, C}, by the difference of averages, e.g. +E[Yt(0)|Sta = +1] − E[Y1(0)|S1a = 1] for a ∈ {G, C}. +This approach does not eliminate the +individual-specific unobservable heteregeneity. +The chained DiD. +Our approach takes advantage of the overlapping panel struc- +ture. Remark that each term in (2) can be decomposed into +E [Yt(1) − Y1(0)|a = 1] = +t−1 +� +τ=1 +E [Yτ+1(Dτ+1) − Yτ(Dτ)|a = 1] +(3) +for a ∈ {G, C}. Thus, identification of ATT(t) is obtained by summing the short- +term DiD as in +ATTCD(t) = +t−1 +� +τ=1 +(E [Yτ+1(Dτ+1) − Yτ(Dτ)|G = 1] − E [Yτ+1(Dτ+1) − Yτ(Dτ)|C = 1]) += +t−1 +� +τ=1 +(E [Yτ+1(Dτ+1) − Yτ(Dτ)|Sτ,τ+1G] − E [Yτ+1(Dτ+1) − Yτ(Dτ)|Sτ,τ+1C]) , +where the second equality holds under the assumption that the sampling process +St,t+1 is independent of (Yt+1 − Yt, D1, D2, ..., DT ). This approach not only allows +eliminating the individual-specific heterogeneity, it also makes use of a weaker iden- +tifying assumption than the one for the cross-section DiD in this setting. +Remark that these assumptions may not be directly comparable in settings with dif- +ferent sampling processes, e.g. if some individuals are observed only once. However, +10 + +replacement of individuals in panel survey data is typically based on some variables +observed before the replacement period but not on their evolution.5 Our application +provides a clear illustration of that argument. +Simple model. +In order to gain intuitions about the identification, estimation +and inference of ATT(t) in this context, we suppose that the outcome variable is +generated by a components of variance process +Yit = αi + δt + +t +� +τ=2 +βτDiτ + εit, +(4) +where ATT(t) = �t +τ=2 βτ is the impact of the treatment evaluated at t. αi is an +individual-specific component, δt is a time-specific component, and εit is a mean-zero +individual-transitory shock that is auto-regressive stationary error process of order +1 represented by εit+1 = ρεit + ηit+1 with ρ ∈ [0, 1) and ηit+1 being a white noise. +Assume further that error terms are homoskedastic with V (εit) = σ2 +ε = σ2 +η/(1 − ρ2) +and V (αi) = σ2 +α, and Dit ⊥⊥ (Yit(0), Yit(1)), for all t > 2. +Identification. +If the sampling process is independent of all the components of +Yit, then both ATTCD and ATTCS identify the parameter of interest. However, if +the unobservable individual-specific component αi is correlated with the sampling +process, identification still holds for ATTCD but the latter admits the bias term +(E[αi|StG = 1] − E[αi|S1G = 1]) − (E[αi|StC = 1] − E[αi|S1C = 1]) . +This bias is non-zero if the compositions of the sampled treatment and control groups +evolve differently through time. This situation happens, for instance, if individuals +with larger αi are more likely to be sampled in the control group in period 1 but +become relatively more likely to be sampled in the treatment group in period t. +Addressing this bias can be difficult, if not impossible, due to the unobservability of +these errors. +5Remark that surveys are usually not specifically designed to use panel econometric methods. +11 + +Therefore, if the sampling process depends on unobservable components of the out- +comes Yit but is not correlated with the first-differences in outcomes Yit+1 − Yit +conditional on treatment status, i.e. +εit+1 − εit in this example, then ATT(t) is +still identified by ATTCD(t) but not necessarily by ATTCS(t). This result implies +that identification with ATTCD(t) is more robust to deviations from an independent +sampling assumption, for instance, because of a dependence between the sampling +process, treatment status, and unobservable individual heterogeneity αi. +Estimation. +Estimators of ATTCD(t) and ATTCS(t) can be obtained by linear re- +gression as follows. For ATTCS(t), one must first modify the outcome variable for +treated individuals to Yit−�n1 +i=1 Yi1Si1Gi/ �n1 +i=1 Si1Gi and to Yit−�n1 +i=1 Yi1Si1Ci/ �n1 +i=1 Si1Ci +for those in the control group, and then regress those modified outcomes onto the +treatment variable and an intercept. The parameter associated with treatment will +yield an estimate of �t +τ=2 βτ. For ATTCD(t), one has to estimate t − 2 two-way +fixed-effects linear regressions specified in (4), one for each subsample of individuals +observed sequentially, and then sum the estimated parameters. +A more direct estimation method for both ATTCD(t) and ATTCS(t) is to substitute +expectations by their sample counterparts in each expression. The corresponding +estimators can be written as the weighted averages +� +ATTCD(t) = 1 +n +n +� +i=1 +t−1 +� +τ=1 +� � +wGiττ+1 (y′ +i − yi) − � +wCiττ+1 (y′ +i − yi) +� +, +� +ATTCS(t) = 1 +n +n +� +i=1 +�� +� +wGit − � +wGi1 +� +yi − +� +� +wCit − � +wCi1 +� +yi +� +, +where yi and y′ +i denote the outcome variable in the current and next period, i.e. τ +and τ + 1, respectively. The weights are defined as +� +waiτ = +Siτai +1 +n +�n +i=1 Siτai +, and +� +waiττ+1 = +Siτ,τ+1ai +1 +n +�n +i=1 Siτ,τ+1ai +, +with ai ∈ {G, C}, and for all τ = 1, ...T − 1. A formal treatment is presented in the +general framework. +12 + +Inference. +When the sampling process does not violate the identifying assump- +tions, the most efficient of the two estimators should be favored. Proposition 1 sheds +light on the relative efficiency of each estimator under standard assumptions in this +simple setting. +Proposition 1 Under the standard assumptions of DiD settings, as presented in the +proof, and assuming Yit to be specified as in (4), for all t > 2, then � +ATTCD(t) has a +smaller asymptotic variance if +2(t − 1) +1 + ρ σ2 +η ≤ +σ2 +η +1 − ρ2 + σ2 +α. +On the one hand, the chained DiD introduces additional noise by adding up individual- +transitory shocks εit. On the other hand, the cross-section DiD’s precision depends +on the variance of unobserved individual heterogeneity αi and serial correlation of +individual shocks εit. It also makes use of twice as many observations to calculate +each empirical expectation since two cohorts are observed at each t, i.e. t − 1, t and +t, t + 1. In comparison, the long DiD does not suffer from any of those two efficiency +losses. According to Proposition 1, � +ATTCD(t) delivers a more precise estimate only if +the sum of the extra individual-transitory shocks has a smaller variance than that of +the individual-specific heterogeneity and individual transitory shock. This condition +is plausible as long as t is not too large, that is not too many incremental effects are +aggregated together, or if σ2 +α is relatively large. As t increases, a new term is added +to the sum and results in a marginal increase of the variance equal to +2 +1+ρσ2 +η. +The serial correlation of εit plays an important role. +As ρ → 0, the variance of +� +ATTCD(t) goes to 2(t − 1)σ2 +η whereas that of of � +ATTCS(t) approaches 2(σ2 +η + σ2 +α). +Conversely, as ρ → 1, the variance of � +ATTCD(t) approaches (t − 1)σ2 +η whereas that +of � +ATTCS(t) goes to +∞. Therefore, � +ATTCD(t) should largely dominate in terms of +precision in settings with autocorrelated idiosyncratic shocks, and where the variance +of unobserved individual heterogeneity is large. We will see later that +� +ATTCD(t) +delivers additional efficiency gains under more general missing data patterns. +13 + +2.2 +General framework +In the general framework, we allow for: 1) heterogeneity in treatment effects; 2) +variation in treatment timing; and 3) general missing data patterns. +We first introduce additional notations. +Let us substitute the treatment group +dummy G by Gg, a binary variable that is equal to one if an individual is first +treated in period g. There are hence several cohorts of treatment groups. The con- +trol group binary variable C denotes individuals who are never treated. Notice that +for each individual �T +g=2 Gg + C = 1. Also, define Dt to be a binary variable equal +to one if an individual is treated in t, and equal to zero otherwise. This variable will +be useful to denote if the individual was first treated for some g ≤ t. Also, define +the generalized propensity score as pg(X) = P(Gg = 1|X, Gg + C = 1). This score +measures the probability of an individual with covariates X to be treated conditional +on being in the treated cohort g or the control group. +There are many parameters of interest in this setting, such as, for example, the +average treatment effect k periods after the treatment date. All possible parameters +consist of aggregates of the most basic parameter: the average treatment effect in +period t for a cohort treated in date g, denoted by +ATT(g, t) = E[Yt(1) − Yt(0)|Gg = 1]. +(5) +This parameter is referred to as the group-time average treatment effect in Callaway and Sant’Anna +(2021), and we will build upon their results to study the collection of such parame- +ters. When only unbalanced panel data is available and t > g + 1, two methods are +possible: (1) the cross-section DiD; and (2) the chained DiD. +In what follows, we assume that individuals are sampled only for two consecutive +periods and then drop out forever, so the long DiD is never feasible. We will introduce +more general missing data patterns in the next subsection. +In order to identify the ATT(g, t) and accommodate varying treatment timing and +treatment effect heterogeneity on observable covariates X, we impose assumptions +as follows. +14 + +Assumption 1 (Sampling) For all t = 1, ..., T , {Yit, Yit+1, Xi, Di1, Di2, ..., DiT }nt +i=1 +is independent and identically distributed (iid) conditional on Sit,t+1 = 1. +Assumption 2 (Conditional Independence of Sampling and Trends) For all +t = 1, ..., T , +Yit+1 − Yit ⊥⊥ Sit,t+1|(Xi, Di1, Di2, ..., DiT ). +(6) +Assumption 3 (Conditional Parallel Trends) For all t = 2, ..., T , g = 2, ..., T , +such that g ≤ t, +E [Yt(0) − Yt−1(0)|X, Gg = 1] = E [Yt(0) − Yt−1(0)|X, C = 1] a.s.. +(7) +Assumption 4 (Irreversibility of Treatment) For all t = 2, ..., T , +Dt−1 = 1 implies that Dt = 1. +(8) +Assumption 5 (Overlap) For all g = 2, ..., T , P(Gg = 1) > 0 and for some ε > 0, +pg(X) < 1 − ε a.s. +Assumption 1 means that we are considering a rotating panel structure. +Condi- +tional on being sampled in two consecutive periods, individuals are assumed to be +iid. However, unlike Assumption 3.3 in Abadie (2005), it does not imply that ob- +servations are representative of the population of interest because we do not as- +sume that the iid draws are taken from the population distribution. Instead, we +focus on the case where the identification with a cross-section DiD can fail by con- +sidering Assumption 2. This assumption constitutes the principal departure from +Callaway and Sant’Anna (2021). It states that the sampling process is statistically +independent of the joint distribution of first-differences of individual outcomes, con- +ditionally on observables and treatment status in any period. In particular, it implies +that E[Yit+1 − Yit|X, a = 1, St,t+1 = 1] for a ∈ {G1, G2, ..., C} corresponds to its pop- +ulation counterpart. However, this needs not be true for E[Yit|X, a = 1, St = 1] for +a ∈ {G1, G2, ..., C}. Notice that these assumptions also imply that the propensity +score pg(X) is independent on the sampling process. +15 + +Assumption 3 is a key identifying assumption in DiD settings with treatment hetero- +geneity. It means that the average outcomes for the treatment and control groups, +conditional of observables, would have followed parallel paths in absence of the treat- +ment. +It is extensively discussed in Abadie (2005) and Callaway and Sant’Anna +(2021). Assumption 4 implies that once an individual is first treated, that individual +will continue to be treated in the following periods. In other words, there is no exit +from the treatment.6 Finally, Assumption 5 ensures that there are positive proba- +bilities to belong to the control and treatment groups for any possible value of X. +Remark that X is assumed to be observed for all individuals. +Data generating process. +Under Assumption 1, the data generating process con- +sists of random draws from the mixture distribution FM(y, y′, g1, ..., gT , c, s1, ..., sT , x) +defined as7 +T +� +t=1 +λt,t+1FYt,Yt+1,G1,...,GT ,C,X|St,t+1(yt, yt+1, g1, ..., gT , C, X|st,t+1 = 1), +where λt,t+1 = P(St,t+1 = 1) is the sampling probability, y and y′ denote the outcome +yt and yt+1, respectively, for an individual sampled at t and t + 1. Expectations un- +der the mixture distribution does not correspond to population expectations. This +difference arises because of different sampling probabilities λt,t+1 = P(St,t+1 = 1) +across time periods and because Assumption 2 does not preclude from some forms +of dependence between the sampling process and the unobservable heterogeneity in +Yit. However, this assumption ensures that expectations of first-differences under the +mixture correspond to population expectations once conditioned on the time peri- +ods. Thereafter, EM[·] denotes expectations with respect to the mixture distribution +FM(·), its empirical counterpart being the sample mean. +6Departures from this assumption are considered in de Chaisemartin and D’Haultfœuille (2022). +7In the application, we will discuss more complicated situations in which the data is generated +by stratified sampling. The same results apply using a suitably reweighted sample (Wooldridge, +2010; Davezies and D’Haultfœuille, 2009). +16 + +An important result of the paper is given in the Theorem 1. We define the weights +wG +ττ−1(g) = +GgSτ,τ−1 +EM[GgSτ,τ−1] +and +wC +ττ−1(g, X) = pg(X)CSτ,τ−1 +1 − pg(X) +/EM[pg(X)CSτ,τ−1 +1 − pg(X) +]. +Theorem 1 Under Assumptions 1 - 5, and for 2 ≤ g ≤ t ≤ T , the long-term +average treatment effect in period t is nonparametrically identified, and given by +ATTCD(g, t) = +t +� +τ=g +∆ATT(g, τ), +where ∆ATT(g, t) = EM +� +wG +ττ−1(g) (Yt − Yt−1) +� +−EM +� +wC +ττ−1(g, X) (Yt − Yt−1) +� +is the +1-period-difference group-time average treatment effects, which measure the increase +of average treatment effect of group g from period t − 1 to period t. Furthermore, +identification is no more guaranteed with the cross-section DiD. +Those identification results suggest the two-step estimator +� +ATT CD(g, t) = 1 +n +n +� +i=1 +t +� +τ=g +� +�wG +iττ−1(g) (Yiτ − Yiτ−1) − �wC +iττ−1(g, X) (Yiτ − Yiτ−1) +� +. +where +�wG +iττ−1(g) = +GigSiτ−1Siτ +1 +n +�n +i=1 GigSiτ−1Siτ +and +�wC +iττ−1(g) = ˆpg(Xi)CiSiτ−1Siτ +1 − ˆpg(Xi) +/ 1 +n +n +� +i=1 +ˆpg(Xi)CiSiτ−1Siτ +1 − ˆpg(Xi) +, +and with ˆpg(·) being an estimated parametric propensity score function, such as logit +or probit, obtained in a first step. +Let us denote � +ATT g≤t the vector of � +ATT CD(g, t)’s for g ≤ t. The next theorem +establishes its joint limiting distribution. +17 + +Theorem 2 Under Assumptions 1 - 5 and a standard assumption on the parametric +estimates of the propensity scores (Assumption 5 in Callaway and Sant’Anna (2021) +or 4.4 in Abadie (2005)), for all 2 ≤ g ≤ t ≤ T , +√n +� +� +ATT g≤t − ATTg≤t +� +d→ N(0, Σ), +as n → ∞ and where the covariance Σ is detailed in the proof. +In the proof of the above theorem, we also show how the multiplier bootstrap proce- +dure proposed by Callaway and Sant’Anna (2021) adapts to this asymptotic result. +The main difference comes from redefining the influence function, but the result +about its asymptotic validity applies without other modification so it is not repeated +here. We also refer the reader to their paper for a complete discussion of summary +parameters which use the ATT(g, t) as building-blocks. We provide some results +for these summary parameters and an extension to using the not yet treated as the +control group in Appendices A.2.3 and A.2.5. +2.3 +General missing data patterns +This framework naturally extends to general missing data patterns beyond rotating +panel structures. For simplicity of exposition, we remove the dependence of g in our +notations. We now consider not only first-differences ∆1ATT(t) but also k-period +differences ∆kATT(t) from t − k to t for k = 1, ..., t − 1. +Table 2 provides an example of different missing data pattern with 4 subsamples and +3 periods, and all treatments take place at t = 1. The first subsample is a balanced +panel (BP). It identifies ∆1ATT(2), ∆1ATT(3), and ∆2ATT(3). The second, third +and fourth subsamples are incomplete panels (IP1, IP2, IP3) which only identifies a +single parameter.8 +8Note that we do not consider refreshment samples because they do not identify any parameter +on their own. They could still be used to address attrition ex-ante (Hoonhout and Ridder, 2019). +18 + +Table 2: Example of a more general missing data pattern +Obs. Indicators +Identified Parameters +Sub-population +S1 +S2 +S3 +Balanced Panel +1 +1 +1 +∆1ATT(2), ∆1ATT(3), ∆2ATT(3) +Incomplete Panel 1 +1 +1 +0 +∆1ATT(2) +Incomplete Panel 2 +0 +1 +1 +∆1ATT(3) +Incomplete Panel 3 +1 +0 +1 +∆2ATT(3) +There are multiple ways to identify the ATT from period 1 to period 3 in this +example. We can identify ATT(3) with (1) BP alone since ATT(3) = ∆2ATT(3) or +(2) ATT(3) = ∆1ATT(2) + ∆1ATT(3); or by combining (2) BP and IP3; or (4) BP +and IP1; or (5) IP1 and IP2; and with (6) IP3 alone. The optimal combination of +the ∆kATT(t) parameters into ATT(t) parameters for general missing data patterns +hence involves solving an (overidentified) linear inverse problem. Doing so will make +use of all subsamples and deliver efficiency gains compared to focusing on one possible +solution, e.g. using only the balanced panel. +The inverse problem arises as follows. Consider the estimates of all possible ∆kATT(t), +for all t ≥ 2 and t − 1 ≥ k ≥ 1, stacked altogether into a vector ∆ATT of length +L∆. ∆kATT(t) is called the k-period-difference group-time average treatment effects, +which measure the increase of average treatment effect of group g from period t − k +to period t. By definition ∆kATT(t) = ATT(t) − ATT(t − k), hence we can write +∆ATT = WATT, +(9) +where ATT is the vector of ATT(t) for t ≥ 2 of length L ≤ L∆ and W is a matrix +where each element takes value in {−1, 0, 1}. Following this reasoning, the example +in Table 2 can be written as + + +∆1ATT(2)BP,IP 1 +∆1ATT(3)BP,IP 2 +∆2ATT(3)BP,IP 3 + + = + + +1 +0 +−1 +1 +0 +1 + + +� +ATT(2) +ATT(3) +� +, +(10) +19 + +where we pool all subsamples that identify each ∆kATT(t).9 We propose to solve this +problem using a GMM approach. Denoting Ω the covariance matrix of ∆ATT, the +optimal GMM estimator of ATT corresponds to (W ′Ω−1W)−1 W ′Ω−1∆ATT since +WAT T is non-random.10 This method allows delivering efficiency gains by using +all individual time-series with at least two observations, without much additional +computational complexity. +Identification, estimation, and inference. +In order to identify the ATT(g, t) +in this general framework, we must modify Assumptions 1 to 2 as follows. +Assumption 6 (Sampling) For all t = 1, ..., T , {Yit−k, Yit, Xi, Di1, Di2, ..., DiT } +nt,k +i=1 +is independent and identically distributed (iid) conditional on Sit−k,t = 1, for k = +1, ..., t − 1. +Assumption 7 (Conditional Independence of Sampling and Trends) For all +t = 1, ..., T and k = 1, ..., t − 1, +Yit − Yit−k ⊥⊥ Sit−k,t|(Xi, Di1, Di2, ..., DiT ). +(11) +Assumption 6 means that we are considering an incomplete panel structure. Con- +ditional on being sampled in the same two periods, individuals are assumed to be +iid. It does not imply that individuals are sampled in two periods only. Further- +more, it does not imply that observations are representative of the population of +interest because we do not assume that the iid draws are taken from the population +distribution. Instead, Assumption 7 states that the sampling process is statistically +independent of the joint distribution of k-period-differences of individual outcomes, +conditionally on observables and treatment status in any period. In particular, it im- +plies that E[Yit −Yit−k|X, a = 1, St−k,t = 1] for a ∈ {G1, G2, ..., C} corresponds to its +9Remark that we could also estimate these parameters separately for each subsample before +stacking them into ∆ATT. That would require to estimate propensity scores conditional on sub- +sample membership. +10Remark that if an element of ATT is not identified, the matrix W ′Ω−1W (or W ′W) will not +be invertible but a (Moore-Penrose) pseudo-inverse can still be used to identify the other elements. +In addition, pseudo-inverses, e.g. Tikhonov’s, will deliver a stable inverse of Ω if ∆ATT is high- +dimensional (Carrasco et al., 2007). +20 + +population counterpart. However, this needs not be true for E[Yit|X, a = 1, St = 1] +for a ∈ {G1, G2, ..., C}. Notice that these assumptions still imply that the propensity +score pg(X) is independent on the sampling process. +Our estimation procedure is as follows: +1. Compute ∆kATT(g, t) = 1/n �n +i=1[( ˆw(g)G +it,t−k − ˆw(g)C +it,t−k)(Yit − Yit−k)], for all +k, g, t, and stack them into a L∆-dimensional vector ∆AT T ; +2. Define the matrix W appropriately; +3. Estimate the asymptotic covariance matrix ˆΩ = n−1ΨΨ′, where Ψ is a L∆ × n +matrix with elements defined in (33). +4. Estimate the optimal GMM estimator � +AT T = (W ′ˆΩ−1W)−1W ′ˆΩ−1∆AT T ;11 +The next theorem establishes the joint limiting distribution of this estimator. +Theorem 3 Under Assumptions 3 - 5, a standard assumption on the parametric +estimates of the propensity scores (Assumption 5 in Callaway and Sant’Anna (2021) +or 4.4 in Abadie (2005)), and Assumptions 6 and 7, for all 2 ≤ g ≤ t ≤ T , +√n +� +� +AT T − AT T +� +d→ N(0, Σ), +as n → ∞ and where the covariance Σ is detailed in the proof. +The theorem shows that this estimator is consistent and asymptotically normal. The +bootstrap procedures for ATT(g, t)’s (Appendix A.2.2) and for summary parameters +(Appendix A.2.4) apply with minor modifications, as discussed in the proof. Al- +though our estimator brings efficiency gains by using all available observations, it +could be further improved by addressing the efficiency losses from first-step plug-in +estimates of propensity scores. This is left for future research.12 +11Note further that this approach embeds the estimator proposed above in the rotating panel +data setting when using only the ∆AT T1(t) and replacing Ω by the identity matrix. +12We identified two means to address this caveat. Frazier and Renault (2017) propose a com- +putationally simple yet general approach that involves targeting and penalization to enforce the +asymptotic efficiency for two-step extremum estimators such as ours, whereas Chaudhuri et al. +(2019) and Sant’Anna and Zhao (2020) propose doubly-robust estimators which allow preserving +efficiency. +21 + +3 +Numerical Simulations +We propose a simulation design adapted from the first section. Let us specify the +outcome variable as a components of variance: +Yit = αi + δt + +t +� +τ=2 +βτDiτ + εit, +(12) +where Diτ ∈ {0, 1} denotes whether individual i has been treated in τ or earlier. Let +us assume that t ∈ {0, ..., T + 1} and treatments can only occur in t ≥ 2 so that +G ∈ {2, ..., T + 1}. The data generating process is characterized by the following +assumptions: +• The individual-specific unobservable heterogeneity is iid gaussian: αi ∼ N(1, σ2 +α), +where σ2 +α = 2 ; +• The time-specific unobservable heterogeneity is iid gaussian: δt ∼ N(1, 1); +• The error term is iid gaussian: εit ∼ N(0, σ2 +ε), where σ2 +ε = 0.5; +• The probability to receive the treatment at time g, conditional on being treated +at g or in the control group, is defined as +Pr(Gig = 1|Xi, αi, Gig + Ci = 1) = +1 +1 + exp (θ0 + θ1Xi + θ2αi × t), +where Xi ∼ N(1, 1) is observable for every i, unlike αi, and θ0 = −1, θ1 = 0.4 +and θ2 = 0 or θ2 = 0.2. In the latter case, the treatment probability varies +with time and the unobserved individual heterogeneity; +• The sampling probability in the consecutive periods t, t + 1 conditional on αi +is given by +Pr(Sitt+1 = 1|αi) = +1 +1 + exp (λ0 + λ1αi × t), +with λ0 = −1, and λ1 = 0 or λ1 = 0.2, so that the sampling process can also +varies with time and the unobserved individual heterogeneity. +22 + +We simulate the sampled data in two steps. First, we generate a population sample +for each period t to represent individuals that are either treated at t or in the control +group. +Second, we sample from this population using the specified process. +We +formalize this procedure as follows: +1. Generate a population of individuals +(a) Draw N = 2 × max +t { +n +Eα[P r(Sitt+1)]} individuals per period in order to have +T + 2 population samples of N individuals, where each individual is char- +acterized by a vector (αi, δt, Xi, εit); +(b) Separately for each population sample g, draw a uniform random number +ξi ∈ [0, 1] per individual. If ξi ≤ Pr(Git = 1|Xi, αi, Git + Ci = 1), then set +(Gig = 1, Ci = 0), otherwise set (Gig = 0, Ci = 1). +(c) Compute Yit from (αi, δt, Xi, εit, Gi0, ..., GiT+1, Ci); +2. Sample from this population +(a) Draw a uniform random number ηit ∈ [0, 1] per individual i and period t. +If ηit ≤ Pr(Sitt+1 = 1|αi), then set Sitt+1 = 1 and Siττ+1 = 0 for τ ̸= t; +(b) Draw (without replacement) n individuals per period t from the popula- +tion for which Sitt+1 = 1; +(c) Compute the different estimators. +(d) Repeat steps 1(b)-2(c) 1,000 times and report the mean and standard +deviation of the estimators. +We consider several simulation designs: +• DGP 1: θ2 = 0 and λ1 = 0. This is the baseline case where the probability of +treatment and the sampling process do not depend on individual heterogeneity +so all estimators are unbiased. +• DGP 2: θ2 = 0.2 and λ1 = 0.2. In this case, both the probability of treatment +and the sampling process depend on individual heterogeneity leading to biased +estimates for the cross-section DiD. +23 + +• DGP 3: θ2 = 0.2 and λ1 = 0.2. In this case, we simulate a stratified sample +where 90% of the individuals are sampled on a rotating basis as before and +those with αi superior to the 90th percentile are always observed (10%). +• DGP 4: θ2 = 0.2 and λ1 = 0.2. We also simulate a stratified sample where +individuals with αi superior to the 60th percentile are always observed (40%) +and the rest is drawn as a rotating subpanel (60%). +For all simulations, we set T = 6, so there is a total of 8 periods. In each period, the +population size is 4800, and we draw 150 individuals such that Sitt+1 = 1. Finally, +we set βτ to take the values {1.75, 1.50, 1.25, 1.00, 0.75, 0.50} for τ = {1, 2, 3, 4, 5, 6}, +that is, the treatment effect is positive and decreasing over time, relative to the +treatment starting date. For each sample, we estimate the chained DiD (using sim- +ulated Sitt+1 = 1) and the cross-section DiD (using Sit = 1 that are obtained from +Sitt+1 = 1). +For DGPs 1 and 2, we also estimate the long DiD assuming that all sampled indi- +viduals are observed for the entire time frame. The simulation results are given in +Table 3. This long DiD here is infeasible but serves as a benchmark to illustrate the +significant loss of information resulting from having an unbalanced panel. However, +the chained DiD delivers unbiased estimates in both cases unlike the cross-section +DiD. +24 + +Table 3: Simulation results for a rotating panel +DGP 1 +DGP 2 +Chained DiD +CS DiD +Long DiD +Chained DiD +CS DiD +Long DiD +β1 +1.748 +1.745 +1.75 +1.752 +1.894 +1.75 +(0.099) +(0.199) +(0.017) +(0.097) +(0.148) +(0.016) +β2 +1.5 +1.498 +1.5 +1.499 +1.774 +1.501 +(0.164) +(0.305) +(0.02) +(0.157) +(0.218) +(0.018) +β3 +1.248 +1.25 +1.251 +1.254 +1.651 +1.25 +(0.231) +(0.355) +(0.023) +(0.224) +(0.257) +(0.02) +β4 +1 +1.006 +1 +1.002 +1.522 +1 +(0.3) +(0.412) +(0.027) +(0.293) +(0.291) +(0.023) +β5 +0.741 +0.765 +0.75 +0.739 +1.369 +0.75 +(0.406) +(0.521) +(0.033) +(0.395) +(0.365) +(0.028) +β6 +0.499 +0.522 +0.499 +0.5 +1.209 +0.503 +(0.586) +(0.711) +(0.046) +(0.603) +(0.515) +(0.04) +Notes: This table shows results obtained from the simulations described above. Simulated βτ take the values +{1.75, 1.50, 1.25, 1.00, 0.75, 0.50} for τ = {1, 2, 3, 4, 5, 6}. +For DGPs 3 and 4, the long DiD is estimated only for individuals that belong to the +balanced subpanel. The simulation results are given in Table 4. We show estimates +from the chained DiD GMM estimator using two weighting matrices: (1) Ch DiD uses +the identity matrix, and (2) CD-GMM uses the optimal weighting matrix presented +earlier. It appears that when the balanced subpanel consists of only 10% of the data, +the chained DiD estimators outperform the long DiD. However, the (asymptotically) +optimal weighting matrix does not always deliver more precise estimates than the +identity matrix in small samples, at least for this simulation design. In comparison to +the identity matrix, It seems that the optimal weights allow mitigating the precision +loss for longer-term effects (e.g. β6) at the cost of losing some precision for smaller- +term effects (e.g. β1). +25 + +Table 4: Simulation results for a stratified sample +DGP 3 +DGP 4 +Ch DiD +CD-GMM +CS DiD +Long DiD +Ch DiD +CD-GMM +CS DiD +Long DiD +β1 +1.753 +1.748 +1.714 +1.753 +1.754 +1.755 +1.715 +1.754 +(0.085) +(0.151) +(0.14) +(0.085) +(0.052) +(0.057) +(0.096) +(0.052) +β2 +1.501 +1.488 +1.413 +0.897 +1.502 +1.503 +1.422 +1.504 +(0.127) +(0.195) +(0.208) +(0.315) +(0.061) +(0.06) +(0.137) +(0.069) +β3 +1.256 +1.237 +1.112 +0.868 +1.252 +1.25 +1.135 +1.25 +(0.177) +(0.234) +(0.239) +(0.311) +(0.072) +(0.062) +(0.148) +(0.068) +β4 +1.005 +0.988 +0.8 +0.812 +1.003 +1.002 +0.855 +1.002 +(0.215) +(0.285) +(0.268) +(0.336) +(0.077) +(0.065) +(0.158) +(0.071) +β5 +0.749 +0.749 +0.459 +0.706 +0.75 +0.751 +0.567 +0.751 +(0.285) +(0.331) +(0.339) +(0.352) +(0.09) +(0.074) +(0.183) +(0.078) +β6 +0.508 +0.52 +0.123 +0.511 +0.509 +0.511 +0.316 +0.511 +(0.412) +(0.398) +(0.479) +(0.393) +(0.121) +(0.094) +(0.237) +(0.097) +Notes: This table shows results obtained from the simulations described above. Simulated βτ take the values +{1.75, 1.50, 1.25, 1.00, 0.75, 0.50} for τ = {1, 2, 3, 4, 5, 6}. +4 +Application: The Employment Effects of an In- +novation Policy in France +We now turn to an application of these methods for estimating the causal impact of +a French innovation policy supporting collaborative R&D projects over the period +2010-2016. +Background. +This innovation policy is made up of different subsidy schemes aimed +at developing R&D collaborations between firms and, often, public organizations. +These schemes aim at subsidizing collaborative projects oriented towards applied +research and experimental development.13 To obtain funding, a firm must set up a +research project in partnership with another institution. The project is then sub- +mitted to one specific subsidy scheme, often following calls for proposals, much like +research grant in academic research. The selection of projects, and the associated +13The schemes are FUI, ISI, PSPC, PIAVE, RAPID and ADEME. Although they share the same +general objective, they support different forms of R&D projects. For example, PSPC projects are +much larger in size than others, FUI projects systematically involve companies and public research +organizations, ADEME projects have environmental objectives, and so on. A detailed description +of the schemes is available in Bellgo et al. (2020). +26 + +funding, is based on a list of criteria including, but not limited to, the innovative +nature of the project, its credibility, maturity, or commercial character. +This innovation policy provides an ideal setting to apply our method because R&D +projects take several years to complete. Evaluating the effectiveness of the policy +hence requires estimating its long-term effects. Unfortunately, one of the main data +sources about firm-level R&D activities comes from a survey with a rotating panel +design with multiple strata. Firms investing a large amount in R&D expenditure, i.e. +large companies, are systematically surveyed, while those spending less, i.e. small +and medium-sized enterprises (SMEs) and intermediate-sized enterprises (ISEs), are +surveyed only two consecutive years and then dropped out of the sample. Further- +more, large firms are always involved in at least one R&D project, so it is not possible +to find a plausible counterfactual for them. The policy evaluation should therefore +focus on SMEs and ISEs, for which we only have an unbalanced panel. +The application presented in this paper focuses on the average treatment effect of +participating in all of these schemes without distinguishing their individual effects. It +is relevant to analyze this average effect to the extent that all the schemes contribute +to the common objective of subsidizing collaborative R&D projects. These results +are a priori the most robust because they are obtained with the greatest number of +observations.14 We estimate the treatment effect of this policy on employment. We +focus on employment because it is the main economic variable for which it is also +possible to consistently observe an almost identical measure for all firms using ad- +ministrative data. Therefore, by focusing on such outcome variable, we can compare +the results obtained with the chained DiD estimator to the “true treatment effect”, +obtained for all the firms in the scope of the study using the long DiD. The complete +results of this policy evaluation, including the effects on a larger number of economic +variables, are available in Bellgo et al. (2020). +Data. +Our data contains information about all R&D projects financed under these +schemes over the period 2010-2016. The data includes an unique identifier for each +14It is almost impossible to precisely estimate the individual effect of the smallest schemes as +they have subsidized a very small number of projects. +27 + +partner participating in a project.15 Using this identifier, we have collected exhaus- +tive firm-level data from administrative sources that provide the main annual indica- +tors on the economic activity of companies over the period 2007-2017. In particular, +we collected the following variables: total workforce and the number of engineers +in the workforce from administrative records on firms’ employment as outcomes of +interest, and other variables to used in the propensity scores.16 +Data on the number of researchers is obtained from the R&D survey.17 This an- +nual survey collects information from about 9000 firms companies each year. The +survey has a stratified sampling design: firms with intramural R&D expenditures +above 750,000 euros are systematically surveyed the following year, while others are +surveyed only two years in a row. The majority of the SMEs and ISEs in the scope +of the study are part of the second stratum. In this context, the application of the +standard long DiD method is not possible, which justifies the use of the method +developed in this article. +Having knowledge of the amount of CIR (research tax credit) paid to companies and +their participation in the French cluster policy is essential. Indeed, the amount of +tax credit granted is a good proxy for a company’s propensity to be active in R&D. +In addition, competitiveness clusters aim to create a network of firms and research +organizations to facilitate the formation of collaborative R&D projects. Participation +in one of these clusters also reveals a firm’s tendency for this form of R&D. These +different variables are therefore suitable candidates to explain the probability of +receiving the treatment in the propensity score.18 It is important to observe some +15This identifier corresponds to the SIREN number, a unique identification number for French +businesses supervised by the French national institute of statistics. +16Firms’ revenues come from annual tax data restated by INSEE (FICUS/FARE datasets). Em- +ployment information comes from administrative records on firms’ employment (DADS datasets). +Data on cluster policy comes from the French competitiveness cluster (“Ple de Comptitivit”) man- +agement database. Finally, data on support for innovation comes from the “Crdit Impt Recherche” +(CIR), a research tax credit, and the “Jeunes Entreprises Innovante” (JEI) scheme, a tax and social +exemption aimed at young innovative firms. The CIR is the main tool for supporting innovation +in France. Contrary to the devices evaluated in this article, the CIR is an indirect tax aid, in the +sense that it is automatically distributed to companies making eligible R&D expenditures and that +apply for it. +17This survey also provides detailed information on R&D expenditures, the financing of these +expenditures, and some outputs. +18More specifically, the propensity score includes the log of R&D grants, the log of the number +28 + +variables comprehensively across firms so they can be included in the propensity +score, as they must be observed in g − 1, t, and t + 1 to compute the elementary +building block constituting each chain link of the chained DiD. +The number of firms that can potentially carry out an innovative R&D activity is very +small compared to the total number of firms in France (about 3,000,000). Therefore, +in order to avoid comparing firms participating in a collaborative R&D project to +other firms which are unlikely to pursue such activity, we restrict the scope of the +study to firms active in R&D at least one year over the whole period considered. This +activity is measured by merging together all the sources of information available to +us for this purpose: the databases of the research tax credit, the JEI scheme, and +the R&D survey. Doing so leaves us with about 30,000 firms in the sample. +The schemes covered by the study are the main support mechanisms for collaborative +R&D in France and involve the highest amounts of public support. However, there +are other alternatives not discussed here. Altogether, the various schemes considered +have provided funding for 1697 projects over the period, and have involved 8724 +partners.19. These projects received a total aid of 3.6 billion euros and involved total +expenditures of 10.4 billion euros from their partners. +Results. +In this application, we focus on the effects on (1) total workforce and +(2) the employment of managers and highly qualified workers. Those variables are +observed exhaustively from administrative data (DADS). We estimate the dynamic +treatment effect on these two observed outcomes by using three estimators: the long +DiD, the chained DiD, and the cross-section DiD. +Although these outcomes are consistently observed through time, attrition can still +occur. For example, firms may disappear over time because of economic difficulties +or because they are acquired by another firm. These companies cannot be taken into +account by the long DiD estimator, unless with an attrition model, whereas they are +naturally accounted for in the chained DiD and cross-section DiD estimators. +of engineers, the log of investment, the log of the variation in R&D grants, the log of the variation +in turnover, an indicator for being in a competitiveness cluster, and an indicator for being in the +IT sector. +19A firm can participate in several projects. +29 + +We balance the data to both facilitate the comparison across estimators and get +closer to our theoretical framework. That is, we keep the firms that are consistently +observed from 2007 to 2017 in the administrative data. +This balanced panel is +referred to as the exhaustive panel. We use the long DiD, the chained DiD and the +cross-section DiD on this exhaustive panel. +Then, we construct an unbalanced version of this panel by discarding all observations +whenever a firm was not sampled in the R&D survey in a given year. Both the chained +DiD and cross section DiD estimators are used on this (artifical) unbalanced panel. +The objective is to see how each estimator is affected by discarding observations, and +compare its performance to the long DiD. +Results are presented in Table 5 for the effects on total workforce and Table 6 for +highly qualified workers. +The first column of each table reports the estimates obtained with the long DiD. We +find that total employment has increased by 5.7% the year after the project started. +The long-term increase amounts 12% five years after the start. For highly qualified +workers, the effect is also positive but not statistically significant during the first two +years. It becomes significant from the third year onwards. +As expected, the estimates obtained using the complete panel are very similar be- +tween the long DiD, the chained DiD and the cross-section DiD estimators (columns +1, 2, and 4 of Tables 5 and 6). The standard errors are also similar between the long +DiD and the chained DiD estimators but they are much higher for the cross section +DiD estimator, which suggests that the variance of the unobserved heterogeneity is +large in this data. On the one hand, the estimated coefficients remain quite similar +and the standard errors are only slightly higher with the chained DiD estimator on +the unbalanced panel (columns 3 of Tables 5 and 6). On the other hand, the esti- +mates are considerably worse when obtained with the cross-section DiD estimator +on the unbalanced panel. The point estimates are different and the standard errors +become too large to appreciate the effects of the policy (columns 5 of Tables 5 and +6). +30 + +Table 5: Effects on total workforce (exhaustively observed outcome) +log(total workforce) +Long DiD +Chained DiD +Cross Section DiD +Exhaustive +Exhaustive +Unbalanced +Exhaustive +Unbalanced +(1) +(2) +(3) +(4) +(5) +β−3 +-0.013 +-0.013 +0.032 +-0.013 +0.075 +[-0.084,0.058] +[-0.079,0.053] +[-0.059,0.123] +[-0.144,0.117] +[-0.609,0.76] +β−2 +-0.023 +-0.023 +-0.008 +-0.023 +0.012 +[-0.091,0.044] +[-0.087,0.041] +[-0.097,0.081] +[-0.13,0.083] +[-0.549,0.574] +β−1 +-0.005 +-0.005 +-0.005 +-0.005 +0.024 +[-0.054,0.044] +[-0.052,0.041] +[-0.066,0.056] +[-0.09,0.08] +[-0.442,0.491] +ref. +0 +0 +0 +0 +0 +β1 +0.027 +0.027 +0.046 +0.028 +0.062 +[-0.026,0.08] +[-0.023,0.078] +[-0.021,0.112] +[-0.103,0.159] +[-0.372,0.495] +β2 +0.057** +0.056** +0.081** +0.057 +0.092 +[-0.006,0.121] +[-0.005,0.118] +[0.002,0.159] +[-0.048,0.163] +[-0.426,0.61] +β3 +0.07** +0.068** +0.089** +0.069 +0.051 +[0,0.14] +[0,0.135] +[0.006,0.172] +[-0.075,0.214] +[-0.506,0.607] +β4 +0.084** +0.077** +0.119*** +0.083 +0.087 +[0.002,0.167] +[0,0.154] +[0.024,0.214] +[-0.091,0.257] +[-0.53,0.704] +β5 +0.123*** +0.111*** +0.149*** +0.119* +0.076 +[0.032,0.213] +[0.025,0.197] +[0.037,0.261] +[-0.032,0.269] +[-0.533,0.686] +Pre- +-0.014 +-0.014 +0.006 +-0.014 +0.037 +trend +[-0.048,0.021] +[-0.049,0.021] +[-0.04,0.053] +[-0.079,0.051] +[-0.294,0.368] +Notes: This table shows the dynamic treatment effects relative to the beginning of the treat- +ment obtained on a panel balanced on exhaustive variables. “Exhaustive” refers to the use +of an exhaustively observed outcome without pretending that this variable is imperfectly ob- +served. “Unbalanced” refers to the use of an exhaustively observed outcome pretending that +this variable is observed from R&D survey, that is, from an unbalanced repeated panel. 95% +confidence intervals are obtained from the multiplier bootstrap. ∗p < 0.10, ∗∗p < 0.05, ∗ ∗ ∗ +p < 0.01 +31 + +Table 6: Effects on highly qualified workers (exhaustively observed outcome) +log(highly qualified workforce) +Long DiD +Chained DiD +Cross Section DiD +Exhaustive +Exhaustive +Unbalanced +Exhaustive +Unbalanced +(1) +(2) +(3) +(4) +(5) +β−3 +0.033 +0.033 +0.088 +0.033 +0.194 +[-0.059,0.126] +[-0.053,0.12] +[-0.059,0.235] +[-0.081,0.148] +[-0.307,0.696] +β−2 +0.018 +0.018 +0.013 +0.018 +0.006 +[-0.066,0.103] +[-0.06,0.096] +[-0.131,0.157] +[-0.08,0.116] +[-0.411,0.423] +β−1 +0.018 +0.018 +0.015 +0.018 +0.019 +[-0.058,0.094] +[-0.054,0.09] +[-0.099,0.129] +[-0.065,0.101] +[-0.328,0.365] +ref. +0 +0 +0 +0 +0 +β1 +0.022 +0.022 +0.02 +0.024 +0.034 +[-0.056,0.1] +[-0.048,0.091] +[-0.09,0.129] +[-0.083,0.13] +[-0.287,0.355] +β2 +0.044 +0.043 +0.067 +0.045 +0.065 +[-0.038,0.126] +[-0.032,0.118] +[-0.051,0.184] +[-0.05,0.141] +[-0.319,0.449] +β3 +0.075* +0.072** +0.061 +0.072 +0.02 +[-0.015,0.164] +[-0.008,0.152] +[-0.066,0.189] +[-0.05,0.194] +[-0.388,0.428] +β4 +0.109** +0.104** +0.108 +0.105 +0.008 +[0.007,0.211] +[0.011,0.196] +[-0.04,0.256] +[-0.04,0.25] +[-0.439,0.456] +β5 +0.125** +0.117*** +0.128* +0.121* +0.069 +[0.014,0.236] +[0.016,0.217] +[-0.032,0.288] +[-0.014,0.257] +[-0.39,0.528] +Pre- +0.023 +0.023 +0.039 +0.023 +0.073 +trend +[-0.022,0.068] +[-0.018,0.064] +[-0.035,0.113] +[-0.033,0.08] +[-0.173,0.319] +Notes: This table shows the dynamic treatment effects relative to the beginning of the treat- +ment obtained on a panel balanced on exhaustive variables. “Exhaustive” refers to the use +of an exhaustively observed outcome without pretending that this variable is imperfectly ob- +served. “Unbalanced” refers to the use of an exhaustively observed outcome pretending that +this variable is observed from R&D survey, that is, from an unbalanced repeated panel. 95% +confidence intervals are obtained from the multiplier bootstrap. ∗p < 0.10, ∗∗p < 0.05, ∗ ∗ ∗ +p < 0.01 +Finally, we apply the chained DiD and cross-section DiD estimators on outcomes +similar to those studied just above but actually coming from the R&D survey. In +this context, it is not possible to apply the standard long DiD estimator because +there are too few observations to calculate differences. For consistency reasons, we +present estimates obtained on the same set of firms as those used for the tables 5 +32 + +and 6.20 The outcome variables do not correspond to the exact same definition of +employment depending on whether they come from the exhaustive administrative +source or from the R&D survey. Total employees headcount from DADS adminis- +trative data corresponds to observations at the legal unit level, whereas the R&D +survey sometimes provide information on employment at the group level.21 The em- +ployment of highly qualified workers from the DASD is close to the number of R&D +researchers and engineers filled in the R&D survey. Highly qualified workers include +engineers but also qualified workers dedicated to other tasks than R&D. Conversely, +R&D researchers and engineers include researchers who are specifically assigned to +research tasks. Despite their difference, these variables measure similar outcomes +and are highly correlated, which justifies their comparison. +Results using outcome variables observed from R&D survey are presented in Table +7. The effects obtained with the chained DiD on total employment are somewhat +less significant than those presented in Table 5, but the coefficients keep the same +order of magnitude (column 1). As might be expected since the policy directly aims +at fostering R&D activities, the effects on the number of researchers are stronger and +more significant (column 2). On the other hand, the effects obtained with the DiD +cross section are much less precise (columns 3 and 4). +20That is, we use the set of data that is balanced on the exhaustive variables, which is then +merged with the R&D survey, and estimate the effects on the variables reported in the R&D +survey. +21A legal unit is a legal entity of public or private law. A firm, in the sense of a group, is an +economic entity that may comprise several legal units thanks to financial links. +33 + +Table 7: Effects on employment variables observed from R&D survey +Unbalanced variables from R&D survey in log +Chained DiD +Cross Section DiD +total workforce +researchers +total workforce +researchers +(1) +(2) +(3) +(4) +β−3 +0.053 +-0.041 +0.104 +0.106 +[-0.066,0.172] +[-0.194,0.112] +[-0.598,0.806] +[-0.249,0.462] +β−2 +0.012 +0.017 +0.064 +0.056 +[-0.111,0.135] +[-0.103,0.137] +[-0.554,0.682] +[-0.246,0.357] +β−1 +0.009 +0.007 +0.02 +0.033 +[-0.074,0.092] +[-0.104,0.119] +[-0.481,0.522] +[-0.219,0.285] +ref. +0 +0 +0 +0 +β1 +0.053 +0.06 +0.075 +0.082 +[-0.032,0.139] +[-0.051,0.17] +[-0.409,0.56] +[-0.171,0.336] +β2 +0.038 +0.124** +0.052 +0.132 +[-0.073,0.149] +[0.001,0.248] +[-0.499,0.603] +[-0.164,0.427] +β3 +0.047 +0.181*** +0.057 +0.139 +[-0.068,0.162] +[0.041,0.32] +[-0.525,0.638] +[-0.178,0.455] +β4 +0.129** +0.245*** +0.081 +0.148 +[0.007,0.25] +[0.093,0.397] +[-0.57,0.733] +[-0.189,0.485] +β5 +0.168*** +0.288*** +0.116 +0.221 +[0.034,0.302] +[0.128,0.448] +[-0.546,0.778] +[-0.144,0.585] +Pre- +0.025 +-0.006 +0.063 +0.065 +trend +[-0.032,0.082] +[-0.075,0.064] +[-0.319,0.444] +[-0.116,0.246] +Notes: This table shows the dynamic treatment effects relative to the beginning of the treat- +ment. The dynamic effects are estimated with outcome variables observed from R&D survey, +that is, from an unbalanced repeated panel. 95% confidence intervals are obtained from the +multiplier bootstrap. ∗p < 0.10, ∗∗p < 0.05, ∗ ∗ ∗ +p < 0.01 +It is not surprising that the coefficients estimated with the cross section DiD esti- +mator is outperformed by the chained DiD estimator. The R&D survey is a good +example of a rotating panel in which the sampling in period t depends on the level +of the outcome variable in t − 1, so that St ̸⊥⊥ Yt−1.22 There is, however, no rea- +son why being surveyed two periods in a row is correlated with the evolution of the +outcome. So we have every reason to think that St,t+1 ⊥ (Yt+1 − Yt). Conversely, +22To be precise, the sampling depends on the level of a firm’s internal R&D expenditure, which +mainly includes researchers’ salaries. +34 + +economic variables are often autocorrelated, so Yt is likely to be correlated to Yt−1 +and the hypothesis St ⊥ Yt is unlikely to be verified. This is particularly true for +employment, which is characterized by a strong hysteresis and significant unobserved +heterogeneity. Under these conditions, the chained DiD estimator is more likely to +perform well. +Finally, we reproduce the results of Tables 5, 6, and 7 by estimating the treatment +effects with the complete original sample from the administrative data, without dis- +carding the individual firms that are not consistently observed throughout the period +to create a balanced panel. The results are presented in Appendix B and confirm +the better performance of the chained DiD estimator compared to the cross-section +DiD one. +5 +Conclusion +In this paper, we have developed a new estimator to identify long-term treatment +effects in unbalanced panel data sets. This is an important issue, not only because +of attrition but also due to how surveys are designed. Common practices are either +to use a long DiD estimator by balancing the data, at the cost of losing precision +and possibly biasing the results, or to use a cross-section DiD estimator at the cost +of not accounting for unobserved heterogeneity. 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(2010): Econometric analysis of cross section and panel data, +MIT press. +40 + +A +Mathematical Appendix +A.1 +Simple setting +Proof 1 (Proof of Proposition 1) In order to identify and estimate the ATT(t) +in this simple setting, we impose the following standard assumptions. +Assumption 8 (Sampling) For all t = 1, ..., T , {Yit, Yit+1, Di1, Di2, ..., DiT }nt +i=1 is +independent and identically distributed (iid) conditional on St,t+1 = 1. +Assumption 9 (Independence of Sampling and Levels) For all t = 1, ..., T , +(Yt, D1, D2, ..., DT ) ⊥⊥ St. +(13) +Assumption 10 (Unconditional Parallel Trends) For all t = 2, ..., T , +E [Yt(0) − Yt−1(0)|G = 1] = E [Yt(0) − Yt−1(0)|C = 1] a.s.. +(14) +Assumption 11 (Irreversibility of Treatment) For all t = 2, ..., T , +Dt−1 = 1 implies that Dt = 1. +(15) +Assumption 12 (Existence of Treatment and Control Groups) +P(G = 1) = 1 − P(C = 1) ∈ (0, 1). +(16) +Substituting (4) and rearranging yields +41 + +� +ATTCD(t) = +t−1 +� +τ=1 +1 +n +n +� +i=1 +� � +wG +iττ+1 (y′ +i − yi) − � +wC +iττ+1 (y′ +i − yi) +� += +t−1 +� +τ=1 +1 +n +n +� +i=1 +� � +wG +iττ+1 (yiτ+1 − yiτ) − � +wC +iττ+1 (yiτ+1 − yiτ) +� += +t−1 +� +τ=1 +1 +n +n +� +i=1 +� � +wG +iττ+1 (δτ+1 − δτ + βt + εiτ+1 − εiτ) − � +wC +iττ+1 (δτ+1 − δτ + εiτ+1 − εiτ) +� += +t−1 +� +τ=1 +� +1 +n +n +� +i=1 +� +wG +iττ+1βτ+1 + 1 +n +n +� +i=1 +� � +wG +iττ+1 − � +wC +iττ+1 +� +(εiτ+1 − εiτ) +� += +t +� +τ=2 +βτ + +t−1 +� +τ=1 +� +1 +n +n +� +i=1 +� +wG +iττ+1 (εiτ+1 − εiτ) − 1 +n +n +� +i=1 +� +wC +iττ+1 (εiτ+1 − εiτ) +� +, +where the second equality follows from the fact that wG +iττ+1 ̸= 0 and wC +iττ+1 ̸= 0 only if +yiτ+1 − yiτ is observed, and the fourth and fifth equalities follow from �n +i=1 � +wG +iττ+1 = +�n +i=1 � +wC +iττ+1 = 1. The second term in the final expression vanishes in expectations +from Assumptions 8 and 10. +The second estimator writes +42 + +� +ATTCS(t) =1 +n +n +� +i=1 +�� � +wG +it − � +wG +i1 +� +yi − +� � +wC +it − � +wC +i1 +� +yi +� +=1 +n +n +� +i=1 +�� � +wG +ityit − � +wG +i1yi1 +� +− +� � +wC +ityit − � +wC +i1yi1 +�� +=1 +n +n +� +i=1 +� � +wG +it − � +wC +it +� +yit − 1 +n +n +� +i=1 +� � +wG +i1 − � +wC +i1 +� +yi1 +=1 +n +n +� +i=1 +� � +wG +it − � +wC +it +� � +αi + δt + +t +� +τ=2 +βτDiτ + εit +� +− 1 +n +n +� +i=1 +� � +wG +i1 − � +wC +i1 +� +(αi + δ1 + εi1) +=1 +n +n +� +i=1 +� +wG +it +t +� +τ=2 +βτ + 1 +n +n +� +i=1 +� � +wG +it − � +wC +it − � +wG +i1 + � +wC +i1 +� +αi + 1 +n +n +� +i=1 +� � +wG +it − � +wC +it +� +δt +− 1 +n +n +� +i=1 +� � +wG +i1 − � +wC +i1 +� +δ1 + 1 +n +n +� +i=1 +� � +wG +it − � +wC +it +� +εi1 − 1 +n +n +� +i=1 +� � +wG +i1 − � +wC +i1 +� +εi1 += +t +� +τ=2 +βτ + 1 +n +n +� +i=1 +� � +wG +it − � +wC +it +� +εit − 1 +n +n +� +i=1 +� � +wG +i1 − � +wC +i1 +� +εi1 ++ 1 +n +n +� +i=1 +� � +wG +it − � +wC +it +� +αi − 1 +n +n +� +i=1 +� � +wG +i1 − � +wC +i1 +� +αi, +where the second and third terms vanish in expectations under Assumption 10. Fi- +nally, the expectation of the last term equates zero under Assumption 9. +We will prove the asymptotic normality of these estimators in the general frame- +work in Theorem 2. In this proof, we only derive their asymptotic variance for the +mentioned example. +43 + +E +� +n +� +� +ATTCD(t) − ATT(t) +�2� += nE + + +� t−1 +� +τ=1 +� +1 +n +n +� +i=1 +� � +wG +iττ+1 − � +wC +iττ+1 +� +(εiτ+1 − εiτ) +��2 + += +t−1 +� +τ=1 +1 +n +n +� +i=1 +E +�� � +wG +iττ+1 − � +wC +iττ+1 +�2 +(εiτ+1 − εiτ)2 +� += +t−1 +� +τ=1 +1 +n +n +� +i=1 +E +�� � +wG +iττ+1 − � +wC +iττ+1 +�2� +E +� +(εiτ+1 − εiτ)2� +, += +t−1 +� +τ=1 +E +�� � +wG +iττ+1 − � +wC +iττ+1 +�2� +E +� +(εiτ+1 − εiτ)2� +, +where the third equality follows from independence of G and εiτ+1 − εiτ. As n → ∞, +the weak law of large numbers implies E +� +� +wG +iττ+1 +2� +p→ +1 +P (SτSτ+1G=1) because Sitt+1ai ∈ +0, 1 and E[Sitt+1aiSjtt+1aj] = 0 for i ̸= j. Therefore, the asymptotic variance for +n → ∞ is +E +� +n +� +� +ATTCD(t) − ATT(t) +�2� += +t−1 +� +τ=1 +� 1 +qp + +1 +q(1 − p) +� � +(ρ − 1)2σ2 +ε + σ2 +η +� += +(t − 1) +qp(1 − p) +2(1 − ρ) +1 − ρ2 σ2 +η += 2 (t − 1) +qp(1 − p) +1 +1 + ρσ2 +η, +when assuming P(SiτSiτ+1) = q ∈ (0, 1) for all τ, i and P(Gi = 1) = p ∈ (0, 1). +Similarly, the variance of the second estimator can be developed into +44 + +E +� +n +� +� +ATTCS(t) − ATT(t) +�2� +=E +�� � +wG +it − � +wC +it +�2� +σ2 +ε + E +�� � +wG +i1 − � +wC +i1 +�2� +σ2 +ε ++ E +�� � +wG +it − � +wC +it +�2 +α2 +i +� ++ E +�� � +wG +i1 − � +wC +i1 +�2 +α2 +i +� +− E +�� � +wG +i1 − � +wC +i1 +� +αi +� +E +�� � +wG +it − � +wC +it +� +αi +� +=2E +�� � +wG +it − � +wC +it +�2� +σ2 +ε + 2E +�� � +wG +it − � +wC +it +�2� +σ2 +α, += +1 +qp(1 − p)(σ2 +ǫ + σ2 +α), +where the second equality follows from the independence of outcomes and sampling as +well as G and αi. As n → ∞, we have E +� +� +wG +iτ +2� +p→ +1 +P (Sτ G=1) = +1 +P (Sτ Sτ+1G=1)+P (Sτ−1Sτ G=1) = +1 +2pq for τ < T , and E +� +� +wG +iτ +2� +p→ +1 +P (Sτ G=1) = +1 +pq for τ = T , because we have two over- +lapping samples in each period except for the first and last. For t < T , the asymptotic +variance as n → ∞ is thus +E +� +n +� +� +ATTCS(t) − ATT(t) +�2� +p→ +1 +qp(1 − p) +� +σ2 +η +1 − ρ2 + σ2 +α +� +. +A.2 +General framework +A.2.1 +Proofs for rotating panel setting +Proof 2 (Proof of Theorem 1) This proof focuses on the identification of pa- +rameters in the general framework with a rotating panel structure. +Let us define +ATTX(g, τ) = E[Yτ(1) − Yτ(0)|X, Gg = 1] to write its first-difference as +∆ATTX(g, τ) = ATTX(g, τ) − ATTX(g, τ − 1) += E[Yτ(1) − Yτ(0)|X, Gg = 1] − E[Yτ−1(1) − Yτ−1(0)|X, Gg = 1] += E[Yτ − Yτ−1|X, Gg = 1, Sτ,τ−1 = 1] − E[Yτ − Yτ−1|C = 1, Sτ,τ−1 = 1] += AX(g, τ) − BX(g, τ) +45 + +where the third equality follows from the conditional parallel trends assumption and +the sampling independence. +We can use the above expression to develop ATT(g, t) into +ATT(g, t) = E (E[Yt(1) − Yt(0)|X, Gg = 1]|Gg = 1) += E (ATTX(g, t)|Gg = 1) += E +� +t +� +τ=g +∆ATTX(g, τ)|Gg = 1 +� += +t +� +τ=g +E (∆ATTX(g, τ)|Gg = 1, Sτ,τ−1 = 1) += +t +� +τ=g +E (AX(g, τ) − BX(g, τ)|Gg = 1, Sτ,τ−1 = 1) +(17) +with +E (AX(g, τ)|Gg = 1, Sτ,τ−1 = 1) = E (Yτ − Yτ−1|Gg = 1, Sτ,τ−1 = 1) += EM +� +(Yτ − Yτ−1) +GgSτ,τ−1 +E[GgSτSτ−1] +� +(18) +by the law of iterated expectations and the definition of FM. Following the proofs +of Theorem 1 and B.1 of Callaway and Sant’Anna (2021), the second term can be +developed into23 +E(BX(g, τ)|Gg = 1, Sτ,τ−1 = 1) = E(E[Yτ − Yτ−1|X, C, Sτ,τ−1]|Gg = 1, Sτ,τ−1 = 1) += E +� +E[ +C +1 − P(Gg = 1|X, Gg + C, Sτ,τ−1)(Yτ − Yτ−1)|X, Gg + C = 1, Sτ,τ−1]|Gg, Sτ,τ−1 +� += +E +� +GgSτ,τ−1E[ +C +1−P (Gg|X,Gg+C,Sτ,τ−1)(Yτ − Yτ−1)|X, Gg + C, Sτ,τ−1]|Gg + C, Sτ,τ−1 +� +P(Gg = 1|Gg + C, Sτ,τ−1) +, +(19) +23We alleviate notations by dropping = 1 from conditioning sets. +46 + +where using the definition of pg yields +... = +E +� +GgSτ,τ−1E[ +C +1−pg(X)(Yτ − Yτ−1)|X, Gg + C, Sτ,τ−1]|Gg + C, Sτ,τ−1 +� +P(Gg = 1|Gg + C, Sτ,τ−1) += +E +� +E[ pg(X)C +1−pg(X)(Yτ − Yτ−1)|X, Gg + C, Sτ,τ−1]|Gg + C, Sτ,τ−1 +� +E(Gg|Gg + C, Sτ,τ−1) += +E +� +(Gg + C)E[ pg(X)C +1−pg(X)(Yτ − Yτ−1)|X, Gg + C, Sτ,τ−1]|Sτ,τ−1 +� +E(Gg|Gg + C, Sτ,τ−1)E((Gg + C)|Sτ,τ−1) += +E +� +(Gg + C)E[ pg(X)C +1−pg(X)(Yτ − Yτ−1)|X, Gg + C, Sτ,τ−1]|Sτ,τ−1 +� +E(Gg|Sτ,τ−1) += +E +� +E[(Gg + C)|X, Sτ,τ−1]E[ pg(X)C +1−pg(X)(Yτ − Yτ−1)|X, Gg + C, Sτ,τ−1]|Sτ,τ−1 +� +E(GgSτ,τ−1) += +E +� +E[ pg(X)C +1−pg(X)(Yτ − Yτ−1)|X, Sτ,τ−1]|Sτ,τ−1 +� +E(Gg|Sτ,τ−1) += +E +� +pg(X)C +1−pg(X)(Yτ − Yτ−1)|Sτ,τ−1 +� +E(Gg|Sτ,τ−1) += +EM +� +pg(X)CSτ,τ−1 +1−pg(X) +(Yτ − Yτ−1) +� +EM(GgSτ,τ−1) +(20) +with +EM +�pg(X)CSτ,τ−1 +1 − pg(X) +� += EM(EM(Gg|X, Gg + C = 1)C +EM(C|X, Gg + C = 1) Sτ,τ−1) += EM(EM(Gg|X)EM(C|X) +EM(C|X) +Sτ,τ−1) += EM(EM(Gg|X)Sτ,τ−1) += EM(GgSτ,τ−1). +(21) +Finally, the proof that identification is not guaranteed with the repeated cross-sectional +estimator (cross-section DiD) presented in Appendix B of Callaway and Sant’Anna +(2021) follows from taking a counterexample. Under Assumption 2, it is possible that +E[Yt|X, C = 1, St = 1] = E[Yt|X, C = 1] but E[Yt|X, Gg = 1, St = 1] = E[Yt|X, Gg = +47 + +1] + αt with α > 0. Following the steps of the proof it is is easy to show that the +cross-section DiD identifies +ATT(g, t) + α(t − g + 1). +Proof 3 (Proof of Theorem 2) This proof is adapted from Callaway and Sant’Anna +(2021) ’s Theorem 2. We proceed in 3 steps. +Parametric propensity scores and notations. +First, we introduce additional +notations and explain Assumption 5 in Callaway and Sant’Anna (2021) about the +estimation of propensity scores.24 +Let Wi = (Yit, Yit+1, Xi, Gi1, Gi2, ..., , GiT , Ci)′ +denote the data for an individual i observed in t and t + 1. +Assumption 5 in +Callaway and Sant’Anna (2021) assumes that the propensity scores, parametrized as +pg(Xi) = Λ(X′ +iπ0 +g) with Λ(·) being a known function (logit or probit), can be paramet- +rically estimated by maximum likelihood. We denote ˆpg(Xi) = Λ(X′ +iˆπg) where ˆπg are +estimated by ML, ˙pg = ∂pg(u)/∂u, and ˙pg(X) = ˙pg(X′ +iπ0 +g). Under this assumption, +the estimated parameter ˆπg is asymptotically linear, i.e., +√n(ˆπg − π0 +g) = +1 +√n +� +i +ξπ +g (Wi) + op(1), +where ξπ +g (Wi) is defined in (3.1) is Callaway and Sant’Anna (2021) and does not +depend on the sampling process since X is observed for all individuals. +Let us now define, +ψgt(Wi) = ψG +gt(Wi) + ψG +gt(Wi), +(22) +where +ψG +gt(Wi) =wG +it,t−1(g) +� +(Yit − Yit−1) − EM +� +wG +it,t−1(g)(Yit − Yit−1) +�� +, +ψC +gt(Wi) =wC +it,t−1(g, X) +� +(Yit − Yit−1) − EM +� +wC +it,t−1(g, X)(Yit − Yit−1) +�� ++ M′ +gtξπ +g (Wi), +24This is a standard assumption in the literature so it is not reproduced here. +48 + +and +Mgt = +EM +� +X( CStSt−1 +1−pg(X))2 ˙pg(X) +� +(Yit − Yit−1) − EM +� +wC +it,t−1(g, X)(Yit − Yit−1) +��� +EM[ pg(X)C +1−pg(X)] +. +is a k dimensional vector, k being the number of covariates in X. +Finally, let +� +∆ATT g≤t and ∆ATTg≤t denote the vectors of all � +∆ATT(g, t) and ∆ATT(g, t) for +any 2 ≤ g ≤ t ≤ T . Similarly, the collection of ψgt across all periods and groups +such that g ≤ t is denoted by Ψg≤t. +Asymptotic result for ∆ATT. +Second, we show the asymptotic result for ∆ATT. +Recall that +� +ATT(g, t) = +t +� +τ=g +� +∆ATT(g, τ), +where +� +∆ATT(g, τ) = ˆEM +� +GgSτ−1Sτ +ˆEM[GgSτ−1Sτ] +(Yτ − Yτ−1) +� +− ˆEM + + +pg(X)CSτ−1Sτ +1−pg(X) +ˆEM +� +pg(X)CSτ−1Sτ +1−pg(X) +�(Yτ − Yτ−1) + + += � +∆ATT g(g, τ) − � +∆ATT C(g, τ), +where ˆE denotes the empirical mean. We will show separately that, for all for all +2 ≤ g ≤ t ≤ T , +√n +� +� +∆ATT g(g, t) − ∆ATTg(g, t) +� += +1 +√n +� +i +ψG +gt(Wi) + op(1), +(23) +and +√n +� +� +∆ATT C(g, t) − ∆ATTC(g, t) +� += +1 +√n +� +i +ψC +gt(Wi) + op(1), +(24) +which together implies +√n +� +� +∆ATT(g, t) − ∆ATT(g, t) +� += +1 +√n +� +i +ψgt(Wi) + op(1) +(25) +49 + +and the asymptotic normality of √n +� +� +∆ATT g≤t − ∆ATTg≤t +� +follows from the mul- +tivariate central limit theorem. +First, we show (23). Let βg = EM[GgSτ−1Sτ] and ˆβg = ˆEM[GgSτ−1Sτ] and note that +√n +� +ˆβg − βg +� += +1 +√n +� +i +(GigSiτ−1Siτ − E[GgSτ−1Sτ]) +p→ 0, as n → +∞. +Then, for all 2 ≤ g ≤ t ≤ T , +√n( � +∆ATT g(g, t)−∆ATTg(g, t)) = √n ˆEM +� +GgSt,t−1 +ˆβg +(Yt − Yt−1) +� +− √nEM +�GgSt,t−1 +βg +(Yt − Yt−1) +� += +√n +ˆβg +( ˆEM [GgSt,t−1(Yt − Yt−1)] − +ˆβg +βg +EM [GgSt,t−1(Yt − Yt−1)]) += +√n +ˆβg +( ˆEM [GgSt,t−1(Yt − Yt−1)] − EM [GgSt,t−1(Yt − Yt−1)]), +and by the continuous mapping theorem, +√n( � +∆ATT g(g, t)−∆ATTg(g, t)) = +√n +βg +( ˆEM [GgSt,t−1(Yt − Yt−1)] − EM [GgSt,t−1(Yt − Yt−1)]) +− √n +� +1 +βg +− 1 +ˆβg +� +EM [GgSt,t−1(Yt − Yt−1)] + op(1) += +√n +βg +( ˆEM [GgSt,t−1(Yt − Yt−1)] − EM [GgSt,t−1(Yt − Yt−1)]) +− +√n +� +ˆβg − βg +� +β2 +g +EM [GgSt,t−1(Yt − Yt−1)] + op(1) += +√n +βg +( ˆEM [GgSt,t−1(Yt − Yt−1)] − +ˆβg +βg +EM [GgSt,t−1(Yt − Yt−1)]) + op(1) += 1 +√n +� +i +Gig +Sit,t−1(Yit − Yit−1) − ∆ATT(g, t) +βg ++ op(1) += 1 +√n +� +i +wG +it,t−1(g) +� +(Yit − Yit−1) − EM +� +wG +t,t−1(g)(Yt − Yit−1) +�� ++ op(1) += 1 +√n +� +i +ψG +gt(Wi) + op(1), +50 + +proving (23). +Let us now turn to (24). For an arbitrary function g, let +wt(g) = g(X)CSt,t−1 +1 − g(X) +and note that +√n( � +∆ATT C(g, t)−∆ATTC(g, t)) = √n +� +ˆEM +� +wt(ˆpg) +ˆEM[wt(ˆpg)] +(Yt − Yt−1) +� +− EM +� +wt(pg) +EM[wt(pg)](Yt − Yt−1) +�� += +√n +ˆEM[wt(ˆpg)] +� +ˆEM [wt(ˆpg)(Yt − Yt−1)] − +ˆEM[wt(ˆpg)] +EM[wt(pg)]EM [wt(pg)(Yt − Yt−1)] +� += +√n +ˆEM[wt(ˆpg)] +� +ˆEM [wt(ˆpg)(Yt − Yt−1)] − EM [wt(pg)(Yt − Yt−1)] +� +− EM [wt(pg)(Yt − Yt−1)] +ˆEM[wt(ˆpg)]EM[wt(pg)] +√n( ˆEM [wt(ˆpg)] − ˆEM [wt(pg)]) += +1 +ˆEM[wt(ˆpg)] +√nAn(ˆpg) − ∆ATTC(g, t) +ˆEM[wt(ˆpg)] +√nBn(ˆpg) += +1 +EM[wt(pg)] +√nAn(ˆpg) − ∆ATTC(g, t) +EM[wt(pg)] +√nBn(ˆpg) + op(1), +where the last equality follows directly from Assumption 5, which implies Lemma +A.2 and Lemma A.3 in Callaway and Sant’Anna (2021). Applying the mean value +theorem yields +An(ˆpg) = ˆEM [wt(pg)(Yt − Yt−1)] − EM [wt(pg)(Yt − Yt−1)] ++ ˆEM +� +X +CSt,t−1 +(1 − pg(X; π))2 ˙pg(X; π) +�′ � +ˆπg − π0 +g +� +, +where π is an intermediate point that satisfies |πgπ0 +g| ≤ |ˆπgπ0 +g| a.s. Thus, by As- +sumption 5, the previously mentioned Lemmas, and the Classical Glivenko-Cantelli’s +51 + +theorem, +An(ˆpg) = ˆEM [wt(pg)(Yt − Yt−1)] − EM [wt(pg)(Yt − Yt−1)] ++ ˆEM +� +X +CSt,t−1 +(1 − pg(X))2 ˙pg(X) +�′ � +ˆπg − π0 +g +� ++ op(n−1/2), +and using the same reasoning we obtain +Bn(ˆpg) = ˆEM [wt(pg)] − EM [wt(pg)] ++ ˆEM +� +X +CSt,t−1 +(1 − pg(X))2 ˙pg(X) +�′ � +ˆπg − π0 +g +� ++ op(n−1/2). +Combining the above results and making use of the same Lemma yields (25) hence +concludes the proof for ∆ATT. The asymptotic covariance is given by Σ∆ = E [Ψg≤τ(Wi)Ψg≤τ(Wi)′]. +Asymptotic result for ATT. +Finally, by making use of (25) we have that +√n +� +� +ATT(g, t) − ATT(g, t) +� += +t +� +τ=g +√n +� +� +∆ATT(g, τ) − ∆ATT(g, τ) +� += 1 +√n +� +i +� +t +� +τ=g +ψgτ(Wi) +� ++ op(1) +d→ N(0, Σ), +where Σ = EM [Φg≤τ(Wi)Φg≤τ(Wi)′] with Φg≤τ(Wi) = �t +τ=g Ψg≤τ(Wi). +Therefore, the influence function of the chained DiD estimator corresponds to the +sum of influence functions of the short-term DiD estimator. +A.2.2 +Bootstrap implementation for rotating panel setting +Bootstrapped confidence bands for � +ATT(g, t). +The algorithm is as follows: +1. Draw a vector of Vb = (V1, ..., Vi, ..., Vn)′, where Vi’s are iid zero mean random +variables with unit variance, such as Bernoulli random variables with Pr(V = +1 − κ) = κ/ +√ +5 with κ = ( +√ +5 + 1)/2 as suggested by Mammen (1993). +52 + +2. Compute the bootstrap draw � +ATT +⋆b +g≤t = � +ATT g≤t + ˆΦg≤τVb where ˆΦg≤τ is a +consistent estimator of Φg≤τ (see below). +3. Compute ˆR⋆b(g, t) = √n(� +ATT +⋆b(g, t) − � +ATT(g, t)) for each element of the +vector � +ATT +⋆b +g≤t. +4. Repeat steps 1-3 B times. +Note: do not re-estimate propensity scores and +parameters for each draw. +5. Compute the bootstrapped covariance for each (g, t) as ˆΣ1/2(g, t) = (q0.75(g, t)− +q0.25(g, t))/(z0.75 − z0.25), where qp(g, t) is the pth sample quantile of ˆR⋆ (across +B draws) and z(g, t) is the pth quantile the standard normal distribution. +6. For each b, compute t-statb +g≤t = max(g,t) | ˆR⋆b(g, t)|ˆΣ−1/2(g, t). +7. Construct ˆc1−α as the empirical (1 − α) quantile of the B boostrap draws of +t-statb +g≤t. +8. Construct the bootstrapped simultaneous confidence band for ATT(g, t) as +ˆC(g, t) = [� +ATT(g, t) ± ˆc1−αˆΣ−1/2(g, t)/√n]. +This procedure requires to compute ˆΦg≤τ, represented here as a K ×n matrix, with n +being the number of observations and K = T (T −1) +2 +being the number of (g, t) element +for any 2 ≤ g ≤ t ≤ T . This is done as follows: +1. For every (g, t), compute the n-dimensional vector ψgt with ith element defined +as ψgt(i) = ψG +gt(i) + ψC +gt(i), where +ψG +gt(i) =wG +it,t−1(g) +� +(Yit − Yit−1) − EM +� +wG +t,t−1(g)(Yt − Yt−1) +�� +, +ψC +gt(i) =wC +it,t−1(g, X) +� +(Yit − Yit−1) − EM +� +wC +t,t−1(g, X)(Yt − Yt−1) +�� ++ M′ +gtξπ +g (i), +where Mgt, a k dimensional vector (k being the number of covariates in X), is +defined as +Mgt = +E +� +X( CStSt−1 +1−pg(X))2 ˙pg(X) +� +(Yit − Yit−1) − E +� +wC +t,t−1(g, X)(Yt − Yit−1) +��� +E[ pg(X)C +1−pg(X)] +, +53 + +with ˆpg(Xi) = Λ(X′ +iˆπg) being the parametric propensity score for covariates Xi +and ˙pg(X) = ∂Λ(X′ +iˆπg)/∂(X′ +iˆπg). Furthermore, ξπ +g (i) is a k-dimensional vector +for each observation i, and is given by +ξπ +g (i) = EM +� (Gg + C) ˙pg(X)2 +pg(Xi)(1 − pg(Xi))XX′ +�−1 +Xi +(Gg + C)(Gg − pg(Xi)) ˙pg(Xi) +pg(Xi)(1 − pg(Xi)) +. +2. Compute φgt = �t +τ=g ψg≤τ for all 2 ≤ g ≤ t ≤ T . +3. Concatenate all φgt’s into a K × n matrix Φg≤t. +A.2.3 +Summary parameters for rotating panel setting +The group-time average treatment effect ATT(g, t) consists of a building-block to +study the dynamic effect of a treatment on different cohorts of treated individ- +uals. +In most applications, the main causal parameters of interest are not the +ATT(g, t) themselves but aggregate parameters of these building-blocks. +In this +section, we briefly mention the three main parameters of interest as proposed in +Callaway and Sant’Anna (2021), and show how the asymptotic results and multi- +plier bootstrap adapt to our setting. +Selective timing. +The causal effect of a policy on the cohort treated in g is given +by +θS(g) = +1 +T − g + 1 +T +� +t=g +ATT(g, t), +and thus, an average causal effect across groups can be written as +θS = +T +� +g=2 +θS(g)Pr(G = g). +Dynamic treatment. +In the presence of dynamic effects, the researcher may be +interested in accounting for the length of exposure to the treatment. The causal +54 + +effects of an exposure length e ∈ {0, 1, 2, ...} across groups is defined as +θD(e) = +T +� +g=2 +T +� +t=g+e +ATT(g, t)Pr(G = g|t = g + e), +and therefore an average across exposure lengths is given by +θD = +1 +T − 1 +T −1 +� +e=1 +θD(e). +Calendar time. +In some applications, the researcher may be interested in how +treatment effects differ with calendar time. Let us consider +θC(t) = +t +� +g=2 +ATT(g, t)Pr(G = g|g ≤ t), +and therefore an average across exposure lengths is given by +θC = +1 +T − 1 +T +� +t=2 +θC(t), +The difference between θS, θD and θC is that the second and third attribute more +weight to the groups with, respectively, longer exposure lengths, and treated in the +earliest periods. +The asymptotic results and bootstrap procedure directly apply to the summary pa- +rameters. The following corollary summarizes these results. +Corollary 1 Under the Assumptions of Theorem 2, for all parameters θ defined +above, including those indexed by some variable, we have +√n(ˆθ − θ) +d→ N(0, Σθ), +as n → ∞, where Σθ is defined in the proof and the bootstrap procedure is defined. +Proof 4 (Proof of Corollary 1) All summary parameters defined in the text can +55 + +be generically written as +θ = +T +� +g=2 +T +� +t=2 +wgtATT(g, t), +where wgt are some random weights. Estimators can be defined as +ˆθ = +T +� +g=2 +T +� +t=2 +ˆwgt � +ATT(g, t), +where estimated weights are such that +√n( ˆwgt − wgt) = +1 +√n +n +� +i=1 +ξw +gt(Wi) + op(1), +with first and second moments given by E[ξw +gt(W)] = 0 and E[ξw +gt(W)ξw +gt(W)′] finite +and positive definite. This condition is satisfied by the sample analogs of weights +appearing in the summary parameters θ’s presented in the main text. The application +of Theorem 2 yields +√n(ˆθ − θ) = 1 +√n +n +� +i=1 +lw(Wi) + op(1) +d→ N(0, E[lw(W)2]) +as n → ∞, and where +lw(Wi) = +T +� +g=2 +T +� +t=2 +� +wgt +t +� +τ=g +ψgτ(Wi) + ξw +gt(Wi)ATT(g, t) +� +, +for ψgt defined in (22) and ξw +gt correspond to the estimation errors of weights. The +same bootstrap procedure can hence be used for ˆθ using a consistent estimate of the +influence function lw. +56 + +A.2.4 +Bootstrap for summary parameters +All estimators of summary parameters defined in the text can be generically written +as +ˆθ = +T +� +g=2 +T +� +t=2 +ˆwgt � +ATT(g, t), +where the weights ˆwgt’s are possibly random. In simple settings they are not. For +instance, let us consider +θs(h) = +T +� +g=2 +T +� +t=2 +wgtATT(g, t), +where wgt = +1 +T−g+1 for t ≥ g and g = h, and 0 otherwise. The algorithm is as follows: +1. Draw a vector of Vb = (V1, ..., Vi, ..., Vn)′, where Vi’s are iid zero mean random +variables with unit variance, such as Bernoulli random variables with Pr(V = +1 − κ) = κ/ +√ +5 with κ = ( +√ +5 + 1)/2 as suggested by Mammen (1993). +2. Compute the bootstrap draw ˆθ⋆b = ˆθ + ˆL′Vb where ˆL is a consistent estimator +of the n-dimensional vector L with ith element given by +L(i) = +T +� +g=2 +T +� +t=2 +wgtφgt(i), +where φgt(i) is defined in the previous algorithm. +3. Compute ˆR⋆b = √n(ˆθ⋆b − ˆθ). +4. Repeat steps 1-3 B times. +5. Compute the bootstrapped covariance as ˆΣ1/2 = (q0.75 − q0.25)/(z0.75 − z0.25), +where qp is the pth sample quantile of ˆR⋆ (across B draws) and z is the pth +quantile of the standard normal distribution. +6. For each b, compute t-statb +g≤t = max(g,t) | ˆR⋆b(g, t)|ˆΣ−1/2(g, t). +7. Construct ˆc1−α as the empirical (1−α) quantile of the B boostrap draws t-statb. +57 + +8. Construct the bootstrapped confidence interval for θ as ˆC = [ˆθ±ˆc1−α ˆΣ−1/2/√n]. +If the weights wgt’s are random, the influence function is changed to +L(i) = +T +� +g=2 +T +� +t=2 +wgtφgt(i) + γw +gt(i)ATT(g, t), +where γw +gt(i) is an error function. For example, let us consider +θs = +T +� +g=2 +T +� +t=2 +wgtATT(g, t), +where wgt = P(G = g) +1 +T−g+1 for t ≥ g , and 0 otherwise. Define ˆwgt = +1 +T−g+1 +1 +n +�n +i=1 Gi +for t ≥ g and 0 otherwise. A consistent estimator of the error function is given by +ˆγw +gt(i) = +1 +T − g + 1 +� +Gi − 1 +n +n +� +i=1 +Gi +� +. +A.2.5 +Not yet treated as the control group +In the previous model, we assumed the existence of a true control group, i.e. a group +of individuals that are never treated. In many applications, this situation is not +realistic. Instead, the researcher can use the individuals that are “not yet treated”, +that is treated in g > t to define a control group. The extension to this setting +being developed in length in Callaway and Sant’Anna (2021), we only explain how +it applies to the chained DiD. +Most importantly, the parallel trend assumption is modified to +Assumption 13 (Conditional Parallel Trends) For all t = 2, ..., T , g = 2, ..., T , +such that g ≤ t, +E [Yt(0) − Yt−1(0)|X, Gg = 1] = E [Yt(0) − Yt−1(0)|X, Dt = 0] a.s.. +(26) +Following minor modifications to Theorem C.1. in Callaway and Sant’Anna (2021), +using the “not yet treated” as the control group only changes the weight wC +ττ−1(g, X) +58 + +used in our Theorem 1 to +wC +ττ−1(g, X) = pg,t(X)(1 − Dt)Sτ,τ−1 +1 − pg,t(X) +/EM[pg,t(X)(1 − Dt)Sτ,τ−1 +1 − pg,t(X) +]. +We observe two changes. +First, the binary variable C becomes 1 − Dt. Second, +generalized propensity score is now also a function of t: pg,t(X) = P(Gg = 1|X, (Gg = +1 ∪ Dt = 1)). The propensity scores must hence be estimated for pairs (g, t) because +the control group evolves through time. The asymptotic properties of the two-step +estimator remain similar, with minor changes to the asymptotic covariance. +A.2.6 +General missing data patterns +Under Assumption 6, the data generating process consists of random draws from the +mixture distribution FM(y, y′, g1, ..., gT , c, s1, ..., sT , x) defined as +T +� +t=1 +λt−k,tFYt−k,Yt,G1,...,GT ,C,X|St−k,t(yt−k, yt, g1, ..., gT , C, X|st−k,t = 1), +where λt−k,t = P(St−k,t = 1) is the probability of being sampled in both t and t − k, +y and y′ denote the outcome yt−k and yt, respectively, for an individual sampled at +t − k and t. Again, expectations under the mixture distribution does not correspond +to population expectations. This difference arises because of different sampling prob- +abilities across time periods and because Assumption 7 does not preclude from some +forms of dependence between the sampling process and the unobservable heterogene- +ity in Yit. +Proof 5 (Proof of Theorem 3) Let us define the vector of parameters Θ = ATT +that includes all θτ = ATT(τ), for all τ > 1 since θ1 = ATT(1) = 0 by construction. +The inverse problem in (9) corresponds to the set of moment equalities +EM [hi(Wi|Θ)] = 0L∆, +(27) +59 + +where hi(Wi|Θ) is a L∆-dimensional vector of which each element is defined by +� +wG +iττ−k(g) (Yiτ − Yiτ−k) − wC +iττ−k(g, X) (Yiτ − Yiτ−k) − θτ + θτ−k +� +, +(28) +possibly for all τ ≥ 2, and 1 ≤ k < τ, with the weights +wG +ττ−k(g) = +GgSτ,τ−k +EM[GgSτ,τ−k] +and +wC +ττ−k(g, X) = pg(X)CSτ,τ−k +1 − pg(X) +/EM[pg(X)CSτ,τ−k +1 − pg(X) +]. +The previous asymptotic results in Theorem 1 and 2 apply to (28) up to minor modifi- +cations under Assumptions 3 - 5, a standard assumption on the parametric estimates +of the propensity scores (Assumption 5 in Callaway and Sant’Anna (2021) or 4.4 in +Abadie (2005)), and Assumptions 6 and 7 so that we can safely assume that con- +sistency and asymptotic normality holds for +� +∆AT T as n → ∞ with covariance +Ω, which is defined later. Here, we focus on the aspects of the proofs which differ, +namely the optimal combination of each “chain link” using GMM. The optimal GMM +estimator consists in minimizing +ˆEM [hi(Wi|Θ)]′ Ω−1 ˆEM [hi(Wi|Θ)] , +(29) +with respect to Θ, using the optimal weighting matrix Ω−1 which corresponds the +inverse of the covariance of hi (Hansen, 1982), hence that of ∆AT T . Let us rewrite +this problem as +max +AT T +−( +� +∆AT T − WAT T )′Ω−1( � +∆AT T − WAT T ), +(30) +then the first-order condition with respect to AT T is given by +− 2( +� +∆AT T ) − WAT T )′Ω−1W = 0, +(31) +60 + +which, in turn, leads to the proposed estimator: +� +AT T = (W ′Ω−1W)−1W ′Ω−1 � +∆AT T . +(32) +The necessary and sufficient rank condition for GMM identification in this linear set- +ting is that the rank of Ω−1W is equal to the number of columns (Newey and McFadden, +1994). This condition is satisfied if both the covariance matrix Ω and the weight +matrix W are non-singular. Remark further that if W is not full row rank then +some ATT(g, t) are not identified by the collection of ∆kATT(g, t)’s identified in the +dataset at hand. +Proving consistency requires introducing standard assumptions for GMM estimators. +We assume that (i) Ω and the weight matrix W are non-singular; (ii) the true value +Θ0 lies within a compact set; and (iii) EM[supΘ ||hi(Wi|Θ)||] < ∞. In addition to our +previous assumptions, applying Theorem 2.6 in Newey and McFadden (1994) yields +the desired consistency result. +Assuming further that (iv) Θ0 lies in the interior +of the compact set; (v) E[||hi(Wi|Θ0)||2, ] < ∞; (vi) W ′Ω−1W non-singular, then +asymptotic normality follows from Theorem 3.4 in Newey and McFadden (1994).25 +Note that the two-step GMM estimator requires estimating Ω. We proceed as follows. +For every (g, t, k), compute the n-dimensional vector ψgtk with ith element defined as +ψgtk(i) = ψG +gtk(i) + ψC +gtk(i), +(33) +where +ψG +gtk(i) =wG +it,t−k(g) +� +(Yit − Yit−k) − EM +� +wG +t,t−k(g)(Yt − Yt−k) +�� +, +ψC +gtk(i) =wC +it,t−k(g, X) +� +(Yit − Yit−k) − EM +� +wC +t,t−k(g, X)(Yt − Yt−k) +�� ++ M′ +gtkξπ +g (i), +where Mgtk, a k dimensional vector (k being the number of covariates in X), is +25All the other sufficient conditions used by these theorems are trivially satisfied in this linear +setting. +61 + +defined as +Mgtk = +E +� +X( CStSt−k +1−pg(X))2 ˙pg(X) +� +(Yit − Yit−k) − E +� +wC +t,t−k(g, X)(Yt − Yit−k) +��� +E[ pg(X)C +1−pg(X)] +, +with ˆpg(Xi) = Λ(X′ +iˆπg) being the parametric propensity score for covariates Xi and +˙pg(X) = ∂Λ(X′ +iˆπg)/∂(X′ +iˆπg). Furthermore, ξπ +g (i) is a k-dimensional vector for each +observation i, and is given by +ξπ +g (i) = EM +� (Gg + C) ˙pg(X)2 +pg(Xi)(1 − pg(Xi))XX′ +�−1 +Xi +(Gg + C)(Gg − pg(Xi)) ˙pg(Xi) +pg(Xi)(1 − pg(Xi)) +. +Concatenate all ψgtk’s into a L∆ × n matrix Ψ, and compute ˆΩ = ˆE[Ψ(i)Ψ(i)′]. +Therefore, the asymptotic covariance of � +AT T is +Σ = (W ′Ω−1W)−1, +(34) +and its corresponding influence function to be used in the bootstrap procedure detailed +in Appendix A.2.2 is the empirical counterpart of +Φ = (W ′Ω−1W)−1W ′Ω−1Ψ. +(35) +Finally, it is easy to show that the choice of the optimal weighting Ω−1 is the same if +the objective is instead to minimize the variance of a linear transformation R′AT T , +where R is a vector of weights, like for all the summary parameters considered in +Appendix A.2.3. In that case, the bootstrap for summary parameters in Appendix +A.2.4 apply with the (general) influence defined as follows. Let the weights in R be +random, the influence function is changed to +L(i) = R′Φ(i) + γw(i)′AT T , +where γw is the error function that depends on the randomness of the weights, as +defined in Appendix A.2.4. +62 + +Online Appendix +B +Appendix for the Application +In this appendix, we present the results obtained for the application when the initial +administrative data are not balanced ex-ante using the exhaustively observed variables. +Results are presented in Tables B.1, B.2, and B.3. This introduces some differences because +there are more individuals in some periods compared to the results presented earlier. +63 + +Table B.1: Effects on total workforce (exhaustively observed outcome with the com- +plete panel data) +log(total workforce) +Long DiD +Chained DiD +Cross Section DiD +Exhaustive +Exhaustive +Unbalanced +Exhaustive +Unbalanced +(1) +(2) +(3) +(4) +(5) +β−3 +-0.001 +0.001 +0.019 +0 +0.002 +[-0.056,0.054] +[-0.07,0.071] +[-0.067,0.105] +[-0.174,0.174] +[-0.46,0.464] +β−2 +-0.027 +-0.027 +-0.018 +-0.022 +0.001 +[-0.079,0.026] +[-0.092,0.038] +[-0.099,0.063] +[-0.173,0.128] +[-0.405,0.408] +β−1 +-0.017 +-0.017 +-0.013 +-0.016 +-0.022 +[-0.061,0.026] +[-0.068,0.033] +[-0.076,0.051] +[-0.14,0.107] +[-0.374,0.33] +ref. +0 +0 +0 +0 +0 +β1 +0.045** +0.045** +0.051* +0.049 +0.052 +[0.002,0.087] +[-0.006,0.095] +[-0.008,0.11] +[-0.073,0.17] +[-0.282,0.386] +β2 +0.066*** +0.077*** +0.088*** +0.057 +0.046 +[0.013,0.118] +[0.013,0.142] +[0.015,0.161] +[-0.086,0.199] +[-0.348,0.44] +β3 +0.069*** +0.085*** +0.089** +0.055 +-0.001 +[0.009,0.128] +[0.01,0.159] +[0.004,0.174] +[-0.085,0.194] +[-0.408,0.406] +β4 +0.074*** +0.085** +0.11*** +0.05 +-0.027 +[0.009,0.139] +[0,0.17] +[0.016,0.204] +[-0.117,0.216] +[-0.473,0.42] +β5 +0.094*** +0.097** +0.134*** +0.052 +-0.07 +[0.019,0.169] +[0.003,0.19] +[0.028,0.24] +[-0.121,0.226] +[-0.545,0.406] +Pre- +-0.015 +-0.015 +-0.004 +-0.013 +-0.006 +trend +[-0.043,0.013] +[-0.05,0.021] +[-0.047,0.039] +[-0.087,0.061] +[-0.238,0.225] +Notes: This table shows the dynamic treatment effects relative to the beginning of the treat- +ment obtained on a panel balanced on exhaustive variables. “Exhaustive” refers to the use +of an exhaustively observed outcome without pretending that this variable is imperfectly ob- +served. “Unbalanced” refers to the use of an exhaustively observed outcome pretending that +this variable is observed from R&D survey, that is, from an unbalanced repeated panel. 95% +confidence intervals are obtained from the multiplier bootstrap. ∗p < 0.10, ∗∗p < 0.05, ∗ ∗ ∗ +p < 0.01 +64 + +Table B.2: Effects on highly qualified workers (exhaustively observed outcome with +the complete panel data) +log(highly qualified workforce) +Long DiD +Chained DiD +Cross Section DiD +Exhaustive +Exhaustive +Unbalanced +Exhaustive +Unbalanced +(1) +(2) +(3) +(4) +(5) +β−3 +0.019 +0.025 +0.07 +0.04 +0.102 +[-0.046,0.085] +[-0.053,0.102] +[-0.059,0.199] +[-0.091,0.171] +[-0.235,0.438] +β−2 +0.024 +0.013 +0.021 +0.023 +-0.026 +[-0.041,0.089] +[-0.062,0.087] +[-0.105,0.148] +[-0.09,0.137] +[-0.327,0.274] +β−1 +0.016 +0.016 +0.012 +0.023 +-0.017 +[-0.043,0.075] +[-0.053,0.085] +[-0.093,0.116] +[-0.076,0.122] +[-0.281,0.247] +ref. +0 +0 +0 +0 +0 +β1 +0.028 +0.028 +0.02 +0.031 +0.019 +[-0.031,0.088] +[-0.04,0.095] +[-0.085,0.125] +[-0.069,0.132] +[-0.228,0.267] +β2 +0.045 +0.047 +0.077 +0.044 +0.044 +[-0.021,0.111] +[-0.028,0.123] +[-0.049,0.202] +[-0.068,0.155] +[-0.239,0.328] +β3 +0.078** +0.077** +0.064 +0.072 +-0.007 +[0.008,0.147] +[-0.006,0.16] +[-0.068,0.197] +[-0.041,0.185] +[-0.3,0.286] +β4 +0.116*** +0.108*** +0.101 +0.103* +-0.052 +[0.042,0.19] +[0.017,0.199] +[-0.038,0.24] +[-0.03,0.235] +[-0.385,0.281] +β5 +0.134*** +0.119*** +0.13** +0.111* +-0.031 +[0.048,0.22] +[0.021,0.218] +[-0.023,0.283] +[-0.029,0.251] +[-0.382,0.32] +Pre- +0.02 +0.018 +0.034 +0.029 +0.019 +trend +[-0.015,0.054] +[-0.022,0.058] +[-0.031,0.1] +[-0.032,0.089] +[-0.151,0.19] +Notes: This table shows the dynamic treatment effects relative to the beginning of the treat- +ment obtained on a panel balanced on exhaustive variables. “Exhaustive” refers to the use +of an exhaustively observed outcome without pretending that this variable is imperfectly ob- +served. “Unbalanced” refers to the use of an exhaustively observed outcome pretending that +this variable is observed from R&D survey, that is, from an unbalanced repeated panel. 95% +confidence intervals are obtained from the multiplier bootstrap. ∗p < 0.10, ∗∗p < 0.05, ∗ ∗ ∗ +p < 0.01 +65 + +Table B.3: Effects on employment variables observed from R&D survey with the +complete panel data +Unbalanced variables from R&D survey in log +Chained DiD +Cross Section DiD +total workforce +researchers +total workforce +researchers +(1) +(2) +(3) +(4) +β−3 +0.036 +-0.03 +0.017 +0.081 +[-0.113,0.185] +[-0.149,0.089] +[-0.471,0.504] +[-0.148,0.31] +β−2 +0.028 +-0.018 +0.079 +0.044 +[-0.114,0.171] +[-0.125,0.088] +[-0.368,0.526] +[-0.164,0.252] +β−1 +0.022 +0.007 +-0.016 +0.037 +[-0.1,0.144] +[-0.082,0.097] +[-0.4,0.369] +[-0.149,0.223] +ref. +0 +0 +0 +0 +β1 +0.05 +0.073 +0.045 +0.095 +[-0.066,0.165] +[-0.023,0.168] +[-0.32,0.409] +[-0.095,0.285] +β2 +0.047 +0.152*** +0.022 +0.141 +[-0.084,0.177] +[0.048,0.257] +[-0.378,0.423] +[-0.059,0.341] +β3 +0.064 +0.214*** +0.014 +0.134 +[-0.073,0.201] +[0.099,0.329] +[-0.402,0.43] +[-0.074,0.341] +β4 +0.12* +0.244*** +-0.035 +0.086 +[-0.029,0.269] +[0.121,0.368] +[-0.501,0.43] +[-0.137,0.309] +β5 +0.142** +0.297*** +-0.046 +0.138 +[-0.013,0.297] +[0.163,0.432] +[-0.508,0.415] +[-0.099,0.374] +Pre- +0.029 +-0.014 +0.027 +0.054 +trend +[-0.042,0.099] +[-0.073,0.046] +[-0.203,0.257] +[-0.065,0.173] +Notes: This table shows the dynamic treatment effects relative to the beginning of the treat- +ment. The dynamic effects are estimated with outcome variables observed from R&D survey, +that is, from an unbalanced repeated panel. 95% confidence intervals are obtained from the +multiplier bootstrap. ∗p < 0.10, ∗∗p < 0.05, ∗ ∗ ∗ +p < 0.01 +66 + diff --git a/UdAzT4oBgHgl3EQfJvt-/content/tmp_files/load_file.txt b/UdAzT4oBgHgl3EQfJvt-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..342cc01079adcb8ccd33b6f782f5b4e6cdb8ffe5 --- /dev/null +++ b/UdAzT4oBgHgl3EQfJvt-/content/tmp_files/load_file.txt @@ -0,0 +1,2540 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf,len=2539 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='01085v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='EM] 3 Jan 2023 The Chained Difference-in-Differences Christophe Bell´ego∗ David Benatia† Vincent Dortet-Bernardet‡ January 4, 2023 Abstract This paper studies the identification, estimation, and inference of long-term (binary) treatment effect parameters when balanced panel data is not avail- able, or consists of only a subset of the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We develop a new estimator: the chained difference-in-differences, which leverages the overlap- ping structure of many unbalanced panel data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This approach consists in efficiently aggregating a collection of short-term treatment effects estimated on multiple incomplete panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Our estimator accommodates (1) multiple time periods, (2) variation in treatment timing, (3) treatment effect heterogeneity, and (4) general missing data patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We establish the asymptotic proper- ties of the proposed estimator and discuss identification and efficiency gains in comparison to existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Finally, we illustrate its relevance through (i) numerical simulations, and (ii) an application about the effects of an inno- vation policy in France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Keywords: Difference-in-Differences, Dynamic treatment effects, Event study, Unbalanced panel, Attrition, Treatment effect heterogeneity, GMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' JEL Codes: C14 C2 C23 O3 J38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' ∗CREST (UMR 9194), ENSAE, Institut Polytechnique de Paris, 5 Avenue Henry Le Chatelier, 91120 Palaiseau, France (e-mail: christophe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='bellego@ensae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='fr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' †HEC Montr´eal, D´epartement d’´Economie Appliqu´ee, 3000 Chemin de la Cˆote-Sainte-Catherine, Montr´eal, QC H3T 2A7, Canada (Corresponding author, e-mail: david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='benatia@hec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='ca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' ‡Direction G´en´erale des Entreprises (DGE), Minist`ere Fran¸cais de l’´Economie et des Finances (e-mail: vincent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='dortet-bernadet@finances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='gouv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='fr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The authors thank Laurent Davezies, Cl´ement de Chaisemartin, Xavier D’Haultefœuille, Yan- nick Guyonvarch, Xavier Jaravel, and all participants to seminars and conferences for insightful discussions and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' R packages are available upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 1 Introduction Many public policies take years before having effects on their targeted outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For instance, most innovation policies have long run objectives such as making new discoveries, enhancing knowledge, and increasing technology development, but in- duce only little innovation in the short-run (Bakker, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' O’Connor and Rice, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Gross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Measuring these long-term effects is however difficult for two principal reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' First, identification of treatment effects from observational data raises significant challenges, whereas randomized controlled experiments are often too costly, or raise ethical concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Second, treatment effects are frequently esti- mated from panel survey data, where the subjects of interest (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' individuals or firms) are not consistently observed over the entire time frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This problem can be caused by attrition (Hausman and Wise, 1979), if individuals drop out of the sample, or because of the survey design itself, if individuals are frequently replaced to prevent attrition (Nijman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In this paper, we study the identification, estimation, and inference of long-term treatment effects in settings where balanced panel data is not available, or con- sists of only a subset of the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We develop a new estimator, the chained difference-in-differences (DiD), which leverages the overlapping structure of many un- balanced panel data sets (Baltagi and Song, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The intuition behind the chained DiD estimator is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Letting Yt be an outcome of interest observed in three dis- tinct time periods t0 < t1 < t2, the long difference Yt2−Yt0 can always be decomposed into a sum of short differences (Yt2 − Yt1) + (Yt1 − Yt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Our estimator generalizes this simple idea by optimally aggregating short-term treatment effect parameters, or “chain links”, obtained from (possibly many) overlapping incomplete panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For instance, one subsample of individuals might be used to estimate the DiD from t0 to t1 while another would be used to estimate the DiD from t1 to t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The sum of both effects identifies the long-term DiD from t0 to t2, which may not be feasible or may suffer from efficiency losses if a large part of the sample is discarded by using only a balanced subsample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Our estimator is designed for multiple time periods, it ac- commodates variations in treatment timing, treatment effect heterogeneity, general missing data patterns, and it may deliver substantial efficiency gains compared to 2 estimators using only subsets of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Building upon the influential work of Callaway and Sant’Anna (2021), we show that our estimator is consistent, asymptotically normal, and computationally simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Their multiplier bootstrap remains asymptotically valid in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We identify three main advantages of our approach: (1) it does not require having a balanced panel subsample as it is the case with a standard DiD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (2) identification rests on a weaker assumption about missing data compared to the cross-section DiD, which treats the sample as repeated cross-sectional data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' and (3) it may also deliver effi- ciency gains compared to other approaches using only subsets of the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The proposed approach is especially relevant with regard to the significant interest for long-term evaluations of interventions, with many applications related to educa- tion and labor economics (Angrist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Kahn, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Oreopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Autor and Houseman, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Garca-Prez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Lechner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Mroz and Savage, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Stevens, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The estimation of treatment effects generally consists in per- forming a DiD focusing only on units that are observed over the entire time frame: the balanced subsample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This approach, hereafter referred to as the long DiD, allows getting rid of both time- and individual-specific unobservable heterogeneity, but may also involve discarding many individuals with missing observations when estimating long-term effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The evaluation of long-term effects is sometimes difficult, if not infeasible, because of such missing data problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Missing observations in panel data sets may exist for two principal reasons: (1) by design or (2) because of attrition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' First, the design of rotating panel surveys attempts to alleviate the burden of admin- istering and responding to statistical surveys, and to prevent attrition, by replacing subjects regularly (Heshmati, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The survey is administered to a cohort of sub- jects in a limited number of periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The cohort is then replaced by another cohort randomly drawn from the population of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The resulting data set is hence composed of a collection of incomplete panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The data is said to have an over- lapping structure if there is at least one period in which two separate cohorts are administered the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The most famous example of rotating panel survey is the Current Population Survey (CPS), where each cohort is interviewed for a total of 8 months (over a period of 16 3 months), and part of the sample is replaced each month by a new subsample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='1 This survey is one of the most widely used data sources in economic and social research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' It has been used or cited in 1000+ articles between 2000 and 2022, including publi- cations in top journals such as the American Economic Review, Journal of Political Economy, Quarterly Journal of Economics, American Sociological Review, and De- mography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2 Other rotating panel surveys, without claiming to be exhaustive, include the Medical Expenditure Panel Survey, the General Social Survey since 2006, and the Consumer Expenditure Survey (Blundell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', 2008) in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', the Labour Force Survey in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', and the Enquˆete Emploi (Labour Force Survey) and Enquˆete sur les Moyens Consacr´es `a la R&D (R&D Survey) in France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Despite their prevalence, the econometrics literature on rotating panels is almost non-existent (Baltagi and Song, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Heshmati (1998) uses rotating panels for production function estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Nijman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (1991) study the optimal choice of the rotation period for estimating a linear combination of period means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Likewise, our estimator consists in a linear combination of parameters corresponding to the long- term average treatment effect parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' To the best of our knowledge, our paper is the first to study identification, estimation, and inference of treatment effects in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Second, individuals may also drop out of the surveys and cause attrition, which can be particularly severe for long panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This attrition raises some concerns be- cause it is associated with selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Attrition can be due to either “ignorable” or “non-ignorable” selection rules (Verbeek and Nijman, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Ignorable attrition implies that missing data occurs completely at random, and as such focusing on a balanced panel subsample does not threaten identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Verbeek and Nijman (1992), among others, propose a test of the ignorability assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Non-ignorable attrition means that missing data is related to either observable or unobservable factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Hirano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (2001) provide important identification results for the addi- tive non-ignorable class of attrition models, which nest several well-known meth- ods (Hausman and Wise, 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Little and Rubin, 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' These results have been extended to multi-periods panels by Hoonhout and Ridder (2019) and apply to our 1The design is detailed at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='bls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='gov/opub/hom/cps/design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='htm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 2This figure is based on a Google Scholar search conducted on December 5, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 4 setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Bhattacharya (2008) studies the properties of a sieves-based semi-parametric estimator of the attrition function initially proposed by Hirano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' These methods are often referred to as models of selection on unobservables because the probability of attrition is allowed to depend on variables that are not observed when an individual drops out, but not on unobservable error terms (Moffit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In- verse propensity weighting is the most popular method for addressing non-ignorable attrition caused by observable factors, even beyond the estimation of average treat- ment effects (Chaudhuri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' There also exist methods using instrumental variables to address attrition due to latent factors (Fr¨olich and Huber, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Other approaches focus on particular structures of attrition, like monotonically missing data (Chaudhuri, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Barnwell and Chaudhuri, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Attrition is generally addressed before estimating treatment effects with the long DiD, either by reweighting the observations in the balanced subsample with in- verse propensity scores (Hirano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', 2003), or by imputing the missing observa- tions (Hirano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' However, discarding a possibly large proportion of the data by focusing on the balanced subsample may lead to significant efficiency losses (Baltagi and Song, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Chaudhuri, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Barnwell and Chaudhuri, 2021), or may lead to discarding the complete dataset, as illustrated in our application using the French R&D survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' If there are too few individuals observed over the entire time horizon, the data is typically treated as repeated cross-sections and treatment effects are estimated with the cross-section DiD (Abadie, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Callaway and Sant’Anna, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='3 The identification of the average treatment effect using the cross-section DiD requires not only a parallel trends assumption on the population, but also that the sampling process is (conditionally) independent of the levels of idiosyncratic shocks to the outcome variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This assumption is violated if, for instance, units with larger un- observed individual shocks are relatively more likely to be sampled in the treatment group in later periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In such a case, the treatment groups observed early on will differ in unobservable ways from those observed later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' An identification problem arises as soon as the sampling process does not affect the observed control groups 3This approach consists in taking the difference of differences of averages, where different sets of units are used to compute each of the four averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 5 in the exact same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' As an illustration, the survey sampling in our application depends on the level of a firm’s internal R&D expenditure by design, which may lead to biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Instead, our approach only requires that the sampling process be (condi- tionally) independent of the trend of those idiosyncratic shocks, as is the case in our application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This weaker assumption is sufficient because the chained DiD allows eliminating individual heterogeneity like the long DiD before taking expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' It is similar to a parallel trends assumption but conditional on being sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' There- fore, the chained DiD is robust to some forms of attrition caused by unobservable heterogeneity, unlike the cross-section DiD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Note that the attrition models discussed earlier can also be used as a first-step in our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Although we do not correct for the efficiency losses of using first-step plug-in estimates in our paper, we identify some suitable options to do so (Frazier and Renault, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Chaudhuri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Sant’Anna and Zhao, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Recent developments in the literature about treatments effects with multiple periods and treatment heterogeneity have revealed that two-way fixed-effects estimators fail to identify the treatment effect parameters of interest (de Chaisemartin and D’Haultfœuille, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Athey and Imbens, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Goodman-Bacon, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Borusyak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Sun and Abraham, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Our paper builds upon Callaway and Sant’Anna (2021) which address this is- sue by generalizing the approach developed in Abadie (2005) to multiple periods and varying treatment timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We contribute further to this literature by extending this framework to settings with incomplete panel data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' An alternative approach would have been to adapt the general framework developed by de Chaisemartin and D’Haultfœuille (2020) to our setting, or the other related papers focused on staggered adoption de- signs and event studies with multiple periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' However, the chained DiD fits very well within the framework developed by Callaway and Sant’Anna (2021) which fo- cuses on estimating all group-time average treatment effects before aggregating all those parameters into summary parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Likewise, our approach focus on (even smaller) building blocks: one-period-difference group-time average treat- ment effects, which measure the increase of average treatment effect of group g from period t−1 to period t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In the general case, these blocks naturally become k-period- difference group-time average treatment effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We illustrate the performance of the chained DiD in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' First, we use sim- 6 ulations to compare the long DiD, chained DiD and cross-section DiD in terms of bias and variance under several data generating processes (DGP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Our simulations include a stratified panel data set, composed of a balanced panel and a rotating panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Second, we study the long-term employment effects of a large-scale innova- tion policy in France giving grants to collaborative R&D projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Technical progress and innovation, stimulated by R&D activities, are key to economic growth (Scherer, 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Howitt and Aghion, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Griffith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', 2004) but firms tend to invest too little because of the public good nature of innovations (Arrow, 1962;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Nelson, 1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Therefore, measuring the long term effects of subsidized R&D is important to inform policymakers and improve future policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This application is especially relevant because the French R&D survey, which con- tains firm-level data on R&D activities, consists of rotating panels but does not in- clude a balanced subsample except for the largest firms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In addition, it is possible to fully observe a limited number of variables for all firms close to those provided by the R&D survey by using administrative data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We are thus able to compare the results of each of the three estimators by focusing on two of those variables: total employment and highly qualified workforce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The long DiD is applied to the complete data and serves as the benchmark estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The chained DiD and cross-section DiD estima- tors are applied to both data sets, and to an “artificial” unbalanced panel which is generated by discarding all observations from the complete administrative data set which are missing in the R&D survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This application is somehow comparable to a simulation exercise but uses real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Our results show that the policy had a positive effect on employment for firms that received a grant to participate to a collaborative R&D project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We find that the estimates as well as their standard errors obtained with the chained DiD estimator are close to those obtained with the long DiD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In contrast, the cross-section estimator delivers biased estimates which also lack sufficient precision to detect any statistically significant effect associated with the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Section 2 presents our method- ology and asymptotic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Numerical simulations are in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Section 4 contains the application to R&D policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Section 5 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 7 2 Identification, Estimation and Inference 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='1 Basic Framework We first present the main insights in a simple framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The main notation is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' There are T periods and each particular time period is denoted by t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In a standard DiD setup, T = 2, no one is treated in t = 1, and all treatments take place in t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' To gain intuition, we first focus on the case where all treatments take place in t = 2 but assume T > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Define G to be a binary variable equal to one if an individual is in the treatment group, and C = 1 −G as a binary variable equal to one for individuals in the control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Also, define Dt to be a binary variable equal to one if an individual is treated in t, and equal to zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Let Yt(0) denote the potential outcome at time t without the treatment and Yt(1) denote its counterpart with the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The observed outcome in each period can be written as Yt = DtYt(1) + (1 − Dt)Yt(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We focus on the long-term average treatment effect on the treated which corresponds to the average treatment effect in period t > 2 on individuals in the treatment group, hence first treated in period 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' It is formally defined by ATT(t) = E [Yt(1) − Yt(0)|G = 1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (1) The identification of ATT(t) with panel data has attracted much research attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In this paper, we are interested in a case where balanced panel data is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The missing data pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We assume that a new random sample of nt individ- uals is drawn at each period t to replace a subsample of previously observed individ- uals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This replacement can occur due to attrition or by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For the moment, we assume that subsamples may differ in size nt but individuals are only observed for two consecutive periods as in many rotating panel design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='4 Therefore, one cannot observe the entire path {Y1, Y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', YT } for any individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The key feature of our 4In practice, rotating panels may involve subgroups sampled over a different number of con- secutive periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The result of this paper continues to apply, so the same estimator and inference procedure can be used even with a more sophisticated sampling process as presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 8 approach is that there is some overlap across subsamples, that is there are at least two different subsamples observed in each period 1 < t < T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This structure is in stark contrast with the literature about attrition in panel data, which typically assumes that there always exists a balanced subsample where individ- uals are observed throughout the entire time frame (Hirano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Hoonhout and Ridder, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Although there is no need for such a balanced subsample here, we extend our method to more general settings in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This extension is illustrated in Sections 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' To characterize the sampling process, we define St to be a binary variable equal to one for individuals observed at t and zero otherwise, and use St,t+1 to denote StSt+1 which indicates if an individual is observed at both t and t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The missing data pattern is summarized in Table 1 for T = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In the general framework, we will assume that treatments can also vary with some observable covariates X and that individuals can be observed in non-consecutive periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Table 1: Missing data pattern in a three-period panel data set Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Indicators Variables Sub-population S1 S2 S3 Y1 Y2 Y3 X Incomplete Panel 1 1 1 0 × × × Incomplete Panel 2 0 1 1 × × × The long DiD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The common approach to identify treatment effects is to consider the long difference-in-differences defined by ATT(t) =E[Yt(1) − Y1(0)|G = 1] − E[Yt(0) − Y1(0)|C = 1], (2) under the standard parallel trend assumption E[Yt(0) − Y1(0)|G = 1] = E[Yt(0) − Y1(0)|C = 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' ATT(t) corresponds to the long-term effect for t > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Unfortunately, individuals are never observed more than two consecutive periods in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Calculating the averages of Yit − Yi1 for t > 2 for the treatment and control groups is hence infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 9 The cross-section DiD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' If the panel consists of incomplete panel data, the iden- tification of the parameter of interest can be achieved by assuming that the sampling process St is independent of (Yt, D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', DT ) in addition to the parallel trend as- sumption (Abadie, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In this case, ATT(t) is identified by the “cross-section DiD” given by ATTCS = (E[Yt|StG = 1] − E[Y1|S1G = 1]) − (E[Yt|StC = 1] − E[Y1|S1C = 1]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The sampling assumption allows replacing the averages of difference, E[Yt(0) − Y1(0)|a = 1] for a ∈ {G, C}, by the difference of averages, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' E[Yt(0)|Sta = 1] − E[Y1(0)|S1a = 1] for a ∈ {G, C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This approach does not eliminate the individual-specific unobservable heteregeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The chained DiD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Our approach takes advantage of the overlapping panel struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Remark that each term in (2) can be decomposed into E [Yt(1) − Y1(0)|a = 1] = t−1 � τ=1 E [Yτ+1(Dτ+1) − Yτ(Dτ)|a = 1] (3) for a ∈ {G, C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Thus, identification of ATT(t) is obtained by summing the short- term DiD as in ATTCD(t) = t−1 � τ=1 (E [Yτ+1(Dτ+1) − Yτ(Dτ)|G = 1] − E [Yτ+1(Dτ+1) − Yτ(Dτ)|C = 1]) = t−1 � τ=1 (E [Yτ+1(Dτ+1) − Yτ(Dτ)|Sτ,τ+1G] − E [Yτ+1(Dτ+1) − Yτ(Dτ)|Sτ,τ+1C]) , where the second equality holds under the assumption that the sampling process St,t+1 is independent of (Yt+1 − Yt, D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', DT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This approach not only allows eliminating the individual-specific heterogeneity, it also makes use of a weaker iden- tifying assumption than the one for the cross-section DiD in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Remark that these assumptions may not be directly comparable in settings with dif- ferent sampling processes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' if some individuals are observed only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' However, 10 replacement of individuals in panel survey data is typically based on some variables observed before the replacement period but not on their evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='5 Our application provides a clear illustration of that argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Simple model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In order to gain intuitions about the identification, estimation and inference of ATT(t) in this context, we suppose that the outcome variable is generated by a components of variance process Yit = αi + δt + t � τ=2 βτDiτ + εit, (4) where ATT(t) = �t τ=2 βτ is the impact of the treatment evaluated at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' αi is an individual-specific component, δt is a time-specific component, and εit is a mean-zero individual-transitory shock that is auto-regressive stationary error process of order 1 represented by εit+1 = ρεit + ηit+1 with ρ ∈ [0, 1) and ηit+1 being a white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Assume further that error terms are homoskedastic with V (εit) = σ2 ε = σ2 η/(1 − ρ2) and V (αi) = σ2 α, and Dit ⊥⊥ (Yit(0), Yit(1)), for all t > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' If the sampling process is independent of all the components of Yit, then both ATTCD and ATTCS identify the parameter of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' However, if the unobservable individual-specific component αi is correlated with the sampling process, identification still holds for ATTCD but the latter admits the bias term (E[αi|StG = 1] − E[αi|S1G = 1]) − (E[αi|StC = 1] − E[αi|S1C = 1]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This bias is non-zero if the compositions of the sampled treatment and control groups evolve differently through time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This situation happens, for instance, if individuals with larger αi are more likely to be sampled in the control group in period 1 but become relatively more likely to be sampled in the treatment group in period t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Addressing this bias can be difficult, if not impossible, due to the unobservability of these errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 5Remark that surveys are usually not specifically designed to use panel econometric methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 11 Therefore, if the sampling process depends on unobservable components of the out- comes Yit but is not correlated with the first-differences in outcomes Yit+1 − Yit conditional on treatment status, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' εit+1 − εit in this example, then ATT(t) is still identified by ATTCD(t) but not necessarily by ATTCS(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This result implies that identification with ATTCD(t) is more robust to deviations from an independent sampling assumption, for instance, because of a dependence between the sampling process, treatment status, and unobservable individual heterogeneity αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Estimators of ATTCD(t) and ATTCS(t) can be obtained by linear re- gression as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For ATTCS(t), one must first modify the outcome variable for treated individuals to Yit−�n1 i=1 Yi1Si1Gi/ �n1 i=1 Si1Gi and to Yit−�n1 i=1 Yi1Si1Ci/ �n1 i=1 Si1Ci for those in the control group, and then regress those modified outcomes onto the treatment variable and an intercept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The parameter associated with treatment will yield an estimate of �t τ=2 βτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For ATTCD(t), one has to estimate t − 2 two-way fixed-effects linear regressions specified in (4), one for each subsample of individuals observed sequentially, and then sum the estimated parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' A more direct estimation method for both ATTCD(t) and ATTCS(t) is to substitute expectations by their sample counterparts in each expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The corresponding estimators can be written as the weighted averages � ATTCD(t) = 1 n n � i=1 t−1 � τ=1 � � wGiττ+1 (y′ i − yi) − � wCiττ+1 (y′ i − yi) � , � ATTCS(t) = 1 n n � i=1 �� � wGit − � wGi1 � yi − � � wCit − � wCi1 � yi � , where yi and y′ i denote the outcome variable in the current and next period, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' τ and τ + 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The weights are defined as � waiτ = Siτai 1 n �n i=1 Siτai , and � waiττ+1 = Siτ,τ+1ai 1 n �n i=1 Siτ,τ+1ai , with ai ∈ {G, C}, and for all τ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='T − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' A formal treatment is presented in the general framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 12 Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' When the sampling process does not violate the identifying assump- tions, the most efficient of the two estimators should be favored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Proposition 1 sheds light on the relative efficiency of each estimator under standard assumptions in this simple setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Proposition 1 Under the standard assumptions of DiD settings, as presented in the proof, and assuming Yit to be specified as in (4), for all t > 2, then � ATTCD(t) has a smaller asymptotic variance if 2(t − 1) 1 + ρ σ2 η ≤ σ2 η 1 − ρ2 + σ2 α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' On the one hand, the chained DiD introduces additional noise by adding up individual- transitory shocks εit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' On the other hand, the cross-section DiD’s precision depends on the variance of unobserved individual heterogeneity αi and serial correlation of individual shocks εit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' It also makes use of twice as many observations to calculate each empirical expectation since two cohorts are observed at each t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' t − 1, t and t, t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In comparison, the long DiD does not suffer from any of those two efficiency losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' According to Proposition 1, � ATTCD(t) delivers a more precise estimate only if the sum of the extra individual-transitory shocks has a smaller variance than that of the individual-specific heterogeneity and individual transitory shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This condition is plausible as long as t is not too large, that is not too many incremental effects are aggregated together, or if σ2 α is relatively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' As t increases, a new term is added to the sum and results in a marginal increase of the variance equal to 2 1+ρσ2 η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The serial correlation of εit plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' As ρ → 0, the variance of � ATTCD(t) goes to 2(t − 1)σ2 η whereas that of of � ATTCS(t) approaches 2(σ2 η + σ2 α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Conversely, as ρ → 1, the variance of � ATTCD(t) approaches (t − 1)σ2 η whereas that of � ATTCS(t) goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Therefore, � ATTCD(t) should largely dominate in terms of precision in settings with autocorrelated idiosyncratic shocks, and where the variance of unobserved individual heterogeneity is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We will see later that � ATTCD(t) delivers additional efficiency gains under more general missing data patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2 General framework In the general framework, we allow for: 1) heterogeneity in treatment effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 2) variation in treatment timing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' and 3) general missing data patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We first introduce additional notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Let us substitute the treatment group dummy G by Gg, a binary variable that is equal to one if an individual is first treated in period g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' There are hence several cohorts of treatment groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The con- trol group binary variable C denotes individuals who are never treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Notice that for each individual �T g=2 Gg + C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Also, define Dt to be a binary variable equal to one if an individual is treated in t, and equal to zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This variable will be useful to denote if the individual was first treated for some g ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Also, define the generalized propensity score as pg(X) = P(Gg = 1|X, Gg + C = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This score measures the probability of an individual with covariates X to be treated conditional on being in the treated cohort g or the control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' There are many parameters of interest in this setting, such as, for example, the average treatment effect k periods after the treatment date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' All possible parameters consist of aggregates of the most basic parameter: the average treatment effect in period t for a cohort treated in date g, denoted by ATT(g, t) = E[Yt(1) − Yt(0)|Gg = 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (5) This parameter is referred to as the group-time average treatment effect in Callaway and Sant’Anna (2021), and we will build upon their results to study the collection of such parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' When only unbalanced panel data is available and t > g + 1, two methods are possible: (1) the cross-section DiD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' and (2) the chained DiD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In what follows, we assume that individuals are sampled only for two consecutive periods and then drop out forever, so the long DiD is never feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We will introduce more general missing data patterns in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In order to identify the ATT(g, t) and accommodate varying treatment timing and treatment effect heterogeneity on observable covariates X, we impose assumptions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 14 Assumption 1 (Sampling) For all t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', T , {Yit, Yit+1, Xi, Di1, Di2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', DiT }nt i=1 is independent and identically distributed (iid) conditional on Sit,t+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Assumption 2 (Conditional Independence of Sampling and Trends) For all t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', T , Yit+1 − Yit ⊥⊥ Sit,t+1|(Xi, Di1, Di2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', DiT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (6) Assumption 3 (Conditional Parallel Trends) For all t = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', T , g = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', T , such that g ≤ t, E [Yt(0) − Yt−1(0)|X, Gg = 1] = E [Yt(0) − Yt−1(0)|X, C = 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='. (7) Assumption 4 (Irreversibility of Treatment) For all t = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', T , Dt−1 = 1 implies that Dt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (8) Assumption 5 (Overlap) For all g = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', T , P(Gg = 1) > 0 and for some ε > 0, pg(X) < 1 − ε a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Assumption 1 means that we are considering a rotating panel structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Condi- tional on being sampled in two consecutive periods, individuals are assumed to be iid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' However, unlike Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='3 in Abadie (2005), it does not imply that ob- servations are representative of the population of interest because we do not as- sume that the iid draws are taken from the population distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Instead, we focus on the case where the identification with a cross-section DiD can fail by con- sidering Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This assumption constitutes the principal departure from Callaway and Sant’Anna (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' It states that the sampling process is statistically independent of the joint distribution of first-differences of individual outcomes, con- ditionally on observables and treatment status in any period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In particular, it implies that E[Yit+1 − Yit|X, a = 1, St,t+1 = 1] for a ∈ {G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', C} corresponds to its pop- ulation counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' However, this needs not be true for E[Yit|X, a = 1, St = 1] for a ∈ {G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Notice that these assumptions also imply that the propensity score pg(X) is independent on the sampling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 15 Assumption 3 is a key identifying assumption in DiD settings with treatment hetero- geneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' It means that the average outcomes for the treatment and control groups, conditional of observables, would have followed parallel paths in absence of the treat- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' It is extensively discussed in Abadie (2005) and Callaway and Sant’Anna (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Assumption 4 implies that once an individual is first treated, that individual will continue to be treated in the following periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In other words, there is no exit from the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='6 Finally, Assumption 5 ensures that there are positive proba- bilities to belong to the control and treatment groups for any possible value of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Remark that X is assumed to be observed for all individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Data generating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Under Assumption 1, the data generating process con- sists of random draws from the mixture distribution FM(y, y′, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', gT , c, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', sT , x) defined as7 T � t=1 λt,t+1FYt,Yt+1,G1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=',GT ,C,X|St,t+1(yt, yt+1, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', gT , C, X|st,t+1 = 1), where λt,t+1 = P(St,t+1 = 1) is the sampling probability, y and y′ denote the outcome yt and yt+1, respectively, for an individual sampled at t and t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Expectations un- der the mixture distribution does not correspond to population expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This difference arises because of different sampling probabilities λt,t+1 = P(St,t+1 = 1) across time periods and because Assumption 2 does not preclude from some forms of dependence between the sampling process and the unobservable heterogeneity in Yit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' However, this assumption ensures that expectations of first-differences under the mixture correspond to population expectations once conditioned on the time peri- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Thereafter, EM[·] denotes expectations with respect to the mixture distribution FM(·), its empirical counterpart being the sample mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 6Departures from this assumption are considered in de Chaisemartin and D’Haultfœuille (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 7In the application, we will discuss more complicated situations in which the data is generated by stratified sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The same results apply using a suitably reweighted sample (Wooldridge, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Davezies and D’Haultfœuille, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 16 An important result of the paper is given in the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We define the weights wG ττ−1(g) = GgSτ,τ−1 EM[GgSτ,τ−1] and wC ττ−1(g, X) = pg(X)CSτ,τ−1 1 − pg(X) /EM[pg(X)CSτ,τ−1 1 − pg(X) ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Theorem 1 Under Assumptions 1 - 5, and for 2 ≤ g ≤ t ≤ T , the long-term average treatment effect in period t is nonparametrically identified, and given by ATTCD(g, t) = t � τ=g ∆ATT(g, τ), where ∆ATT(g, t) = EM � wG ττ−1(g) (Yt − Yt−1) � −EM � wC ττ−1(g, X) (Yt − Yt−1) � is the 1-period-difference group-time average treatment effects, which measure the increase of average treatment effect of group g from period t − 1 to period t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Furthermore, identification is no more guaranteed with the cross-section DiD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Those identification results suggest the two-step estimator � ATT CD(g, t) = 1 n n � i=1 t � τ=g � �wG iττ−1(g) (Yiτ − Yiτ−1) − �wC iττ−1(g, X) (Yiτ − Yiτ−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' where �wG iττ−1(g) = GigSiτ−1Siτ 1 n �n i=1 GigSiτ−1Siτ and �wC iττ−1(g) = ˆpg(Xi)CiSiτ−1Siτ 1 − ˆpg(Xi) / 1 n n � i=1 ˆpg(Xi)CiSiτ−1Siτ 1 − ˆpg(Xi) , and with ˆpg(·) being an estimated parametric propensity score function, such as logit or probit, obtained in a first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Let us denote � ATT g≤t the vector of � ATT CD(g, t)’s for g ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The next theorem establishes its joint limiting distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 17 Theorem 2 Under Assumptions 1 - 5 and a standard assumption on the parametric estimates of the propensity scores (Assumption 5 in Callaway and Sant’Anna (2021) or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='4 in Abadie (2005)), for all 2 ≤ g ≤ t ≤ T , √n � � ATT g≤t − ATTg≤t � d→ N(0, Σ), as n → ∞ and where the covariance Σ is detailed in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In the proof of the above theorem, we also show how the multiplier bootstrap proce- dure proposed by Callaway and Sant’Anna (2021) adapts to this asymptotic result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The main difference comes from redefining the influence function, but the result about its asymptotic validity applies without other modification so it is not repeated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We also refer the reader to their paper for a complete discussion of summary parameters which use the ATT(g, t) as building-blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We provide some results for these summary parameters and an extension to using the not yet treated as the control group in Appendices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='3 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='3 General missing data patterns This framework naturally extends to general missing data patterns beyond rotating panel structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For simplicity of exposition, we remove the dependence of g in our notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We now consider not only first-differences ∆1ATT(t) but also k-period differences ∆kATT(t) from t − k to t for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Table 2 provides an example of different missing data pattern with 4 subsamples and 3 periods, and all treatments take place at t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The first subsample is a balanced panel (BP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' It identifies ∆1ATT(2), ∆1ATT(3), and ∆2ATT(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The second, third and fourth subsamples are incomplete panels (IP1, IP2, IP3) which only identifies a single parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='8 8Note that we do not consider refreshment samples because they do not identify any parameter on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' They could still be used to address attrition ex-ante (Hoonhout and Ridder, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 18 Table 2: Example of a more general missing data pattern Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Indicators Identified Parameters Sub-population S1 S2 S3 Balanced Panel 1 1 1 ∆1ATT(2), ∆1ATT(3), ∆2ATT(3) Incomplete Panel 1 1 1 0 ∆1ATT(2) Incomplete Panel 2 0 1 1 ∆1ATT(3) Incomplete Panel 3 1 0 1 ∆2ATT(3) There are multiple ways to identify the ATT from period 1 to period 3 in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We can identify ATT(3) with (1) BP alone since ATT(3) = ∆2ATT(3) or (2) ATT(3) = ∆1ATT(2) + ∆1ATT(3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' or by combining (2) BP and IP3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' or (4) BP and IP1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' or (5) IP1 and IP2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' and with (6) IP3 alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The optimal combination of the ∆kATT(t) parameters into ATT(t) parameters for general missing data patterns hence involves solving an (overidentified) linear inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Doing so will make use of all subsamples and deliver efficiency gains compared to focusing on one possible solution, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' using only the balanced panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The inverse problem arises as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Consider the estimates of all possible ∆kATT(t), for all t ≥ 2 and t − 1 ≥ k ≥ 1, stacked altogether into a vector ∆ATT of length L∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' ∆kATT(t) is called the k-period-difference group-time average treatment effects, which measure the increase of average treatment effect of group g from period t − k to period t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' By definition ∆kATT(t) = ATT(t) − ATT(t − k), hence we can write ∆ATT = WATT, (9) where ATT is the vector of ATT(t) for t ≥ 2 of length L ≤ L∆ and W is a matrix where each element takes value in {−1, 0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Following this reasoning, the example in Table 2 can be written as \uf8ee \uf8ef\uf8f0 ∆1ATT(2)BP,IP 1 ∆1ATT(3)BP,IP 2 ∆2ATT(3)BP,IP 3 \uf8f9 \uf8fa\uf8fb = \uf8ee \uf8ef\uf8f0 1 0 −1 1 0 1 \uf8f9 \uf8fa\uf8fb � ATT(2) ATT(3) � , (10) 19 where we pool all subsamples that identify each ∆kATT(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='9 We propose to solve this problem using a GMM approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Denoting Ω the covariance matrix of ∆ATT, the optimal GMM estimator of ATT corresponds to (W ′Ω−1W)−1 W ′Ω−1∆ATT since WAT T is non-random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='10 This method allows delivering efficiency gains by using all individual time-series with at least two observations, without much additional computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Identification, estimation, and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In order to identify the ATT(g, t) in this general framework, we must modify Assumptions 1 to 2 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Assumption 6 (Sampling) For all t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', T , {Yit−k, Yit, Xi, Di1, Di2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', DiT } nt,k i=1 is independent and identically distributed (iid) conditional on Sit−k,t = 1, for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Assumption 7 (Conditional Independence of Sampling and Trends) For all t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', T and k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', t − 1, Yit − Yit−k ⊥⊥ Sit−k,t|(Xi, Di1, Di2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', DiT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (11) Assumption 6 means that we are considering an incomplete panel structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Con- ditional on being sampled in the same two periods, individuals are assumed to be iid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' It does not imply that individuals are sampled in two periods only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Further- more, it does not imply that observations are representative of the population of interest because we do not assume that the iid draws are taken from the population distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Instead, Assumption 7 states that the sampling process is statistically independent of the joint distribution of k-period-differences of individual outcomes, conditionally on observables and treatment status in any period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In particular, it im- plies that E[Yit −Yit−k|X, a = 1, St−k,t = 1] for a ∈ {G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', C} corresponds to its 9Remark that we could also estimate these parameters separately for each subsample before stacking them into ∆ATT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' That would require to estimate propensity scores conditional on sub- sample membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 10Remark that if an element of ATT is not identified, the matrix W ′Ω−1W (or W ′W) will not be invertible but a (Moore-Penrose) pseudo-inverse can still be used to identify the other elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In addition, pseudo-inverses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Tikhonov’s, will deliver a stable inverse of Ω if ∆ATT is high- dimensional (Carrasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 20 population counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' However, this needs not be true for E[Yit|X, a = 1, St = 1] for a ∈ {G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Notice that these assumptions still imply that the propensity score pg(X) is independent on the sampling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Our estimation procedure is as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Compute ∆kATT(g, t) = 1/n �n i=1[( ˆw(g)G it,t−k − ˆw(g)C it,t−k)(Yit − Yit−k)], for all k, g, t, and stack them into a L∆-dimensional vector ∆AT T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Define the matrix W appropriately;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Estimate the asymptotic covariance matrix ˆΩ = n−1ΨΨ′, where Ψ is a L∆ × n matrix with elements defined in (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Estimate the optimal GMM estimator � AT T = (W ′ˆΩ−1W)−1W ′ˆΩ−1∆AT T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='11 The next theorem establishes the joint limiting distribution of this estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Theorem 3 Under Assumptions 3 - 5, a standard assumption on the parametric estimates of the propensity scores (Assumption 5 in Callaway and Sant’Anna (2021) or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='4 in Abadie (2005)), and Assumptions 6 and 7, for all 2 ≤ g ≤ t ≤ T , √n � � AT T − AT T � d→ N(0, Σ), as n → ∞ and where the covariance Σ is detailed in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The theorem shows that this estimator is consistent and asymptotically normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The bootstrap procedures for ATT(g, t)’s (Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2) and for summary parameters (Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='4) apply with minor modifications, as discussed in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Al- though our estimator brings efficiency gains by using all available observations, it could be further improved by addressing the efficiency losses from first-step plug-in estimates of propensity scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This is left for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='12 11Note further that this approach embeds the estimator proposed above in the rotating panel data setting when using only the ∆AT T1(t) and replacing Ω by the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 12We identified two means to address this caveat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Frazier and Renault (2017) propose a com- putationally simple yet general approach that involves targeting and penalization to enforce the asymptotic efficiency for two-step extremum estimators such as ours, whereas Chaudhuri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (2019) and Sant’Anna and Zhao (2020) propose doubly-robust estimators which allow preserving efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 21 3 Numerical Simulations We propose a simulation design adapted from the first section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Let us specify the outcome variable as a components of variance: Yit = αi + δt + t � τ=2 βτDiτ + εit, (12) where Diτ ∈ {0, 1} denotes whether individual i has been treated in τ or earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Let us assume that t ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', T + 1} and treatments can only occur in t ≥ 2 so that G ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', T + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The data generating process is characterized by the following assumptions: The individual-specific unobservable heterogeneity is iid gaussian: αi ∼ N(1, σ2 α), where σ2 α = 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The time-specific unobservable heterogeneity is iid gaussian: δt ∼ N(1, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The error term is iid gaussian: εit ∼ N(0, σ2 ε), where σ2 ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The probability to receive the treatment at time g, conditional on being treated at g or in the control group, is defined as Pr(Gig = 1|Xi, αi, Gig + Ci = 1) = 1 1 + exp (θ0 + θ1Xi + θ2αi × t), where Xi ∼ N(1, 1) is observable for every i, unlike αi, and θ0 = −1, θ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='4 and θ2 = 0 or θ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In the latter case, the treatment probability varies with time and the unobserved individual heterogeneity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The sampling probability in the consecutive periods t, t + 1 conditional on αi is given by Pr(Sitt+1 = 1|αi) = 1 1 + exp (λ0 + λ1αi × t), with λ0 = −1, and λ1 = 0 or λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2, so that the sampling process can also varies with time and the unobserved individual heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 22 We simulate the sampled data in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' First, we generate a population sample for each period t to represent individuals that are either treated at t or in the control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Second, we sample from this population using the specified process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We formalize this procedure as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Generate a population of individuals (a) Draw N = 2 × max t { n Eα[P r(Sitt+1)]} individuals per period in order to have T + 2 population samples of N individuals, where each individual is char- acterized by a vector (αi, δt, Xi, εit);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (b) Separately for each population sample g, draw a uniform random number ξi ∈ [0, 1] per individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' If ξi ≤ Pr(Git = 1|Xi, αi, Git + Ci = 1), then set (Gig = 1, Ci = 0), otherwise set (Gig = 0, Ci = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (c) Compute Yit from (αi, δt, Xi, εit, Gi0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', GiT+1, Ci);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Sample from this population (a) Draw a uniform random number ηit ∈ [0, 1] per individual i and period t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' If ηit ≤ Pr(Sitt+1 = 1|αi), then set Sitt+1 = 1 and Siττ+1 = 0 for τ ̸= t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (b) Draw (without replacement) n individuals per period t from the popula- tion for which Sitt+1 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (c) Compute the different estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (d) Repeat steps 1(b)-2(c) 1,000 times and report the mean and standard deviation of the estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We consider several simulation designs: DGP 1: θ2 = 0 and λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This is the baseline case where the probability of treatment and the sampling process do not depend on individual heterogeneity so all estimators are unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' DGP 2: θ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2 and λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In this case, both the probability of treatment and the sampling process depend on individual heterogeneity leading to biased estimates for the cross-section DiD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 23 DGP 3: θ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2 and λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In this case, we simulate a stratified sample where 90% of the individuals are sampled on a rotating basis as before and those with αi superior to the 90th percentile are always observed (10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' DGP 4: θ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2 and λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We also simulate a stratified sample where individuals with αi superior to the 60th percentile are always observed (40%) and the rest is drawn as a rotating subpanel (60%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For all simulations, we set T = 6, so there is a total of 8 periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In each period, the population size is 4800, and we draw 150 individuals such that Sitt+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Finally, we set βτ to take the values {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='75, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='50, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='00, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='50} for τ = {1, 2, 3, 4, 5, 6}, that is, the treatment effect is positive and decreasing over time, relative to the treatment starting date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For each sample, we estimate the chained DiD (using sim- ulated Sitt+1 = 1) and the cross-section DiD (using Sit = 1 that are obtained from Sitt+1 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For DGPs 1 and 2, we also estimate the long DiD assuming that all sampled indi- viduals are observed for the entire time frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The simulation results are given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This long DiD here is infeasible but serves as a benchmark to illustrate the significant loss of information resulting from having an unbalanced panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' However, the chained DiD delivers unbiased estimates in both cases unlike the cross-section DiD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 24 Table 3: Simulation results for a rotating panel DGP 1 DGP 2 Chained DiD CS DiD Long DiD Chained DiD CS DiD Long DiD β1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='748 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='745 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='752 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='894 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='75 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='099) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='199) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='017) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='097) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='148) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='016) β2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='498 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='499 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='774 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='501 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='164) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='305) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='02) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='157) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='218) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='018) β3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='248 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='251 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='254 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='651 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='25 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='231) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='355) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='023) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='224) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='257) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='02) β4 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='006 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='522 1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='3) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='412) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='027) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='293) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='291) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='023) β5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='741 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='765 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='739 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='369 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='75 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='406) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='521) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='033) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='395) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='365) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='028) β6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='522 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='209 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='503 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='586) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='711) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='046) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='603) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='515) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='04) Notes: This table shows results obtained from the simulations described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Simulated βτ take the values {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='75, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='50, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='00, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='50} for τ = {1, 2, 3, 4, 5, 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For DGPs 3 and 4, the long DiD is estimated only for individuals that belong to the balanced subpanel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The simulation results are given in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We show estimates from the chained DiD GMM estimator using two weighting matrices: (1) Ch DiD uses the identity matrix, and (2) CD-GMM uses the optimal weighting matrix presented earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' It appears that when the balanced subpanel consists of only 10% of the data, the chained DiD estimators outperform the long DiD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' However, the (asymptotically) optimal weighting matrix does not always deliver more precise estimates than the identity matrix in small samples, at least for this simulation design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In comparison to the identity matrix, It seems that the optimal weights allow mitigating the precision loss for longer-term effects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' β6) at the cost of losing some precision for smaller- term effects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' β1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 25 Table 4: Simulation results for a stratified sample DGP 3 DGP 4 Ch DiD CD-GMM CS DiD Long DiD Ch DiD CD-GMM CS DiD Long DiD β1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='753 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='748 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='714 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='511 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='412) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='398) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='479) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='393) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='121) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='094) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='237) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='097) Notes: This table shows results obtained from the simulations described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Simulated βτ take the values {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='75, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='50, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='00, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='50} for τ = {1, 2, 3, 4, 5, 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 4 Application: The Employment Effects of an In- novation Policy in France We now turn to an application of these methods for estimating the causal impact of a French innovation policy supporting collaborative R&D projects over the period 2010-2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This innovation policy is made up of different subsidy schemes aimed at developing R&D collaborations between firms and, often, public organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' These schemes aim at subsidizing collaborative projects oriented towards applied research and experimental development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='13 To obtain funding, a firm must set up a research project in partnership with another institution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The project is then sub- mitted to one specific subsidy scheme, often following calls for proposals, much like research grant in academic research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The selection of projects, and the associated 13The schemes are FUI, ISI, PSPC, PIAVE, RAPID and ADEME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Although they share the same general objective, they support different forms of R&D projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For example, PSPC projects are much larger in size than others, FUI projects systematically involve companies and public research organizations, ADEME projects have environmental objectives, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' A detailed description of the schemes is available in Bellgo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 26 funding, is based on a list of criteria including, but not limited to, the innovative nature of the project, its credibility, maturity, or commercial character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This innovation policy provides an ideal setting to apply our method because R&D projects take several years to complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Evaluating the effectiveness of the policy hence requires estimating its long-term effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Unfortunately, one of the main data sources about firm-level R&D activities comes from a survey with a rotating panel design with multiple strata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Firms investing a large amount in R&D expenditure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' large companies, are systematically surveyed, while those spending less, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' small and medium-sized enterprises (SMEs) and intermediate-sized enterprises (ISEs), are surveyed only two consecutive years and then dropped out of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Further- more, large firms are always involved in at least one R&D project, so it is not possible to find a plausible counterfactual for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The policy evaluation should therefore focus on SMEs and ISEs, for which we only have an unbalanced panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The application presented in this paper focuses on the average treatment effect of participating in all of these schemes without distinguishing their individual effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' It is relevant to analyze this average effect to the extent that all the schemes contribute to the common objective of subsidizing collaborative R&D projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' These results are a priori the most robust because they are obtained with the greatest number of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='14 We estimate the treatment effect of this policy on employment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We focus on employment because it is the main economic variable for which it is also possible to consistently observe an almost identical measure for all firms using ad- ministrative data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Therefore, by focusing on such outcome variable, we can compare the results obtained with the chained DiD estimator to the “true treatment effect”, obtained for all the firms in the scope of the study using the long DiD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The complete results of this policy evaluation, including the effects on a larger number of economic variables, are available in Bellgo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Our data contains information about all R&D projects financed under these schemes over the period 2010-2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The data includes an unique identifier for each 14It is almost impossible to precisely estimate the individual effect of the smallest schemes as they have subsidized a very small number of projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 27 partner participating in a project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='15 Using this identifier, we have collected exhaus- tive firm-level data from administrative sources that provide the main annual indica- tors on the economic activity of companies over the period 2007-2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In particular, we collected the following variables: total workforce and the number of engineers in the workforce from administrative records on firms’ employment as outcomes of interest, and other variables to used in the propensity scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='16 Data on the number of researchers is obtained from the R&D survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='17 This an- nual survey collects information from about 9000 firms companies each year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The survey has a stratified sampling design: firms with intramural R&D expenditures above 750,000 euros are systematically surveyed the following year, while others are surveyed only two years in a row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The majority of the SMEs and ISEs in the scope of the study are part of the second stratum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In this context, the application of the standard long DiD method is not possible, which justifies the use of the method developed in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Having knowledge of the amount of CIR (research tax credit) paid to companies and their participation in the French cluster policy is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Indeed, the amount of tax credit granted is a good proxy for a company’s propensity to be active in R&D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In addition, competitiveness clusters aim to create a network of firms and research organizations to facilitate the formation of collaborative R&D projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Participation in one of these clusters also reveals a firm’s tendency for this form of R&D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' These different variables are therefore suitable candidates to explain the probability of receiving the treatment in the propensity score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='18 It is important to observe some 15This identifier corresponds to the SIREN number, a unique identification number for French businesses supervised by the French national institute of statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 16Firms’ revenues come from annual tax data restated by INSEE (FICUS/FARE datasets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Em- ployment information comes from administrative records on firms’ employment (DADS datasets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Data on cluster policy comes from the French competitiveness cluster (“Ple de Comptitivit”) man- agement database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Finally, data on support for innovation comes from the “Crdit Impt Recherche” (CIR), a research tax credit, and the “Jeunes Entreprises Innovante” (JEI) scheme, a tax and social exemption aimed at young innovative firms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The CIR is the main tool for supporting innovation in France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Contrary to the devices evaluated in this article, the CIR is an indirect tax aid, in the sense that it is automatically distributed to companies making eligible R&D expenditures and that apply for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 17This survey also provides detailed information on R&D expenditures, the financing of these expenditures, and some outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 18More specifically, the propensity score includes the log of R&D grants, the log of the number 28 variables comprehensively across firms so they can be included in the propensity score, as they must be observed in g − 1, t, and t + 1 to compute the elementary building block constituting each chain link of the chained DiD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The number of firms that can potentially carry out an innovative R&D activity is very small compared to the total number of firms in France (about 3,000,000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Therefore, in order to avoid comparing firms participating in a collaborative R&D project to other firms which are unlikely to pursue such activity, we restrict the scope of the study to firms active in R&D at least one year over the whole period considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This activity is measured by merging together all the sources of information available to us for this purpose: the databases of the research tax credit, the JEI scheme, and the R&D survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Doing so leaves us with about 30,000 firms in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The schemes covered by the study are the main support mechanisms for collaborative R&D in France and involve the highest amounts of public support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' However, there are other alternatives not discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Altogether, the various schemes considered have provided funding for 1697 projects over the period, and have involved 8724 partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' These projects received a total aid of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='6 billion euros and involved total expenditures of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='4 billion euros from their partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In this application, we focus on the effects on (1) total workforce and (2) the employment of managers and highly qualified workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Those variables are observed exhaustively from administrative data (DADS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We estimate the dynamic treatment effect on these two observed outcomes by using three estimators: the long DiD, the chained DiD, and the cross-section DiD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Although these outcomes are consistently observed through time, attrition can still occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For example, firms may disappear over time because of economic difficulties or because they are acquired by another firm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' These companies cannot be taken into account by the long DiD estimator, unless with an attrition model, whereas they are naturally accounted for in the chained DiD and cross-section DiD estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' of engineers, the log of investment, the log of the variation in R&D grants, the log of the variation in turnover, an indicator for being in a competitiveness cluster, and an indicator for being in the IT sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 19A firm can participate in several projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 29 We balance the data to both facilitate the comparison across estimators and get closer to our theoretical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' That is, we keep the firms that are consistently observed from 2007 to 2017 in the administrative data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This balanced panel is referred to as the exhaustive panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We use the long DiD, the chained DiD and the cross-section DiD on this exhaustive panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Then, we construct an unbalanced version of this panel by discarding all observations whenever a firm was not sampled in the R&D survey in a given year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Both the chained DiD and cross section DiD estimators are used on this (artifical) unbalanced panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The objective is to see how each estimator is affected by discarding observations, and compare its performance to the long DiD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Results are presented in Table 5 for the effects on total workforce and Table 6 for highly qualified workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The first column of each table reports the estimates obtained with the long DiD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We find that total employment has increased by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='7% the year after the project started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The long-term increase amounts 12% five years after the start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For highly qualified workers, the effect is also positive but not statistically significant during the first two years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' It becomes significant from the third year onwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' As expected, the estimates obtained using the complete panel are very similar be- tween the long DiD, the chained DiD and the cross-section DiD estimators (columns 1, 2, and 4 of Tables 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The standard errors are also similar between the long DiD and the chained DiD estimators but they are much higher for the cross section DiD estimator, which suggests that the variance of the unobserved heterogeneity is large in this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' On the one hand, the estimated coefficients remain quite similar and the standard errors are only slightly higher with the chained DiD estimator on the unbalanced panel (columns 3 of Tables 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' On the other hand, the esti- mates are considerably worse when obtained with the cross-section DiD estimator on the unbalanced panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The point estimates are different and the standard errors become too large to appreciate the effects of the policy (columns 5 of Tables 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 30 Table 5: Effects on total workforce (exhaustively observed outcome) log(total workforce) Long DiD Chained DiD Cross Section DiD Exhaustive Exhaustive Unbalanced Exhaustive Unbalanced (1) (2) (3) (4) (5) β−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='075 [-0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='021] [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='049,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='021] [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='053] [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='079,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='051] [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='294,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='368] Notes: This table shows the dynamic treatment effects relative to the beginning of the treat- ment obtained on a panel balanced on exhaustive variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' “Exhaustive” refers to the use of an exhaustively observed outcome without pretending that this variable is imperfectly ob- served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' “Unbalanced” refers to the use of an exhaustively observed outcome pretending that this variable is observed from R&D survey, that is, from an unbalanced repeated panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 95% confidence intervals are obtained from the multiplier bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' ∗p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='10, ∗∗p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='05, ∗ ∗ ∗ p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='01 31 Table 6: Effects on highly qualified workers (exhaustively observed outcome) log(highly qualified workforce) Long DiD Chained DiD Cross Section DiD Exhaustive Exhaustive Unbalanced Exhaustive Unbalanced (1) (2) (3) (4) (5) β−3 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='022,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='068] [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='018,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='064] [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='035,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='113] [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='033,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='08] [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='173,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='319] Notes: This table shows the dynamic treatment effects relative to the beginning of the treat- ment obtained on a panel balanced on exhaustive variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' “Exhaustive” refers to the use of an exhaustively observed outcome without pretending that this variable is imperfectly ob- served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' “Unbalanced” refers to the use of an exhaustively observed outcome pretending that this variable is observed from R&D survey, that is, from an unbalanced repeated panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 95% confidence intervals are obtained from the multiplier bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' ∗p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='10, ∗∗p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='05, ∗ ∗ ∗ p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='01 Finally, we apply the chained DiD and cross-section DiD estimators on outcomes similar to those studied just above but actually coming from the R&D survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In this context, it is not possible to apply the standard long DiD estimator because there are too few observations to calculate differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For consistency reasons, we present estimates obtained on the same set of firms as those used for the tables 5 32 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='20 The outcome variables do not correspond to the exact same definition of employment depending on whether they come from the exhaustive administrative source or from the R&D survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Total employees headcount from DADS adminis- trative data corresponds to observations at the legal unit level, whereas the R&D survey sometimes provide information on employment at the group level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='21 The em- ployment of highly qualified workers from the DASD is close to the number of R&D researchers and engineers filled in the R&D survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Highly qualified workers include engineers but also qualified workers dedicated to other tasks than R&D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Conversely, R&D researchers and engineers include researchers who are specifically assigned to research tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Despite their difference, these variables measure similar outcomes and are highly correlated, which justifies their comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Results using outcome variables observed from R&D survey are presented in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The effects obtained with the chained DiD on total employment are somewhat less significant than those presented in Table 5, but the coefficients keep the same order of magnitude (column 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' As might be expected since the policy directly aims at fostering R&D activities, the effects on the number of researchers are stronger and more significant (column 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' On the other hand, the effects obtained with the DiD cross section are much less precise (columns 3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 20That is, we use the set of data that is balanced on the exhaustive variables, which is then merged with the R&D survey, and estimate the effects on the variables reported in the R&D survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 21A legal unit is a legal entity of public or private law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' A firm, in the sense of a group, is an economic entity that may comprise several legal units thanks to financial links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 33 Table 7: Effects on employment variables observed from R&D survey Unbalanced variables from R&D survey in log Chained DiD Cross Section DiD total workforce researchers total workforce researchers (1) (2) (3) (4) β−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} 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+page_content='116,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='246] Notes: This table shows the dynamic treatment effects relative to the beginning of the treat- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The dynamic effects are estimated with outcome variables observed from R&D survey, that is, from an unbalanced repeated panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 95% confidence intervals are obtained from the multiplier bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' ∗p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='10, ∗∗p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='05, ∗ ∗ ∗ p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='01 It is not surprising that the coefficients estimated with the cross section DiD esti- mator is outperformed by the chained DiD estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The R&D survey is a good example of a rotating panel in which the sampling in period t depends on the level of the outcome variable in t − 1, so that St ̸⊥⊥ Yt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='22 There is, however, no rea- son why being surveyed two periods in a row is correlated with the evolution of the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' So we have every reason to think that St,t+1 ⊥ (Yt+1 − Yt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Conversely, 22To be precise, the sampling depends on the level of a firm’s internal R&D expenditure, which mainly includes researchers’ salaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 34 economic variables are often autocorrelated, so Yt is likely to be correlated to Yt−1 and the hypothesis St ⊥ Yt is unlikely to be verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This is particularly true for employment, which is characterized by a strong hysteresis and significant unobserved heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Under these conditions, the chained DiD estimator is more likely to perform well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Finally, we reproduce the results of Tables 5, 6, and 7 by estimating the treatment effects with the complete original sample from the administrative data, without dis- carding the individual firms that are not consistently observed throughout the period to create a balanced panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The results are presented in Appendix B and confirm the better performance of the chained DiD estimator compared to the cross-section DiD one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 5 Conclusion In this paper, we have developed a new estimator to identify long-term treatment effects in unbalanced panel data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This is an important issue, not only because of attrition but also due to how surveys are designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Common practices are either to use a long DiD estimator by balancing the data, at the cost of losing precision and possibly biasing the results, or to use a cross-section DiD estimator at the cost of not accounting for unobserved heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We introduce a new method that simply consists of aggregating short-term DiD estimators obtained from two periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Our theoretical results show that this estimator identifies the average treatment effects of interest, is consistent and asymptotically normal, accounts for treatment heterogeneity and varying treatment timing, as well as general missing data patterns, and delivers efficiency gains by making use of all observations in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' An application to an innovation policy implemented in France reveals that, indeed, this estimator allows identifying statistically significant long-term treatment effects where previous methods fail to do so.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Nijman (1992): “Testing for selectivity bias in panel data models,” International Economic Review, 681–703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' ——— (1996): “Incomplete panels and selection bias,” in The econometrics of panel data, Springer, 449–490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Wooldridge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (2010): Econometric analysis of cross section and panel data, MIT press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 40 A Mathematical Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='1 Simple setting Proof 1 (Proof of Proposition 1) In order to identify and estimate the ATT(t) in this simple setting, we impose the following standard assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Assumption 8 (Sampling) For all t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', T , {Yit, Yit+1, Di1, Di2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', DiT }nt i=1 is independent and identically distributed (iid) conditional on St,t+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Assumption 9 (Independence of Sampling and Levels) For all t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', T , (Yt, D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', DT ) ⊥⊥ St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (13) Assumption 10 (Unconditional Parallel Trends) For all t = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', T , E [Yt(0) − Yt−1(0)|G = 1] = E [Yt(0) − Yt−1(0)|C = 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='. (14) Assumption 11 (Irreversibility of Treatment) For all t = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', T , Dt−1 = 1 implies that Dt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (15) Assumption 12 (Existence of Treatment and Control Groups) P(G = 1) = 1 − P(C = 1) ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='(16) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='iττ+1 (εiτ+1 − εiτ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' where the second equality follows from the fact that wG iττ+1 ̸= 0 and wC iττ+1 ̸= 0 only if yiτ+1 − yiτ is observed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' and the fourth and fifth equalities follow from �n i=1 � wG iττ+1 = �n i=1 � wC iττ+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The second term in the final expression vanishes in expectations from Assumptions 8 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='The second estimator writes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='ATTCS(t) =1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='n ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='wG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='it − � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='wC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='it ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='yit − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='n ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='wG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='i1 − � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='wC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='i1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='αi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' where the second and third terms vanish in expectations under Assumption 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Fi- nally, the expectation of the last term equates zero under Assumption 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We will prove the asymptotic normality of these estimators in the general frame- work in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In this proof, we only derive their asymptotic variance for the mentioned example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 43 E � n � � ATTCD(t) − ATT(t) �2� = nE \uf8ee \uf8f0 � t−1 � τ=1 � 1 n n � i=1 � � wG iττ+1 − � wC iττ+1 � (εiτ+1 − εiτ) ��2\uf8f9 \uf8fb = t−1 � τ=1 1 n n � i=1 E �� � wG iττ+1 − � wC iττ+1 �2 (εiτ+1 − εiτ)2 � = t−1 � τ=1 1 n n � i=1 E �� � wG iττ+1 − � wC iττ+1 �2� E � (εiτ+1 − εiτ)2� , = t−1 � τ=1 E �� � wG iττ+1 − � wC iττ+1 �2� E � (εiτ+1 − εiτ)2� , where the third equality follows from independence of G and εiτ+1 − εiτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' As n → ∞, the weak law of large numbers implies E � � wG iττ+1 2� p→ 1 P (SτSτ+1G=1) because Sitt+1ai ∈ 0, 1 and E[Sitt+1aiSjtt+1aj] = 0 for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Therefore, the asymptotic variance for n → ∞ is E � n � � ATTCD(t) − ATT(t) �2� = t−1 � τ=1 � 1 qp + 1 q(1 − p) � � (ρ − 1)2σ2 ε + σ2 η � = (t − 1) qp(1 − p) 2(1 − ρ) 1 − ρ2 σ2 η = 2 (t − 1) qp(1 − p) 1 1 + ρσ2 η, when assuming P(SiτSiτ+1) = q ∈ (0, 1) for all τ, i and P(Gi = 1) = p ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' the variance of the second estimator can be developed into 44 E � n � � ATTCS(t) − ATT(t) �2� =E �� � wG it − � wC it �2� σ2 ε + E �� � wG i1 − � wC i1 �2� σ2 ε + E �� � wG it − � wC it �2 α2 i � + E �� � wG i1 − � wC i1 �2 α2 i � − E �� � wG i1 − � wC i1 � αi � E �� � wG it − � wC it � αi � =2E �� � wG it − � wC it �2� σ2 ε + 2E �� � wG it − � wC it �2� σ2 α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' = 1 qp(1 − p)(σ2 ǫ + σ2 α),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' where the second equality follows from the independence of outcomes and sampling as well as G and αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' As n → ∞, we have E � � wG iτ 2� p→ 1 P (Sτ G=1) = 1 P (Sτ Sτ+1G=1)+P (Sτ−1Sτ G=1) = 1 2pq for τ < T , and E � � wG iτ 2� p→ 1 P (Sτ G=1) = 1 pq for τ = T , because we have two over- lapping samples in each period except for the first and last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For t < T , the asymptotic variance as n → ∞ is thus E � n � � ATTCS(t) − ATT(t) �2� p→ 1 qp(1 − p) � σ2 η 1 − ρ2 + σ2 α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2 General framework A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='1 Proofs for rotating panel setting Proof 2 (Proof of Theorem 1) This proof focuses on the identification of pa- rameters in the general framework with a rotating panel structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Let us define ATTX(g, τ) = E[Yτ(1) − Yτ(0)|X, Gg = 1] to write its first-difference as ∆ATTX(g, τ) = ATTX(g, τ) − ATTX(g, τ − 1) = E[Yτ(1) − Yτ(0)|X, Gg = 1] − E[Yτ−1(1) − Yτ−1(0)|X, Gg = 1] = E[Yτ − Yτ−1|X, Gg = 1, Sτ,τ−1 = 1] − E[Yτ − Yτ−1|C = 1, Sτ,τ−1 = 1] = AX(g, τ) − BX(g, τ) 45 where the third equality follows from the conditional parallel trends assumption and the sampling independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We can use the above expression to develop ATT(g, t) into ATT(g, t) = E (E[Yt(1) − Yt(0)|X, Gg = 1]|Gg = 1) = E (ATTX(g, t)|Gg = 1) = E � t � τ=g ∆ATTX(g, τ)|Gg = 1 � = t � τ=g E (∆ATTX(g, τ)|Gg = 1, Sτ,τ−1 = 1) = t � τ=g E (AX(g, τ) − BX(g, τ)|Gg = 1, Sτ,τ−1 = 1) (17) with E (AX(g, τ)|Gg = 1, Sτ,τ−1 = 1) = E (Yτ − Yτ−1|Gg = 1, Sτ,τ−1 = 1) = EM � (Yτ − Yτ−1) GgSτ,τ−1 E[GgSτSτ−1] � (18) by the law of iterated expectations and the definition of FM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Following the proofs of Theorem 1 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='1 of Callaway and Sant’Anna (2021), the second term can be developed into23 E(BX(g, τ)|Gg = 1, Sτ,τ−1 = 1) = E(E[Yτ − Yτ−1|X, C, Sτ,τ−1]|Gg = 1, Sτ,τ−1 = 1) = E � E[ C 1 − P(Gg = 1|X, Gg + C, Sτ,τ−1)(Yτ − Yτ−1)|X, Gg + C = 1, Sτ,τ−1]|Gg, Sτ,τ−1 � = E � GgSτ,τ−1E[ C 1−P (Gg|X,Gg+C,Sτ,τ−1)(Yτ − Yτ−1)|X, Gg + C, Sτ,τ−1]|Gg + C, Sτ,τ−1 � P(Gg = 1|Gg + C, Sτ,τ−1) , (19) 23We alleviate notations by dropping = 1 from conditioning sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 46 where using the definition of pg yields .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' = E � GgSτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1E[ C 1−pg(X)(Yτ − Yτ−1)|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Gg + C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1]|Gg + C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1 � P(Gg = 1|Gg + C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1) = E � E[ pg(X)C 1−pg(X)(Yτ − Yτ−1)|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Gg + C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1]|Gg + C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1 � E(Gg|Gg + C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1) = E � (Gg + C)E[ pg(X)C 1−pg(X)(Yτ − Yτ−1)|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Gg + C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1]|Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1 � E(Gg|Gg + C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1)E((Gg + C)|Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1) = E � (Gg + C)E[ pg(X)C 1−pg(X)(Yτ − Yτ−1)|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Gg + C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1]|Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1 � E(Gg|Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1) = E � E[(Gg + C)|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1]E[ pg(X)C 1−pg(X)(Yτ − Yτ−1)|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Gg + C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1]|Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1 � E(GgSτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1) = E � E[ pg(X)C 1−pg(X)(Yτ − Yτ−1)|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1]|Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1 � E(Gg|Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1) = E � pg(X)C 1−pg(X)(Yτ − Yτ−1)|Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1 � E(Gg|Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1) = EM � pg(X)CSτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1 1−pg(X) (Yτ − Yτ−1) � EM(GgSτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1) (20) with EM �pg(X)CSτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1 1 − pg(X) � = EM(EM(Gg|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Gg + C = 1)C EM(C|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Gg + C = 1) Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1) = EM(EM(Gg|X)EM(C|X) EM(C|X) Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1) = EM(EM(Gg|X)Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1) = EM(GgSτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='τ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (21) Finally, the proof that identification is not guaranteed with the repeated cross-sectional estimator (cross-section DiD) presented in Appendix B of Callaway and Sant’Anna (2021) follows from taking a counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Under Assumption 2, it is possible that E[Yt|X, C = 1, St = 1] = E[Yt|X, C = 1] but E[Yt|X, Gg = 1, St = 1] = E[Yt|X, Gg = 47 1] + αt with α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Following the steps of the proof it is is easy to show that the cross-section DiD identifies ATT(g, t) + α(t − g + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Proof 3 (Proof of Theorem 2) This proof is adapted from Callaway and Sant’Anna (2021) ’s Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We proceed in 3 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Parametric propensity scores and notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' First, we introduce additional notations and explain Assumption 5 in Callaway and Sant’Anna (2021) about the estimation of propensity scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='24 Let Wi = (Yit, Yit+1, Xi, Gi1, Gi2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', , GiT , Ci)′ denote the data for an individual i observed in t and t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Assumption 5 in Callaway and Sant’Anna (2021) assumes that the propensity scores, parametrized as pg(Xi) = Λ(X′ iπ0 g) with Λ(·) being a known function (logit or probit), can be paramet- rically estimated by maximum likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We denote ˆpg(Xi) = Λ(X′ iˆπg) where ˆπg are estimated by ML, ˙pg = ∂pg(u)/∂u, and ˙pg(X) = ˙pg(X′ iπ0 g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Under this assumption, the estimated parameter ˆπg is asymptotically linear, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', √n(ˆπg − π0 g) = 1 √n � i ξπ g (Wi) + op(1), where ξπ g (Wi) is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='1) is Callaway and Sant’Anna (2021) and does not depend on the sampling process since X is observed for all individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Let us now define, ψgt(Wi) = ψG gt(Wi) + ψG gt(Wi), (22) where ψG gt(Wi) =wG it,t−1(g) � (Yit − Yit−1) − EM � wG it,t−1(g)(Yit − Yit−1) �� , ψC gt(Wi) =wC it,t−1(g, X) � (Yit − Yit−1) − EM � wC it,t−1(g, X)(Yit − Yit−1) �� + M′ gtξπ g (Wi), 24This is a standard assumption in the literature so it is not reproduced here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 48 and Mgt = EM � X( CStSt−1 1−pg(X))2 ˙pg(X) � (Yit − Yit−1) − EM � wC it,t−1(g, X)(Yit − Yit−1) ��� EM[ pg(X)C 1−pg(X)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' is a k dimensional vector, k being the number of covariates in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Finally, let � ∆ATT g≤t and ∆ATTg≤t denote the vectors of all � ∆ATT(g, t) and ∆ATT(g, t) for any 2 ≤ g ≤ t ≤ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Similarly, the collection of ψgt across all periods and groups such that g ≤ t is denoted by Ψg≤t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Asymptotic result for ∆ATT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Second, we show the asymptotic result for ∆ATT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Recall that � ATT(g, t) = t � τ=g � ∆ATT(g, τ), where � ∆ATT(g, τ) = ˆEM � GgSτ−1Sτ ˆEM[GgSτ−1Sτ] (Yτ − Yτ−1) � − ˆEM \uf8ee \uf8f0 pg(X)CSτ−1Sτ 1−pg(X) ˆEM � pg(X)CSτ−1Sτ 1−pg(X) �(Yτ − Yτ−1) \uf8f9 \uf8fb = � ∆ATT g(g, τ) − � ∆ATT C(g, τ), where ˆE denotes the empirical mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We will show separately that, for all for all 2 ≤ g ≤ t ≤ T , √n � � ∆ATT g(g, t) − ∆ATTg(g, t) � = 1 √n � i ψG gt(Wi) + op(1), (23) and √n � � ∆ATT C(g, t) − ∆ATTC(g, t) � = 1 √n � i ψC gt(Wi) + op(1), (24) which together implies √n � � ∆ATT(g, t) − ∆ATT(g, t) � = 1 √n � i ψgt(Wi) + op(1) (25) 49 and the asymptotic normality of √n � � ∆ATT g≤t − ∆ATTg≤t � follows from the mul- tivariate central limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' First, we show (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Let βg = EM[GgSτ−1Sτ] and ˆβg = ˆEM[GgSτ−1Sτ] and note that √n � ˆβg − βg � = 1 √n � i (GigSiτ−1Siτ − E[GgSτ−1Sτ]) p→ 0, as n → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' for all 2 ≤ g ≤ t ≤ T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' √n( � ∆ATT g(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' t)−∆ATTg(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' t)) = √n ˆEM � GgSt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1 ˆβg (Yt − Yt−1) � − √nEM �GgSt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1 βg (Yt − Yt−1) � = √n ˆβg ( ˆEM [GgSt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(Yt − Yt−1)] − ˆβg βg EM [GgSt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(Yt − Yt−1)]) = √n ˆβg ( ˆEM [GgSt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(Yt − Yt−1)] − EM [GgSt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(Yt − Yt−1)]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' and by the continuous mapping theorem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' √n( � ∆ATT g(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' t)−∆ATTg(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' t)) = √n βg ( ˆEM [GgSt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(Yt − Yt−1)] − EM [GgSt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(Yt − Yt−1)]) − √n � 1 βg − 1 ˆβg � EM [GgSt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(Yt − Yt−1)] + op(1) = √n βg ( ˆEM [GgSt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(Yt − Yt−1)] − EM [GgSt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(Yt − Yt−1)]) − √n � ˆβg − βg � β2 g EM [GgSt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(Yt − Yt−1)] + op(1) = √n βg ( ˆEM [GgSt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(Yt − Yt−1)] − ˆβg βg EM [GgSt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(Yt − Yt−1)]) + op(1) = 1 √n � i Gig Sit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(Yit − Yit−1) − ∆ATT(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' t) βg + op(1) = 1 √n � i wG it,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(g) � (Yit − Yit−1) − EM � wG t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(g)(Yt − Yit−1) �� + op(1) = 1 √n � i ψG gt(Wi) + op(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 50 proving (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Let us now turn to (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For an arbitrary function g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' let wt(g) = g(X)CSt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1 1 − g(X) and note that √n( � ∆ATT C(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' t)−∆ATTC(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' t)) = √n � ˆEM � wt(ˆpg) ˆEM[wt(ˆpg)] (Yt − Yt−1) � − EM � wt(pg) EM[wt(pg)](Yt − Yt−1) �� = √n ˆEM[wt(ˆpg)] � ˆEM [wt(ˆpg)(Yt − Yt−1)] − ˆEM[wt(ˆpg)] EM[wt(pg)]EM [wt(pg)(Yt − Yt−1)] � = √n ˆEM[wt(ˆpg)] � ˆEM [wt(ˆpg)(Yt − Yt−1)] − EM [wt(pg)(Yt − Yt−1)] � − EM [wt(pg)(Yt − Yt−1)] ˆEM[wt(ˆpg)]EM[wt(pg)] √n( ˆEM [wt(ˆpg)] − ˆEM [wt(pg)]) = 1 ˆEM[wt(ˆpg)] √nAn(ˆpg) − ∆ATTC(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' t) ˆEM[wt(ˆpg)] √nBn(ˆpg) = 1 EM[wt(pg)] √nAn(ˆpg) − ∆ATTC(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' t) EM[wt(pg)] √nBn(ˆpg) + op(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' where the last equality follows directly from Assumption 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' which implies Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='3 in Callaway and Sant’Anna (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Applying the mean value theorem yields An(ˆpg) = ˆEM [wt(pg)(Yt − Yt−1)] − EM [wt(pg)(Yt − Yt−1)] + ˆEM � X CSt,t−1 (1 − pg(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' π))2 ˙pg(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' π) �′ � ˆπg − π0 g � , where π is an intermediate point that satisfies |πgπ0 g| ≤ |ˆπgπ0 g| a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Thus, by As- sumption 5, the previously mentioned Lemmas, and the Classical Glivenko-Cantelli’s 51 theorem, An(ˆpg) = ˆEM [wt(pg)(Yt − Yt−1)] − EM [wt(pg)(Yt − Yt−1)] + ˆEM � X CSt,t−1 (1 − pg(X))2 ˙pg(X) �′ � ˆπg − π0 g � + op(n−1/2), and using the same reasoning we obtain Bn(ˆpg) = ˆEM [wt(pg)] − EM [wt(pg)] + ˆEM � X CSt,t−1 (1 − pg(X))2 ˙pg(X) �′ � ˆπg − π0 g � + op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Combining the above results and making use of the same Lemma yields (25) hence concludes the proof for ∆ATT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The asymptotic covariance is given by Σ∆ = E [Ψg≤τ(Wi)Ψg≤τ(Wi)′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Asymptotic result for ATT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Finally, by making use of (25) we have that √n � � ATT(g, t) − ATT(g, t) � = t � τ=g √n � � ∆ATT(g, τ) − ∆ATT(g, τ) � = 1 √n � i � t � τ=g ψgτ(Wi) � + op(1) d→ N(0, Σ), where Σ = EM [Φg≤τ(Wi)Φg≤τ(Wi)′] with Φg≤τ(Wi) = �t τ=g Ψg≤τ(Wi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Therefore, the influence function of the chained DiD estimator corresponds to the sum of influence functions of the short-term DiD estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2 Bootstrap implementation for rotating panel setting Bootstrapped confidence bands for � ATT(g, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The algorithm is as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Draw a vector of Vb = (V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', Vi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', Vn)′, where Vi’s are iid zero mean random variables with unit variance, such as Bernoulli random variables with Pr(V = 1 − κ) = κ/ √ 5 with κ = ( √ 5 + 1)/2 as suggested by Mammen (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 52 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Compute the bootstrap draw � ATT ⋆b g≤t = � ATT g≤t + ˆΦg≤τVb where ˆΦg≤τ is a consistent estimator of Φg≤τ (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Compute ˆR⋆b(g, t) = √n(� ATT ⋆b(g, t) − � ATT(g, t)) for each element of the vector � ATT ⋆b g≤t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Repeat steps 1-3 B times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Note: do not re-estimate propensity scores and parameters for each draw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Compute the bootstrapped covariance for each (g, t) as ˆΣ1/2(g, t) = (q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='75(g, t)− q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='25(g, t))/(z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='75 − z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='25), where qp(g, t) is the pth sample quantile of ˆR⋆ (across B draws) and z(g, t) is the pth quantile the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For each b, compute t-statb g≤t = max(g,t) | ˆR⋆b(g, t)|ˆΣ−1/2(g, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Construct ˆc1−α as the empirical (1 − α) quantile of the B boostrap draws of t-statb g≤t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Construct the bootstrapped simultaneous confidence band for ATT(g, t) as ˆC(g, t) = [� ATT(g, t) ± ˆc1−αˆΣ−1/2(g, t)/√n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This procedure requires to compute ˆΦg≤τ, represented here as a K ×n matrix, with n being the number of observations and K = T (T −1) 2 being the number of (g, t) element for any 2 ≤ g ≤ t ≤ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This is done as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For every (g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' compute the n-dimensional vector ψgt with ith element defined as ψgt(i) = ψG gt(i) + ψC gt(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' where ψG gt(i) =wG it,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(g) � (Yit − Yit−1) − EM � wG t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(g)(Yt − Yt−1) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' ψC gt(i) =wC it,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' X) � (Yit − Yit−1) − EM � wC t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' X)(Yt − Yt−1) �� + M′ gtξπ g (i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' where Mgt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' a k dimensional vector (k being the number of covariates in X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' is defined as Mgt = E � X( CStSt−1 1−pg(X))2 ˙pg(X) � (Yit − Yit−1) − E � wC t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−1(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' X)(Yt − Yit−1) ��� E[ pg(X)C 1−pg(X)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 53 with ˆpg(Xi) = Λ(X′ iˆπg) being the parametric propensity score for covariates Xi and ˙pg(X) = ∂Λ(X′ iˆπg)/∂(X′ iˆπg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Furthermore, ξπ g (i) is a k-dimensional vector for each observation i, and is given by ξπ g (i) = EM � (Gg + C) ˙pg(X)2 pg(Xi)(1 − pg(Xi))XX′ �−1 Xi (Gg + C)(Gg − pg(Xi)) ˙pg(Xi) pg(Xi)(1 − pg(Xi)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Compute φgt = �t τ=g ψg≤τ for all 2 ≤ g ≤ t ≤ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Concatenate all φgt’s into a K × n matrix Φg≤t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='3 Summary parameters for rotating panel setting The group-time average treatment effect ATT(g, t) consists of a building-block to study the dynamic effect of a treatment on different cohorts of treated individ- uals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In most applications, the main causal parameters of interest are not the ATT(g, t) themselves but aggregate parameters of these building-blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In this section, we briefly mention the three main parameters of interest as proposed in Callaway and Sant’Anna (2021), and show how the asymptotic results and multi- plier bootstrap adapt to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Selective timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The causal effect of a policy on the cohort treated in g is given by θS(g) = 1 T − g + 1 T � t=g ATT(g, t), and thus, an average causal effect across groups can be written as θS = T � g=2 θS(g)Pr(G = g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Dynamic treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In the presence of dynamic effects, the researcher may be interested in accounting for the length of exposure to the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The causal 54 effects of an exposure length e ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='} across groups is defined as θD(e) = T � g=2 T � t=g+e ATT(g, t)Pr(G = g|t = g + e), and therefore an average across exposure lengths is given by θD = 1 T − 1 T −1 � e=1 θD(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Calendar time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In some applications, the researcher may be interested in how treatment effects differ with calendar time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Let us consider θC(t) = t � g=2 ATT(g, t)Pr(G = g|g ≤ t), and therefore an average across exposure lengths is given by θC = 1 T − 1 T � t=2 θC(t), The difference between θS, θD and θC is that the second and third attribute more weight to the groups with, respectively, longer exposure lengths, and treated in the earliest periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The asymptotic results and bootstrap procedure directly apply to the summary pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The following corollary summarizes these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Corollary 1 Under the Assumptions of Theorem 2, for all parameters θ defined above, including those indexed by some variable, we have √n(ˆθ − θ) d→ N(0, Σθ), as n → ∞, where Σθ is defined in the proof and the bootstrap procedure is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Proof 4 (Proof of Corollary 1) All summary parameters defined in the text can 55 be generically written as θ = T � g=2 T � t=2 wgtATT(g, t), where wgt are some random weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Estimators can be defined as ˆθ = T � g=2 T � t=2 ˆwgt � ATT(g, t), where estimated weights are such that √n( ˆwgt − wgt) = 1 √n n � i=1 ξw gt(Wi) + op(1), with first and second moments given by E[ξw gt(W)] = 0 and E[ξw gt(W)ξw gt(W)′] finite and positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This condition is satisfied by the sample analogs of weights appearing in the summary parameters θ’s presented in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The application of Theorem 2 yields √n(ˆθ − θ) = 1 √n n � i=1 lw(Wi) + op(1) d→ N(0, E[lw(W)2]) as n → ∞, and where lw(Wi) = T � g=2 T � t=2 � wgt t � τ=g ψgτ(Wi) + ξw gt(Wi)ATT(g, t) � , for ψgt defined in (22) and ξw gt correspond to the estimation errors of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The same bootstrap procedure can hence be used for ˆθ using a consistent estimate of the influence function lw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 56 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='4 Bootstrap for summary parameters All estimators of summary parameters defined in the text can be generically written as ˆθ = T � g=2 T � t=2 ˆwgt � ATT(g, t), where the weights ˆwgt’s are possibly random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In simple settings they are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For instance, let us consider θs(h) = T � g=2 T � t=2 wgtATT(g, t), where wgt = 1 T−g+1 for t ≥ g and g = h, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The algorithm is as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Draw a vector of Vb = (V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', Vi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', Vn)′, where Vi’s are iid zero mean random variables with unit variance, such as Bernoulli random variables with Pr(V = 1 − κ) = κ/ √ 5 with κ = ( √ 5 + 1)/2 as suggested by Mammen (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Compute the bootstrap draw ˆθ⋆b = ˆθ + ˆL′Vb where ˆL is a consistent estimator of the n-dimensional vector L with ith element given by L(i) = T � g=2 T � t=2 wgtφgt(i), where φgt(i) is defined in the previous algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Compute ˆR⋆b = √n(ˆθ⋆b − ˆθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Repeat steps 1-3 B times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Compute the bootstrapped covariance as ˆΣ1/2 = (q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='75 − q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='25)/(z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='75 − z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='25), where qp is the pth sample quantile of ˆR⋆ (across B draws) and z is the pth quantile of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For each b, compute t-statb g≤t = max(g,t) | ˆR⋆b(g, t)|ˆΣ−1/2(g, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Construct ˆc1−α as the empirical (1−α) quantile of the B boostrap draws t-statb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 57 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Construct the bootstrapped confidence interval for θ as ˆC = [ˆθ±ˆc1−α ˆΣ−1/2/√n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' If the weights wgt’s are random, the influence function is changed to L(i) = T � g=2 T � t=2 wgtφgt(i) + γw gt(i)ATT(g, t), where γw gt(i) is an error function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For example, let us consider θs = T � g=2 T � t=2 wgtATT(g, t), where wgt = P(G = g) 1 T−g+1 for t ≥ g , and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Define ˆwgt = 1 T−g+1 1 n �n i=1 Gi for t ≥ g and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' A consistent estimator of the error function is given by ˆγw gt(i) = 1 T − g + 1 � Gi − 1 n n � i=1 Gi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='5 Not yet treated as the control group In the previous model, we assumed the existence of a true control group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' a group of individuals that are never treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In many applications, this situation is not realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Instead, the researcher can use the individuals that are “not yet treated”, that is treated in g > t to define a control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The extension to this setting being developed in length in Callaway and Sant’Anna (2021), we only explain how it applies to the chained DiD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Most importantly, the parallel trend assumption is modified to Assumption 13 (Conditional Parallel Trends) For all t = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', T , g = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', T , such that g ≤ t, E [Yt(0) − Yt−1(0)|X, Gg = 1] = E [Yt(0) − Yt−1(0)|X, Dt = 0] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='. (26) Following minor modifications to Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' in Callaway and Sant’Anna (2021), using the “not yet treated” as the control group only changes the weight wC ττ−1(g, X) 58 used in our Theorem 1 to wC ττ−1(g, X) = pg,t(X)(1 − Dt)Sτ,τ−1 1 − pg,t(X) /EM[pg,t(X)(1 − Dt)Sτ,τ−1 1 − pg,t(X) ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We observe two changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' First, the binary variable C becomes 1 − Dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Second, generalized propensity score is now also a function of t: pg,t(X) = P(Gg = 1|X, (Gg = 1 ∪ Dt = 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The propensity scores must hence be estimated for pairs (g, t) because the control group evolves through time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The asymptotic properties of the two-step estimator remain similar, with minor changes to the asymptotic covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='6 General missing data patterns Under Assumption 6, the data generating process consists of random draws from the mixture distribution FM(y, y′, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', gT , c, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', sT , x) defined as T � t=1 λt−k,tFYt−k,Yt,G1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=',GT ,C,X|St−k,t(yt−k, yt, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=', gT , C, X|st−k,t = 1), where λt−k,t = P(St−k,t = 1) is the probability of being sampled in both t and t − k, y and y′ denote the outcome yt−k and yt, respectively, for an individual sampled at t − k and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Again, expectations under the mixture distribution does not correspond to population expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This difference arises because of different sampling prob- abilities across time periods and because Assumption 7 does not preclude from some forms of dependence between the sampling process and the unobservable heterogene- ity in Yit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Proof 5 (Proof of Theorem 3) Let us define the vector of parameters Θ = ATT that includes all θτ = ATT(τ), for all τ > 1 since θ1 = ATT(1) = 0 by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The inverse problem in (9) corresponds to the set of moment equalities EM [hi(Wi|Θ)] = 0L∆, (27) 59 where hi(Wi|Θ) is a L∆-dimensional vector of which each element is defined by � wG iττ−k(g) (Yiτ − Yiτ−k) − wC iττ−k(g, X) (Yiτ − Yiτ−k) − θτ + θτ−k � , (28) possibly for all τ ≥ 2, and 1 ≤ k < τ, with the weights wG ττ−k(g) = GgSτ,τ−k EM[GgSτ,τ−k] and wC ττ−k(g, X) = pg(X)CSτ,τ−k 1 − pg(X) /EM[pg(X)CSτ,τ−k 1 − pg(X) ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The previous asymptotic results in Theorem 1 and 2 apply to (28) up to minor modifi- cations under Assumptions 3 - 5, a standard assumption on the parametric estimates of the propensity scores (Assumption 5 in Callaway and Sant’Anna (2021) or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='4 in Abadie (2005)), and Assumptions 6 and 7 so that we can safely assume that con- sistency and asymptotic normality holds for � ∆AT T as n → ∞ with covariance Ω, which is defined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Here, we focus on the aspects of the proofs which differ, namely the optimal combination of each “chain link” using GMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The optimal GMM estimator consists in minimizing ˆEM [hi(Wi|Θ)]′ Ω−1 ˆEM [hi(Wi|Θ)] , (29) with respect to Θ, using the optimal weighting matrix Ω−1 which corresponds the inverse of the covariance of hi (Hansen, 1982), hence that of ∆AT T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Let us rewrite this problem as max AT T −( � ∆AT T − WAT T )′Ω−1( � ∆AT T − WAT T ), (30) then the first-order condition with respect to AT T is given by − 2( � ∆AT T ) − WAT T )′Ω−1W = 0, (31) 60 which, in turn, leads to the proposed estimator: � AT T = (W ′Ω−1W)−1W ′Ω−1 � ∆AT T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (32) The necessary and sufficient rank condition for GMM identification in this linear set- ting is that the rank of Ω−1W is equal to the number of columns (Newey and McFadden, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This condition is satisfied if both the covariance matrix Ω and the weight matrix W are non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Remark further that if W is not full row rank then some ATT(g, t) are not identified by the collection of ∆kATT(g, t)’s identified in the dataset at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Proving consistency requires introducing standard assumptions for GMM estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We assume that (i) Ω and the weight matrix W are non-singular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (ii) the true value Θ0 lies within a compact set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' and (iii) EM[supΘ ||hi(Wi|Θ)||] < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In addition to our previous assumptions, applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='6 in Newey and McFadden (1994) yields the desired consistency result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Assuming further that (iv) Θ0 lies in the interior of the compact set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (v) E[||hi(Wi|Θ0)||2, ] < ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (vi) W ′Ω−1W non-singular, then asymptotic normality follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='4 in Newey and McFadden (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='25 Note that the two-step GMM estimator requires estimating Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' We proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' For every (g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' compute the n-dimensional vector ψgtk with ith element defined as ψgtk(i) = ψG gtk(i) + ψC gtk(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (33) where ψG gtk(i) =wG it,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−k(g) � (Yit − Yit−k) − EM � wG t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−k(g)(Yt − Yt−k) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' ψC gtk(i) =wC it,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−k(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' X) � (Yit − Yit−k) − EM � wC t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='t−k(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' X)(Yt − Yt−k) �� + M′ gtkξπ g (i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' where Mgtk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' a k dimensional vector (k being the number of covariates in X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' is 25All the other sufficient conditions used by these theorems are trivially satisfied in this linear setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 61 defined as Mgtk = E � X( CStSt−k 1−pg(X))2 ˙pg(X) � (Yit − Yit−k) − E � wC t,t−k(g, X)(Yt − Yit−k) ��� E[ pg(X)C 1−pg(X)] , with ˆpg(Xi) = Λ(X′ iˆπg) being the parametric propensity score for covariates Xi and ˙pg(X) = ∂Λ(X′ iˆπg)/∂(X′ iˆπg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Furthermore, ξπ g (i) is a k-dimensional vector for each observation i, and is given by ξπ g (i) = EM � (Gg + C) ˙pg(X)2 pg(Xi)(1 − pg(Xi))XX′ �−1 Xi (Gg + C)(Gg − pg(Xi)) ˙pg(Xi) pg(Xi)(1 − pg(Xi)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Concatenate all ψgtk’s into a L∆ × n matrix Ψ, and compute ˆΩ = ˆE[Ψ(i)Ψ(i)′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Therefore, the asymptotic covariance of � AT T is Σ = (W ′Ω−1W)−1, (34) and its corresponding influence function to be used in the bootstrap procedure detailed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2 is the empirical counterpart of Φ = (W ′Ω−1W)−1W ′Ω−1Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' (35) Finally, it is easy to show that the choice of the optimal weighting Ω−1 is the same if the objective is instead to minimize the variance of a linear transformation R′AT T , where R is a vector of weights, like for all the summary parameters considered in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' In that case, the bootstrap for summary parameters in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='4 apply with the (general) influence defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Let the weights in R be random, the influence function is changed to L(i) = R′Φ(i) + γw(i)′AT T , where γw is the error function that depends on the randomness of the weights, as defined in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 62 Online Appendix B Appendix for the Application In this appendix, we present the results obtained for the application when the initial administrative data are not balanced ex-ante using the exhaustively observed variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' Results are presented in Tables B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='1, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' This introduces some differences because there are more individuals in some periods compared to the results presented earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 63 Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='1: Effects on total workforce (exhaustively observed outcome with the com- plete panel data) log(total workforce) Long DiD Chained DiD Cross Section DiD Exhaustive Exhaustive Unbalanced Exhaustive Unbalanced (1) (2) (3) (4) (5) β−3 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='021] [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='047,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='039] [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='087,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='061] [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='238,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='225] Notes: This table shows the dynamic treatment effects relative to the beginning of the treat- ment obtained on a panel balanced on exhaustive variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' “Exhaustive” refers to the use of an exhaustively observed outcome without pretending that this variable is imperfectly ob- served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' “Unbalanced” refers to the use of an exhaustively observed outcome pretending that this variable is observed from R&D survey, that is, from an unbalanced repeated panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 95% confidence intervals are obtained from the multiplier bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' ∗p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='10, ∗∗p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='05, ∗ ∗ ∗ p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='01 64 Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='2: Effects on highly qualified workers (exhaustively observed outcome with the complete panel data) log(highly qualified workforce) Long DiD Chained DiD Cross Section DiD Exhaustive Exhaustive Unbalanced Exhaustive Unbalanced (1) (2) (3) (4) (5) β−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='102 [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='046,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='085] [-0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='015,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='054] [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='022,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='058] [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='031,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='1] [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='032,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='089] [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='151,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='19] Notes: This table shows the dynamic treatment effects relative to the beginning of the treat- ment obtained on a panel balanced on exhaustive variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' “Exhaustive” refers to the use of an exhaustively observed outcome without pretending that this variable is imperfectly ob- served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' “Unbalanced” refers to the use of an exhaustively observed outcome pretending that this variable is observed from R&D survey, that is, from an unbalanced repeated panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 95% confidence intervals are obtained from the multiplier bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' ∗p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='10, ∗∗p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='05, ∗ ∗ ∗ p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='01 65 Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='3: Effects on employment variables observed from R&D survey with the complete panel data Unbalanced variables from R&D survey in log Chained DiD Cross Section DiD total workforce researchers total workforce researchers (1) (2) (3) (4) β−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='081 [-0.' metadata={'source': 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dynamic treatment effects relative to the beginning of the treat- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' The dynamic effects are estimated with outcome variables observed from R&D survey, that is, from an unbalanced repeated panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' 95% confidence intervals are obtained from the multiplier bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content=' ∗p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='10, ∗∗p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfJvt-/content/2301.01085v1.pdf'} +page_content='05, ∗ ∗ ∗ p < 0.' metadata={'source': 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-0,0 +1,1546 @@ +Draft version January 10, 2023 +Typeset using LATEX twocolumn style in AASTeX62 +CGM2 + CASBaH: The Mass Dependence of H I Lyα-Galaxy Clustering and the Extent of the CGM +Matthew C. Wilde,1 Kirill Tchernyshyov,1 Jessica K. Werk,1 Todd M. Tripp,2 Joseph N. Burchett,3, 4 +J. Xavier Prochaska,3, 5, 6 Nicolas Tejos,7 Nicolas Lehner,8 Rongmon Bordoloi,9 John M. O’Meara,10 +Jason Tumlinson,11 and J. Christopher Howk12 +1University of Washington, Department of Astronomy, Seattle, WA 98195, USA +2Department of Astronomy, University of Massachusetts, 710 North Pleasant Street, Amherst, MA 01003-9305, USA +3University of California, Santa Cruz; 1156 High St., Santa Cruz, CA 95064, USA +4Department of Astronomy , New Mexico State University, PO Box 30001, MSC 4500, Las Cruces, NM 88001 +5Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU) The University of Tokyo; 5-1-5 Kashiwanoha, Kashiwa, +277-8583, Japan +6Division of Science, National Astronomical Observatory of Japan,2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan +7Instituto de F´ısica, Pontificia Universidad Cat´olica de Valpara´ıso, Casilla 4059, Valpara´ıso, Chile +8Department of Physics and Astronomy, University of Notre Dame, Notre Dame, IN 46556 +9North Carolina State University, Department of Physics, Raleigh, NC 27695-8202 +10W. M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743, USA +11Space Telescope Science Institute, Baltimore, MD, USA +12Department of Physics and Astronomy, University of Notre Dame, Notre Dame, IN 46556, USA +ABSTRACT +We combine datasets from the CGM2 and CASBaH surveys to model a transition point, Rcross, +between circumgalactic and intergalactic media (CGM and IGM, respectively). +In total, our data +consist of 7244 galaxies at z < 0.5 with precisely measured spectroscopic redshifts, all having impact +parameters of 0.01 − 20 comoving Mpc from 28 QSO sightlines with high-resolution UV spectra that +cover H I Lyα. Our best-fitting model is an exclusionary two-component model that combines a 3D +absorber-galaxy cross correlation function with a simple Gaussian profile at inner radii to represent +the CGM. By design, this model gives rise to a determination of Rcross as a function of galaxy stellar +mass, which can be interpreted as the boundary between the CGM and IGM. For galaxies with 108 ≤ +M⋆/M⊙ ≤ 1010.5, we find that Rcross(M⋆) ≈ 2 ± 0.6Rvir. Additionally, we find excellent agreement +between Rcross(M⋆) and the theoretically-determined splashback radius for galaxies in this mass range. +Overall, our results favor models of galaxy evolution at z < 0.5 that distribute T ≈ 104K gas to distances +beyond the virial radius. +1. INTRODUCTION +The formation and evolution of galaxies involves a +complex interplay between gravitational collapse of gas +from the intergalactic medium (IGM), galaxy mergers, +and feedback due to stellar evolution and active galactic +nuclei (AGN) that drive gaseous outflows and change the +ionization state of the galaxies’ gaseous halos. Together, +these processes drive the “cosmic baryon cycle” which +takes place largely in the region of a galaxy referred to +as the circumgalactic medium (CGM). Indeed, under- +standing the CGM is critical for developing a complete +theory of galaxy evolution, as highlighted by the recent +decadal survey (National Acadamy of Sciences 2021). +Corresponding author: Matthew C. Wilde +mwilde@uw.edu +In particular, the extent of the gaseous CGM relative to +the extent of the dark matter halo is a subject of great +interest for models that aim to reproduce the properties +of gaseous halos. +The existence of the CGM, first predicted by Bahcall +& Spitzer (1969), was initially revealed by detection of +Mg II and H I absorption at large projected distances +(R⊥ > 20 kpc) from L∗ galaxies (Bergeron 1986; Morris +et al. 1993; Bergeron & Boiss´e 1991; Lanzetta et al. 1995; +Chen et al. 2005), and subsequently traced via higher- +energy metal-line transitions such as Si III, C IV and +O VI that are observed to correlate with galaxies and +their global properties (e.g. Tripp & Savage 2000; Tripp +et al. 2008; Prochaska et al. 2011; Tumlinson et al. 2011; +Werk et al. 2013). Within 0.5 Rvir of L ∼ L* galaxies, +the metal line incidence is found to be 60 − 90 % for a +range of ionized metal species (Werk et al. 2013). Con- +arXiv:2301.02718v1 [astro-ph.GA] 6 Jan 2023 + +2 +versely, Berg et al. (2022) find an 80% chance of find- +ing a massive galaxy nearby to any high-metallicity ab- +sorber. The CGM of M⋆ > 108 M⊙ galaxies is now well- +established to be metal-enriched (Liang & Chen 2014; +Bordoloi et al. 2014; Prochaska et al. 2017; Berg et al. +2022), and to extend to at least 1 Rvir, and very likely +beyond it (Wakker & Savage 2009; Burchett et al. 2015; +Finn et al. 2016; Wilde et al. 2021; Borthakur 2022). +Generally, hydrodynamical simulations of galaxy evo- +lution, +which exhibit complex interactions between +gravitational collapse from the cosmological large scale +structure and subsequent feedback from supernovae +and AGN-driven winds that heat and enrich the CGM +and IGM (EAGLE, Schaye et al. 2015; IllustrisTNG, +Pillepich et al. 2018; SIMBA, Dav´e et al. 2019; and +CAMELS, Villaescusa-Navarro et al. 2022), are con- +sistent with the range of observations of the CGM in +absorption. +Yet these models still rely on simplistic +implementations of the “sub-grid” physics in order to +model entire galaxies (e.g. Ford et al. 2013; Hummels +et al. 2013), and physical properties of the CGM are +dependent on the simulation resolution (Hummels et al. +2019; Peeples et al. 2019). More sensitive observations +of the CGM, including the ability to detect the diffuse +gas in emission, are needed both to break degenera- +cies in these models, e.g., between heating and cooling +mechanisms, and to develop a flexible parametric model +of the CGM (Singh et al. 2021). +The two-point correlation function between H I ab- +sorption along QSO sightlines and galaxies has proven +to be an essential tool to understand the connection of +galaxies to the IGM (e.g. Morris et al. 1993; Chen et al. +2005; Ryan-Weber 2006; Prochaska et al. 2011; Tejos +et al. 2014; Prochaska et al. 2019). The primary advan- +tages of leveraging the clustering of these two entities +over one-to-one association analyses is that it provides +results for large scales (1-10 Mpc) as well as the rel- +atively smaller scales where the baryonic processes as- +sociated with the CGM play out, and the correlation +function statistically characterizes absorber-galaxy rela- +tionships when multiple galaxies are close to the sight- +line and a one-to-one assignment is ambiguous. Since +H I traces both enriched material from galaxies as well +as primordial accretion from the IGM, observations of +the CGM, IGM, and galaxies in the same volume are +fundamental to both testing the predictions of galaxy +evolution models and providing a means to differentiate +between them (e.g. Fumagalli et al. 2011; Oppenheimer +et al. 2012; Stinson et al. 2012; Ford et al. 2013; Hum- +mels et al. 2013; Butsky et al. 2020; Singh et al. 2021). +Understanding the physical profile and size of the +CGM sheds light on the non-linear processes of galaxy +formation: on what spatial scale(s) do virialization, ac- +cretion, and feedback transform these galactic atmo- +spheres? Astronomers have long used some version of +the virial radius as an estimator for the size of galaxy +halos, but this estimate is somewhat arbitrary and is +based on the distibution of unobservable dark matter. +By observing the radial gas profile around galaxies out +to large scales, we can effectively map the gaseous halo, +which in turn constrains the physics of galaxy-scale feed- +back processes. Observationally determining the galac- +tic atmosphere’s extent has additional implications for +constraining galaxy evolution and assembly models. For +example, the galaxy baryon and metal budgets require a +scale to integrate the total mass (e.g. Peeples et al. 2014; +Werk et al. 2014). Furthermore, the gaseous halo likely +plays an important role in the quenching of dwarf satel- +lite galaxies as they become stripped by ram-pressure +in a low-density CGM (Putman et al. 2021), and it is +useful to constrain where this occurs, i.e., the extent of +the CGM, and how this depends on central galaxy mass. +The presence of H I absorption beyond the virial ra- +dius is now widely accepted for a range of galaxy stellar +masses (e.g. Prochaska et al. 2011; Tejos et al. 2012, +2014; Wilde et al. 2021; Bouma et al. 2021; Borthakur +2022). In Wilde et al. (2021) (Paper I) we found an em- +pirical relation between galaxy stellar mass and the ex- +tent of the CGM as indicated by H I covering fractions. +For galaxies with stellar masses 108 < M⋆/M⊙ < 1011.5, +we found that the CGM extends to two times the virial +radius. In this paper, we focus on the functional forms +of the mass dependence of the H I-traced CGM using a +power-law model similar to the 2-halo correlation func- +tion. We also investigate other two-component models +that differentiate the CGM from the IGM. We combine +the CGM2 Survey, which focuses on sightlines at low +galaxy impact parameters (< 1 Mpc), with the COS +Absorption Survey of Baryon Harbors (CASBaH) that +probes larger spatial scales (< 20 Mpc). In doing so, +we greatly increase the absorber-galaxy sample from +543 spectroscopically-confirmed absorber-galaxy pairs +to 7244 pairs spanning 0.003 < z < 0.48. +Our goal +is to provide the most reliable constraints to date on the +spatial extent of the CGM as traced by H I absorption. +The paper is structured as follows: In Section 2, we +briefly review each of the galaxy-absorber surveys and +discuss their combined properties. In Section 3, we in- +troduce two models of the H I-galaxy correlation func- +tions and cover our main results in Section 4. We com- +pare our results with simulations and previous results +and discuss their implications for galaxy evolution mod- +els in Section 5. Finally, we summarize our results in +Section 6. + +3 +2. DATA - COMBINING CGM2 AND CASBAH +Both surveys feature far-ultraviolet spectroscopy of +QSOs with HST, using both the Cosmic Origins Spectro- +graph (COS, Green et al. 2012) and the Space Telescope +Imaging Spectrograph (STIS, Woodgate et al. 1998), and +deep, ground-based optical spectroscopy of foreground +galaxies in the QSO fields. CASBaH is well suited to the +study of the interface between the CGM and the IGM, +at scales ≳ 1 Mpc. CGM2 provides a relatively more +complete mapping of the inner CGM at scales ≲ 1 Mpc. +By combining CGM2 and CASBaH data, we leverage +the strengths of each survey, as described below. Figure +(1) shows the distributions of galaxy stellar masses and +impact parameters versus redshift from both surveys out +to z = 0.5. Together, the surveys allow us to probe the +CGM as it transitions into the IGM for a large sample +of galaxies. +2.1. CGM2 +The CGM2 survey, first presented in Wilde et al. +(2021), includes precise spectroscopic redshifts and bulk +galaxy properties (e.g. stellar masses, M∗, and star +formation rates, SFR) from a combination of Gemini +GMOS spectra and deep, broadband photometry for +∼1000 galaxies in the foreground of 22 QSOs, each +with S/N ≈10 HST/COS G130M+G160M spectra. By +matching galaxy and absorber redshifts in ±500 km s−1 +windows, the CGM2 survey is ultimately a large col- +lection of measurements pertaining to the CGM of z +< 1 galaxies over a wide range of stellar masses, 108 +≲ M⋆/M⊙ ≲ 1011.5. The data acquisition and analysis +are explained in detail in Wilde et al. (2021). Here we +present a brief overview of the survey data relevant to +the present analysis. +The CGM2 +galaxy spectra were obtained using +Gemini-GMOS spectrographs on the twin Gemini North +and South telescopes (Hook et al. 2004; Gimeno et al. +2016). Galaxy redshifts were inferred from the template +fitting code, Redrock1 (v0.14) and manually inspected +with VETRR2. The typical statistical uncertainly of our +redshifts is σz ∼ 50-100 km s−1 (z ≃ 0.00016-0.00030). +Photometry of the CGM2 galaxy catalog was obtained +from the Gemini-GMOS pre-imaging in g and i bands +as well as all available bands from DESI Legacy Imag- +ing Surveys Data Release 8 (DR8) (Dey et al. 2019), +WISE (Cutri et al. 2013), Pan-STARRS Data Release +2 (Chambers et al. 2016), and SDSS DR14 (Abolfathi +et al. 2018). +1 https://github.com/desihub/redrock +2 https://github.com/mattcwilde/vetrr +0.0 +0.2 +0.4 +0.6 +z +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +log R [cMpc] +CASBaH +CGM2 +0.0 +0.2 +0.4 +0.6 +z +7 +8 +9 +10 +11 +12 +logM [M ] +CASBaH +CGM2 +Figure 1. Top: Distribution of the combined CGM2 (blue +dots) and CASBaH (purple dots) data sets in both logarith- +mic impact parameter, and redshift. The data are roughly +uniform in redshift space but we can see the relative contri- +butions of the data sets in impact parameter space; CGM2 is +highly concentrated at lower impact parameters while CAS- +BaH explores much greater impact parameters. +Bottom: +Galaxy stellar mass distribution as a function of redshift for +the two data sets. + +4 +The 22 QSOs included in the CGM2 survey have +HST/COS spectra selected from the COS-Halos (GO11598, +GO13033; Tumlinson et al. 2013) and COS-Dwarfs +(GO12248; Bordoloi et al. 2014) surveys. +In general, +the CGM2 QSO targets have zQSO > 0.6 and avail- +able HST imaging, which permits detailed analysis of +absorption-hosting galaxies with z < 0.5. All COS spec- +tra include both the G130M and G160M gratings, and +have a S/N ≃ 8 − 12 per resolution element (FWHM +≃ 16-18 km s−1) or better over 1150-1800 ˚A. The COS +data and their reduction are presented in detail in Tum- +linson et al. (2013) and Bordoloi et al. (2014) and follows +the same method used by Tripp et al. (2011), Meiring +et al. (2011), Tumlinson et al. (2011) and Thom et al. +(2012). +2.2. CASBaH +The CASBaH program was designed to take advan- +tage of the multitude of resonance transitions at rest- +frame wavelengths < 912 ˚A to probe the physical condi- +tions, metallicity, and physics of the multiphase CGM. A +wide variety of elements and ionization stages have res- +onance lines only at λ < 912 ˚A (see, e.g., Verner et al. +1994), so observations of this wavelength range provide +new diagnostics and precise constraints using banks of +adjacent ions such as N i through N v, O i through +O vi, and Ne ii through Ne viii (see Tripp et al. 2011, +for examples of lines detected by CASBaH). The Ne viii +770.4, 780.3 ˚A doublet has received particular attention +as a probe of warm-hot gas at ≈ 105 − 106 K (e.g., Sav- +age et al. 2005; Burchett et al. 2019; Wijers et al. 2020). +In many contexts such as the Milky Way interstellar +medium, these lines are inaccessible because they are +blocked by the H i Lyman limit. CASBaH overcomes +this limitation by observing QSO absorbers with suffi- +cient redshift to bring the lines into the observable band +of HST. +The motivation and design of the CASBaH program +is summarized in section 1 of Haislmaier et al. (2021), +and the CASBaH galaxy redshift survey is presented +in Prochaska et al. (2019). Briefly, CASBaH obtained +both HST/COS and HST/STIS spectra of nine QSOs at +0.92 < zQSO < 1.48, with two primary selection crite- +ria. First, since some of the most important target lines +(e.g., Ne viii) are weak, the QSOs were required to be +UV-bright so that good signal-to-noise and sensitivity to +weak lines would be attained. Second, the targets were +required to have zQSO > 0.9 to provide a total redshift +path that is sufficient to accumulate a statistically use- +ful sample of absorbers of interest. No considerations +were given to known foreground galaxies or absorbers, +so the targets were not selected in a way that would fa- +vor particular types of foreground absorbers or galaxies, +except that sightlines with known black Lyman limits at +λob > 1150 ˚A were excluded to avoid using HST time on +sightlines that would not contribute useful pathlengths +to the samples (see Burchett et al. 2019). The CASBaH +UV spectra were reduced in the same way as the CGM2 +data. +The CASBaH galaxy-redshift survey (Prochaska et al. +2019) measured thousands of redshifts in the fields of +seven of the CASBaH QSOs using the Keck DEIMOS +and MMT Hectospec spectrographs, with typical red- +shift uncertainties of ≈ 30 km s−1. +The survey used +a wedding-cake strategy with the Hectospec covering +galaxies in the ≈ 1◦ fields centered on the QSOs and +the DEIMOS survey providing a deeper survey with a +smaller field of view (81.5 arcmin2) (see Prochaska et al. +2019). +Using the CASBaH galaxy database, supple- +mented with data from public surveys such SDSS, we +selected a sample of 6701 galaxies with spectroscopic +redshifts z < 0.481 and comoving impact parameters +less than 13 cMpc, appropriate for the H I analysis pre- +sented here. +2.3. Synergy of CGM2 + CASBaH +The CASBaH and CGM2 surveys have complemen- +tary designs. +On the one hand, CGM2 is built on +COS-Halos and thus favors at least one L∗ galaxy close +to the sightline. +CGM2 also covers a smaller FOV. +On the other hand, CASBaH is a blind survey that +covers a larger FOV. Consequently, CASBaH provides +more information about galaxies and large-scale struc- +tures at larger impact parameters, but as a blind sur- +vey, it is cross-section weighted in favor of galaxies at +larger impact parameters. Also, since CASBaH avoided +sightlines with black Lyman limits in the HST band +(i.e., at λob ≥ 1150 ˚A), it will not include galaxies at +zgal > 0.26 that harbor absorbers with N(H i) ≳ 1017 +cm−2. Thus, CGM2 probes the inner CGM including +higher N(H i) absorbers, while CASBaH complements +CGM2 by adding very large samples of galaxies and +structures at larger distances. +2.4. Galaxy Properties +To estimate the galaxy properties for both surveys, we +used CIGALE (Noll et al. 2009; Boquien et al. 2019) to +fit the spectral energy distribution (SED) and retrieve +stellar mass and star formation rates (SFR). We used +the Bruzual & Charlot (2003) stellar population mod- +els, assuming a Chabrier (2003) initial mass function +(IMF). We chose a grid of metallicities ranging from +0.001-2.5Z⊙. A delayed star formation history (SFH) +model was employed with an exponential burst. +The + +5 +e-folding time of the main stellar population models +ranged from 0.1-8 Gyr. We varied the age of the oldest +stars in the galaxy from 2-12 Gyr. We included an op- +tional late burst with an e-folding time of 50 Myr and +an age of 20 Myr. The burst mass fraction varied from +0.0 or 0.1 to turn this feature on or off. Nebular emis- +sion and reprocessed dust models (Dale et al. 2014) were +also included with the default values. The dust models +have slopes ranging from 1−2.5 and the nebular models +include no active galactic nuclei. +We employed the Calzetti et al. (1994) dust attenua- +tion law, but we also included a “bump” in the UV (see +discussion in Prochaska et al. 2019) at 217.5 nm with a +FWHM of 35.6 nm. The bump amplitude is set at 1.3 +and the power law slope is -0.13 (Lo Faro et al. 2017). +We varied the color excess of the stellar continuum from +the young population, E(B-V), from 0.12-1.98. Finally, +we used a reduction factor of 0.44 to the color excess for +the old population compared to the young stars. +CIGALE then provides us with Bayesian estimates for +the stellar mass and SFR for each galaxy in the com- +bined catalog. In order to calculate the virial radius we +used the abundance matching method of Moster et al. +(2013) with the modifications used in Burchett et al. +(2016). We adopt the convention of using Rvir = R200m, +the radius within which the average mass density is 200 +times the mean matter density of the universe, as the +virial radius (Rvir) of a galaxy halo. +2.5. Combining the CGM2 and CASBaH Surveys +In order to combine the surveys, we modified both +catalogs to ensure the same matching criteria between +galaxies and absorbers. In the original CGM2 survey, +we measured the 2σ upper limit on absorption within +δv = ±30 km s−1 of the galaxies redshift using the nor- +malized error of the quasar flux when no absorption sys- +tem was found within our |δv| < 500 km s−1 window. +In order to match the CASBaH survey, we adjusted this +to a 3σ upper limit. This did not change our results in a +meaningful way. The original CASBaH survey used a ve- +locity window of |δv| < 400 km s−1 to match the galaxies +to absorption systems. We adjusted the window for this +work to |δv| < 500 km s−1 to match the CGM2 survey. +As in Paper I, we restrict our H I measurements to those +less than z < 0.481 since at this redshift, the Lyman-α +line redshifts out of the G160 grating band, and thus we +are only sensitive to higher order transitions at higher +redshifts. +Having made these two small changes to each survey, +both could be combined to give us a total survey that in- +cludes 7244 galaxies spanning ∼ 0.01−8 comoving Mpc +in impact parameter around 28 QSO sightlines. +The +distributions of impact parameter, redshift, and stellar +mass are shown in Figure 1. In this paper, we will fo- +cus on galaxies with 8 < log M⋆/M⊙ < 10.5, a stellar +mass range with good coverage in both surveys, which +trims our galaxy sample to 6136 galaxies from CASBaH +and 453 galaxies from CGM2 for a total sample of 6589 +absorber-galaxy pairs. The number of absorber-galaxy +pairs is summarized in Table 1. +3. MODELING ABSORBER-GALAXY +CLUSTERING +We model the CGM using an absorber-galaxy cross- +correlation analysis. This technique is based on model- +ing the covering fraction, fc, as a binomial probability +distribution of detections. To ensure high completeness +in the absorber sample, based on the S/N of the data, +we require a total column density NHI ≥ 1014 cm−2 to +consider the sightline to have a “detection”. Likewise, +a non-detection is the case where we do not detect gas +above this threshold. The models used here are based +on the models employed in Paper I, which was inspired +by the model developed by Hennawi & Prochaska (2007) +and Prochaska et al. (2019). A more detailed explana- +tion can be found in those three papers. +In Paper I, +we found a mass dependence of the extent of the CGM +based on dividing the data into three mass bins. In this +work, we wish to quantify the mass dependence of the +clustering as well as determine the redshift dependence +given our data. +3.1. Single Power-Law Model +The single power-law model consists of two terms: the +base rate of detection due to the random incidence of ab- +sorbers greater than this threshold and an excess above +this base rate due to the clustering of galaxy-absorber +pairs. +Much like Prochaska et al. (2019), we define the 3D +absorber-galaxy cross-correlation function, ξag(r) as +ξag(r) = +� r +r0 +�−γ +. +(1) +To model the galaxy mass dependence of the cluster- +ing, we add a new mass dependence to the clustering +scale, r0, +r0,m(m) = r0 +�M⋆ +M0 +�β +. +(2) +As before, we examine the projected 2-D correlation +function, which is obtained by integrating the 3-D cor- +relation function over the line of sight +χ⊥(r) = +1 +∆r∥ +� +r∥ +ξag( +� +r2 +∥ + r2 +⊥ )dr∥, +(3) + +6 +Table 1. Number of Absorber-Galay Pairs +Survey +107−11.3M∗/M⊙ +108−10.5M∗/M⊙ +108−9M∗/M⊙ +109−10M∗/M⊙ +1010−10.5M∗/M⊙ +(1) +(2) +(3) +(4) +(5) +(6) +CGM2 +543 +453 +103 +271 +79 +CASBaH +6701 +6136 +1265 +3545 +1326 +Total +7244 +6589 +1368 +3816 +1405 +Note—Summary of absorber-galaxy pairs used in this manuscript. (1) The number of absorber-galaxy +pairs in each survey and total of the combined surveys; (2) the number of absorber-galaxy pairs in +the entire mass range; (3) the mass range used to perfom the model fitting; (4, 5, 6) the number of +absorber-galaxy pairs within each mass bin used for model verification. +Figure 2. Corner plots showing the posterior parameter probabilities for the parameters in the single power-law clustering +model. We find a non-zero, positive mass dependence term in the two-halo absorber-galaxy clustering, β2h. + +ro = 3.60+0.26 +-0.27 +-0.05 +B2h += 0.08+0.03 +-0.03 +α = 0.74+0.25 +-0.24 +Co = 30.14+2:22 +-2.14 +Co +2h +ro7 +where r∥ is the line-of-sight distance, r⊥ is the transverse +distance, and ∆r∥ is the size of the redshift window. +For simplicity of notation, r is equivalent to r⊥ in the +following analysis. +In the following definitions, we label the single power +law clustering terms “2-halo,” as the galaxy clustering +method we adopt here describes the clustering of sepa- +rate dark matter halos. This approach distinguishes the +“two-halo” only method from the two-component model +we develop later in this manuscript. +In order to model fc, we assume that the number of +detected absorbers above the column-density threshold +has a Poisson distribution. We consider two cases: (1) +one or more absorbers detected, and (2) the case where +no absorbers are detected. In this framework the prob- +ability of seeing no absorbers is +P miss = λ0 exp(−λ) +0! +(4) +where we denote the rate of incidence (see below) as λ. +The probability of finding one or more absorbers is just +the complement of Equation 4, +fc = 1 − P miss. +(5) +We model the rate of absorber incidence as the pro- +jected correlation function, the 2-halo term, as the ex- +cess over the probability of intersecting an absorber with +NHI > 1014 cm−2 in the redshift window, +λ = (1 + χ2h +⊥ ) ⟨dN/dz⟩δz, +(6) +where ⟨dN/dz⟩ is the base rate of detection due to the +random incidence of absorbers greater than this thresh- +old and deltaz is the line-of-sight redshift window. +In addition to parameterizing the mass dependence +as in Equation (2), we also parameterize the redshift +dependence of ⟨dN/dz⟩ as follows: +dN(NHI ≥ N14 +HI, z) +dz += C0(1 + z)α, +(7) +where N 14 +HI denotes absorbers with column densities of +1014 cm−2, C0 is the random rate of incidence at z = +0, and δz is the redshift window. We adopt a redshift +window to be ±500 km s−1 in velocity units. +Thus, we have a rate of incidence of the form +λ = (1 + [χ2h +⊥ (r, m|r2h +0 , γ2h, β2h)]) ⟨dN(z|C0, α)/dz⟩ δz. +(8) +Finally, we construct the likelihood function, +L = +� +i +P hit(ri, zi, mi|θ) +� +j +P miss(rj, zj, mj|θ), +(9) +where θ = [r2h +0 , γ2h, β2h, C0, α]. +In constructing our Bayesian model, we must choose +priors. For the single power law parameters, we chose +the priors based on the results of cross-correlation anal- +ysis by Tejos et al. (2014) except for our new mass de- +pendent term, β2h, which was motivated by physical +arguments: +• r2h +0 +∼ N(µ = 3.2, σ = 0.3), r2h +0 +> 0 +• γ2h ∼ N(µ = 1.7, σ = 0.1), γ2h > 0 +• β2h > 0, +where N is the normal distribution with mean µ and +variance σ2. +The priors for the redshift dependence were chosen +based on the findings in Kim et al. (2021): +• C0 ∼ Lognormal(µ = 1.25, σ = 0.11) , C0 > 0 +• α ∼ N(µ = 0.97, σ = 0.87) , −3 < α < 3 +We note that we chose to use the more recent results +of Kim et al. (2021) in modeling the redshift evolution +instead of that from Danforth et al. (2016), as were used +in Paper I. +As in Paper I, we apply the Bayesian Markov Chain +Monte Carlo (MCMC) sampler emcee (Foreman-Mackey +et al. 2013) to generate samples from the posterior prob- +ability distribution function to estimate the parameters +of interest and their distributions, using Equation (9) +and the priors described above. +In Figure 2, we show the posterior distributions of +our single power-law model with M0 = 109.5M⊙. These +were fit only to data with 8 < log M⋆/M⊙ < 10.5, as +above this range there is a change in the virial radius +due to the M⋆−Mhalo relation from abundance matching +(Moster et al. 2013). Below this mass range we find a +very flat covering fraction profile, which does not show +a clustering signal. +3.2. Two-component Models +The single power-law model used in galaxy-galaxy +clustering and adapted above to model the galaxy- +absorber clustering makes no assumption of a CGM or +overlapping (in projection) gaseous halos. However, the +existence of the CGM is now well-established (Tumlin- +son et al. 2017). +In particular, the trends of ionized +metal species with impact parameter around L* and +sub-L* galaxies from z = 0 − 3.5 distinctly show that +metal-enriched gaseous atmospheres are a fundamental +component of galaxies (e.g. Werk et al. 2013; Lehner +et al. 2014; Bordoloi et al. 2014; Borthakur et al. 2015; + +8 +Rudie et al. 2019). In the following section, we therefore +assume the existence of the CGM and use a simple Gaus- +sian profile to model the excess clustering signal due to +the presence of the CGM. In addition, we investigated +several other functional forms of the CGM component, +which we describe in §3.2.2. We find that the particular +functional form of this component has little impact on +the results. +3.2.1. The Gaussian CGM Two-Component Model +We now add a third term to the detection rate: a +Gaussian 1-halo component. +The detection rate now +consists of a baseline random incidence rate, an enhance- +ment due to large-scale absorber-galaxy clustering, and +an additional enhancement due to the CGM. We em- +ploy an exclusion model where the contribution from +the 2-halo term terminates at the distance it reaches +the 1-halo component. This scheme, shown in Figure +3, also allows us to determine a natural estimate of the +extent of the CGM: the crossing point of the 1-and 2- +halo components. More explicitly, within some radius, +the galaxy has a CGM that we define as the gas of that +galaxy and any other satellite galaxies within its halo. +Our formalism then defines the Rcross where this CGM +component exceeds the 2-halo. +10 +1 +100 +R [cMpc] +100 +101 +102 +103 +104 +3D Correlation Function +Rcross +G(r)1h +(r)2h +Figure 3. A schematic depiction of our two-component ex- +clusion model and the determination of Rcross. The 2-halo +component cuts off interior to Rcross. +The model is similar to that single power-law we intro- +duced before with a few key differences. We introduce +a Gaussian one-halo term defined as: +G(r)1h = Ae−(r/σ)2. +(10) +Where the two models intersect, Rcross, we can solve +for σ as +σ = +� +1 +2 +R2cross +ln(A) + γ ln(Rcross/r0). +(11) +It should be noted that Rcross here is the 3-D distance +and not the projected distance. In order to characterize +the mass dependence of Rcross we define +Rcross = Rcross,0 +�M⋆ +M0 +�β1h +, +(12) +where Rcross,0 is the 1-halo term extent for a galaxy at +the fixed pivot mass M0. The galaxy mass dependence +of σ includes contributions from the mass dependencies +of Rcross and r0. +This parameterization allows us to compare the mass +dependence of the 1-halo term, β1h with that of the 2- +halo term, β2h. +In order to solve for the projected clustering signal, +ξ, we first make some definitions to ease the notation. +We use s = r∥ in the remainder of the analysis. The +integration is performed over different portions of the +line of sight distance, s, corresponding to the 1 and 2- +halo components. We define the line of sight crossing +point scross as +scross = +� +max(R2cross − r2 +⊥, 0), +(13) +and we can then integrate Equation 10 to seval = +min(scross, smax), where smax is the maximum interval +we wish to integrate over, which in our case is [−500, 500] +km s−1. Thus we have +χ(r⊥) ∝ 2 +� seval +0 +G(r⊥, s)1hds + 2 +� smax +seval +ξ(r⊥, s)2hds +(14) +where the factor of 2 comes from the fact that both +components are symmetric. Here we integrate the one- +halo component over the more nearby regime out to seval +and only integrate the 2-halo term beyond seval out to +the maximum line of sight distance, thus excluding the +regimes in which the models do not apply. For the two- +component model, we choose fairly weak priors on un- +known parameters based on physical arguments while +following the same priors as described above for the pa- +rameters in the single power-law model: +• β1h > −3 +• A > 0 +• Rcross > 0 + +9 +Figure 4. Posterior probabilities for the parameters in the two-component clustering model. We again recover a non-zero, +positive mass dependence term in the two-halo absorber-galaxy clustering, β2h but find an even stronger one-halo CGM clustering +mass dependence β1h ≃ 0.14 ± 0.07. +We can then follow the same MCMC fitting proce- +dure described above to determine the posteriors for +the parameters in this model as well as the crossing ra- +dius, Rcross. These are shown in Figure 4. As before, +we only fit data with 8 < log M⋆/M⊙ < 10.5 and use +M0 = 109.5M⊙. +3.2.2. Other Two-Component Models +While the single power-law clustering model does +an adequate job reproducing the data on large spatial +scales, its contribution is insufficient at R⊥ ≲ 200 kpc +as can be seen in Figure 5 (pink curve). Furthermore, +the primary goal of our study is to find the boundary +between the CGM and IGM, and thus including a CGM +component is essential for this purpose. +We explored +several candidate functional forms for this CGM com- +ponent. +We first investigated a two-component model where +each component is represented by a power law, inspired +by the 1-halo and 2-halo terms that are used to model +the clustering of galaxies. The 3D and projected forms + +0.07+0.03 +22h +-0.03 +2.0 +519.26 +-0.06 +ross +-0.07 +S +公 +3.5 +00 +QQ +Q: +Rcross +B2h +Co +Blh10 +102 +103 +104 +R [ckpc] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Covering Fraction +logM /M +=8 +9 +G+ +Rcross +Observed +102 +103 +104 +R [ckpc] +logM /M +=9 +10 +102 +103 +104 +R [ckpc] +logM /M +=10 +10.5 +Figure 5. Comparison of our two models to the empirical covering fraction as a function of impact parameter in comoving kpc +in mass bins of 108−9M⊙, 109−10M⊙ and 1010−10.5M⊙. The data are shown in black with 1σ error bars. The single power-law +model is shown in pink while the two-component model is shown in purple. The vertical dotted line denotes Rcross in each mass +bin. Both models recreate the covering fraction of the data in all mass bins except for the lowest mass bin where the clustering +signal disappears. The two-component model provides a better match to the data for galaxies of M⋆ > 109M⊙ at the lowest +impact parameters where the single power law model underestimates the covering fraction. +of the two absorber-galaxy correlation functions are +given by Equations 1 and 3, respectively, and the two- +component correlation function is the sum of these parts. +We also considered a model where the two-component +correlation function is, in 3D, the maximum of the two +power laws. This is similar to our chosen model, but +with an inner power law rather than an inner Gaussian +profile. +To rise above the outer power law component at small +radii, the inner power law has to be steeper. In practice, +the two power law indices turned out to be similar, yield- +ing essentially the same result as a single power law fit. +This outcome is not unexpected: the enhancement in +the incidence rate or surface density of gas near galaxies +often does not resemble a steepening power law at small +radii (Zhu et al. 2014; Lan 2020). +In those studies, the enhancement is better described +by a function that declines gradually (compared to a +power law) at small radii and quickly at large radii. The +top-hat function, which has amplitude A inside a bound- +ary and amplitude 0 outside the boundary, is an extreme +example of this class. Our adopted Gaussian profile al- +lows a smoother transition between the CGM-like and +outer components of the model. However, we note that +a fit to the data combining a inner 3D top-hat with an +outer power law yields an Rcross(M∗) that is effectively +indistinguishable from the one that emerges from the +Gaussian component model. +3.3. Model Comparison +In addition to comparing the two models to each other, +Figure 5 compares the models to the empirical covering +fraction as a function of impact parameter and mass. +The data are shown in black with 1σ error bars. The sin- +gle power-law model is shown in pink while the two com- +ponent model is shown in purple. Both models recreate +the covering fractions in all mass bins at all values of R⊥ +except for one data point in the log M∗/M⊙ = 9 − 10 +bin at R⊥ ≈ 200 kpc. Moreover, the two models make +different predictions at low R⊥ except for in the lowest +mass bin (log M∗/M⊙ < 9) where there is no discernible +excess above the clustering signal. This does not pre- +clude the presence of a CGM around these galaxies, but +rather suggests that we require more data at lower R⊥ +for galaxies with log M∗/M⊙ < 9 to be able to constrain +Rcross at these masses. +The two-halo only model under-predicts the observed +signal for galaxies at intermediate masses (log M∗/M⊙ = +9−10). The two component model does better for galax- +ies of M⋆ = 109−10M⊙ at the lowest impact parame- +ters where the single power law model underestimates +the covering fraction, although not significantly so. For +Rcross < 300 kpc, one detects 52 H I systems where +46 systems are predicted. Assuming Poisson statistics, +the two-halo only model is consistent with the data at +1σ level. +Analogous to the one-halo term of galaxy- +galaxy clustering, the data themselves do not require an +enhanced covering fraction of H I absorption that we +identify as the CGM. +We find the 1-halo component has a stronger clus- +tering mass dependence, β1h ≃ 0.14 ± 0.07, than the +two-halo term, β2h ≃ 0.08 ± 0.03. +We also find the +2-halo clustering terms in each model to be internally +consistent with each other as seen in Figure 6. +4. RESULTS + +11 +Figure 6. Comparison of the two-halo 3D cross correlation posteriors between the two-component model (r0 = 3.99+0.28 +−0.24 cMpc, +γ = 1.62 ± 0.07) and the single power-law model (r0 = 3.58+0.28 +−0.24 cMpc, γ = 1.55 ± 0.05). The two models are consistent with +each other within the 1σ limits and have a power-law slope consistent with the absorber-galaxy 3D cross correlation found in +the literature (e.g. Tejos et al. 2014) of γ = 1.7 ± 0.1. +4.1. Clustering Mass Dependence +As seen in Figure 2, we find the clustering parameters +to be r0 = 3.6 ± 0.3 cMpc, γ = 1.6 ± 0.5. r0 and γ are +consistent with those found in Tejos et al. (2014) who +find r0 = 3.7±0.1 cMpc and γ = 1.7±0.3. We also find +a mass dependence of the absorber-galaxy clustering of +β2h = 0.07+0.3 +−0.2. +We find the the two component model better fits the +data as can be seen in Figure 5. Specifically, the two +component model better matches the covering fraction +for galaxies of M⋆ > 109−10M⊙ at the lower impact +parameters where the single power law model underes- +timates the covering fraction. In addition, we find the +two-component model reproduces the mass dependence +of the 2-halo clustering term, β2h ≃ 0.07 while also pro- +ducing a stronger mass dependence of the 1-halo clus- +tering term, β1h ≃ 0.14. +4.2. Physically-Motivated Extent of the CGM +As mentioned above, using the two-component model +produces an estimate of Rcross, a natural metric for the +extent of the CGM. This 3-D distance demarcates where +the contribution to the clustering begins to be domi- +nated by the CGM above the expected two-halo clus- +tering due to isolated galaxy halos traced by H I. Rcross + +0.18 +0.12 +0.00 +-0.06 +ro +Y +B2h12 +Figure 7. +A comparison of Rcross with the virial radius +(Rvir, grey filled region) as well as the splashback radius +(Rsplash, pink shaded region) of the galaxy sample. +The +filled regions in Rvir and Rsplash denote the redshift range +for the galaxies in our sample (0.1 ≲ z ≲ 0.48). The filled +blue region represents the 1σ limits of the distribution in +Rcross while the blue line denotes the median of this distri- +bution. The black crosses correspond to the values published +in Paper I. The vertical dotted lines denote the mass range +of 8 < log(M⋆/M⊙) < 10.5 to which we limited the fitting +in our MCMC analysis in Figure 5. +can be viewed as the maximum radius to which an en- +hancement from the CGM could extend without over- +predicting the data at large radii. +In Figure 7, we see Rcross (blue) compared with +the spread in virial radii of the galaxy sample (grey +filled region). +The filled blue region represents the +1σ limits of the distribution in Rcross while the blue +line denotes the median of this distribution. +We find +Rcross is ∼ 2 ± 0.6Rvir for galaxies in the range 8 < +log(M⋆/M⊙) < 10.5. The black crosses correspond to +the values published in Paper I defined as the extent +where there is 50%chance to see H I absorption above +1014 cm−2. The vertical dotted lines denote the mass +range of 8 < log(M⋆/M⊙) < 10.5 that was used in our +MCMC analysis. Above this range, we see a change in +the relation of the virial radius with stellar mass, and +below this mass range, we find little to no correlation +between absorbers and galaxies. 5). +We also calculated the splashback radius, Rsp, us- +ing the method from Diemer (2018) and encoded in the +COLOSSUS3 package. This radius denotes the location +at which particles reach the apocenter of their first orbit. +We find excellent agreement of Rcross with the results in +Paper I and Rcross neatly matches the splashback radius +3 https://bdiemer.bitbucket.io/colossus/ +for galaxies in this mass range. We discuss these results +in more detail below. +5. DISCUSSION +Both of the models we investigate do an adequate job +of recreating the cross correlation signal at all impact +parameters and masses 108 < M⋆ < 1010.5M⊙ as seen +in Figure 5. It is not entirely clear that the single power +law model has any physically-consistent meaning, how- +ever. +Effectively, it would seem to signify that every +time one measures H I absorption at the same redshift +as a particular galaxy (|∆v| < 500 km s−1), the absorp- +tion is always due to another galaxy’s CGM . Note, we +would conclude this for all galaxies, i.e. +each has no +CGM and only neighbors with a CGM. This is clearly +impossible. The two-halo-only model for the CGM effec- +tively breaks down when the galaxies lie within the halo +under consideration, i.e. when they “mix.” We cannot +and do not try to distinguish between the two. How- +ever, our formalism does allow one to identify the outer +extent of this “mixing.” +The two-component model asserts that galaxies with +M⋆ > 108M⊙ have a CGM, an assumption that is moti- +vated by previous survey results (e.g. Werk et al. 2013). +Additionally, this model is able to better recreate the +data – from the combined datasets of CGM2 + CAS- +BaH, which together represent the largest sample of +galaxies with confirmed spectroscopic redshifts in the +foregrounds of UV-bright QSOs with high-resolution ab- +sorption spectroscopy – both at smaller impact param- +eters and at M⋆ > 109 M⊙. +The much larger number of galaxies at larger impact +parameters drives the fit of the models to the data. +There is, however, a > 1σ inconsistency between the +two-halo only model and the data at R⊥ ∼ 200 and for +both models at R⊥ ∼ 600 in the logM⋆ = 9 − 10M⊙ +mass range. +The latter inconsistency may be due to +cosmic variance or the assumption that the absorber- +galaxy measurements are independent and are not cor- +related, which would increase the scale of the error bars +at R⊥ ∼ 600. +5.1. Comparing the mass dependence of the single and +two-component models +Our galaxy sample includes a large number of galaxies +at low (< 500 kpc) impact parameters which allows us +to better model the regime in which the two-halo galaxy +clustering becomes dominated by the signal of galaxies +that inhabit the same dark matter halo, the one-halo +term. By separating these two terms in the manner pre- +sented here, we can disentangle the large scale clustering +as well as the contribution of the CGM to the 3D corre- +lation of absorbers and galaxies. + +Rvir +103. +Rlm Wilde+21 +[ckpc] +R +102. +7 +8 +6 +10 +11 +log10(M*)13 +Our analysis finds nearly identical terms for the mass +dependence of the clustering at large scales, β2h as well +as the contribution of absorbers at random, C0 and α. +We do find a stronger mass dependence in the one-halo +term, β1h than at larger scales. +This can be seen in +Figure 5 where the correlation steepens in higher mass +bins. +5.2. Absorber-Galaxy Bias +Our covering fraction analyses provide an estimate of +the galaxy-absorber correlation function, ξag (eq. +1). +Here, we test if the mass dependence of ξag outside the +CGM is consistent with absorption systems and galax- +ies simply being two independent tracers of the same +underlying dark matter distribution. +Assuming both +tracers have linear bias, ξag should be equal to babgξDM, +where ba and bg are the absorber and galaxy bias, respec- +tively, and ξDM is the dark matter 3D correlation func- +tion. +Following Tinker et al. (2010) (hereafter, T10), +we assume the dark matter correlation function can be +described by a power-law function of radius with index +γ = 1.62. We fix the power-law index in the ξag deter- +mined by fitting a single power-law to the data to this +same value, with which it is consistent. With the above +assumptions, ξag = (r/r0(M))−γ = babgξDM(r). +The +radial dependence cancels, leaving the proportionality +r0(M)γ ∝ babg. +We show a scaled r0(M)γ in Figure 8 along with the +galaxy bias as a function of stellar mass from T10 and +implemented in the COLOSSUS package (Diemer 2018). +If ba is constant and the assumptions stated above +hold, r0(M)γ should have the same mass dependence +as galaxy bias. While there is a visually apparent differ- +ence between the galaxy bias and the best-fit r0(M)γ, +this difference is not significant at a 2σ level and so is +merely suggestive. If the difference is real, it could be a +consequence of the H I mass per dark matter mass be- +ing a function of overdensity. Up to the overdensities at +which Mstar = 1010.5 M⊙ galaxies tend to be found, this +function would be increasing: H I would be less common +in low density regions than in higher density filaments. +This behavior would be consistent with theoretical ex- +pectations (e.g., Hui & Gnedin 1997; Schaye 2001; Dav´e +et al. 2010) and observations (e.g., Rudie et al. 2012; +Burchett et al. 2020). +5.3. Comparison to Previous Work +One of the key aspects of this analysis is determining +the mass dependence of the extent of the NHI > 1014 +cm−2 for which our model provides a direct metric, +Rcross(M⋆). +We compare our resulting Rcross(M⋆) to +the method and results from Paper I in Figure 7. The +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +Log10M * /M +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +Scaled bias += 0.07 + / +0.03, = 1.62 +Tinker2010 galaxy bias +Figure 8. A comparison of the slopes of the relative bias as +a function of mass derived from our analysis (orange) com- +pared to Tinker et al. (2010) (T10, black). The dashed lines +correspond to the ranges spanned by the 1σ limits in in β2h. +The relative bias, r0(M) ∝ (M⋆/M0)γβ, are normalized to +the value of T10 at log M⋆/M⊙ = 9.5. We find a steeper +mass dependence than T10, but the significance of the dif- +ference is less than 2σ. +result of Paper I, R14 +CGM, which are based only on the +CGM2 survey are shown as black crosses in the mass bins +they span in that paper. We also compare the method +used in that paper to determine R14 +CGM, the radius at +which the probability of detecting NHI > 1014 cm−2 is +> 50%, calculated with the two-component model us- +ing the combined CGM2 + CASBaH surveys and find +it to be consistent within 1σ with our newer model for +Rcross(M⋆). We find that our mass dependent estimate +of the extent of the CGM, Rcross(M⋆) corroborates the +findings of Paper I that the NHI > 1014 cm−2 extends +to approximately twice the virial radius (∼ 2 ± 0.6Rvir). +One of the main strengths of the CGM2+ CASBaH +sample is the large number of galaxies at small projected +separations (<1 Mpc). This allows us to investigate the +smaller scale regime in more detail within the context +of similar studies such as Tejos et al. (2014) (hereafter, +T14) who uses a single power law model to measure the +two-point correlation between H I and galaxies above +NHI > 1014 cm−2. +In this work they break up their +measurements into SF vs non-SF samples while we do +not. +Our sample however is dominated by the more +common SF galaxies and we will compare our results +to their SF sample. +Comparing our cross-correlation +results with T14, we find good agreement between the +results in T14, rT14 +0 += 3.8 ± 0.2 Mpc, γ = 1.7 ± 0.1 +and the results from both models presented here, r0 = +3.99+0.28 +−0.24 Mpc, γ = 1.62±0.07) and the single power-law +model (r0 = 3.58+0.28 +−0.24 Mpc, γ = 1.55 ± 0.05. We find + +14 +a mass dependence of this cross-correlation, however as +parameterized by β2h. +Our results are slightly in tension with Momose et al. +(2021) who find galaxies in the 109−10M⊙ range domi- +nate their H I-galaxy cross correlation signal. We find +the largest mass bin sample to have the most elevated +covering fractions at low impact parameter. +5.4. Physical Extent of Galaxy Halos +Astronomers often use the viral radius as a means to +describe the characteristic size of galaxy halos and it is +convenient to compare this to the extent of the gaseous +galactic atmosphere as we have done here and in Paper +I. The virial radius is typically defined in terms of the +spherical overdensity mass definition which is based on +the radius which encloses an overdensity of 200 times the +critical or mean density, i.e., R200c and R200m. Because +the mean and critical densities are decreasing over cos- +mic time, this can lead to a pseudo-evolution as pointed +out in Diemer et al. (2013). In addition, subhalos show +evidence of being stripped outside the virial radius of +clusters (Behroozi et al. 2014). +An alternative physically motivated halo scale is the +splashback radius, Rsp (Diemer & Kravtsov 2014; Ad- +hikari et al. 2014; More et al. 2015). This radius effec- +tively distinguishes infalling material from matter orbit- +ing in the halo. We compare our results to the splash- +back radius in Figure 7 and find that our estimate of +the extent of the H I CGM, Rcross, neatly aligns with +Rsp over the mass range 108 < M⋆/M⊙ < 1010.5. This +result implies that Rsp is a better approximation of the +CGM extent than the more commonly used viral radius. +O’Neil et al. (2021) compared Rsp as estimated from +dark matter and gas profiles in the IllustrisTNG simula- +tions and found that the gas Rsp is consistently smaller +than the dark matter Rsp. +However, they were look- +ing at much more massive halos Mhalo > 1013 in which +shocks dominate the gas distribution. Nonetheless, the +fact that Rcross ≈ Rsp at the mass ranges considered +here (Mhalo 1010−12M⊙) is intriguing. The halo mass +accretion rate generally sets whether Rsp exceeds Rvir; a +rapid accretion rate will impact the growth of the grav- +itational potential well, leading to Rsp < Rvir. If the +location of Rcross reflects the extent of orbiting gas in a +halo, then our observational results imply a halo mass +accretion rate that is slow enough to keep the apocenters +of orbiting structures at large radii. +Another way of defining the extent of the CGM is to +use the boundary of the pressure-supported CGM. For +galaxies with halo masses ≳ 1011.5M⊙ (M⋆ ≈ 109.8M⊙), +this pressure support comes from fact that the gas that +has fallen into the gravitational potential well is virially +shocked and cannot cool within a Hubble time (Binney +1977; Rees & Ostriker 1977; Silk 1977). For the galax- +ies in our survey, which are predominately below this +halo mass, however, the gas would rapidly cool and thus +this pressure support might come from galactic winds. +Fielding et al. (2017) and Lochhaas et al. (2018) show +that supernovae winds with reasonable mass loading ef- +ficiencies could shock the gas to distances past the virial +radius and account for the survival of cool gas at these +large radii. Using a more comprehensive model of the +multiphase CGM, Fielding & Bryan (2022) show that +SF in the galactic disk can slow cooling and accretion as +part of a global preventive self-regulation mechanism. In +addition, the winds can transport cold clouds to large +radii, consistent with these constraints from our com- +bined survey data. +6. SUMMARY +Herein, we have examined the associations of galax- +ies with Lyα absorption z < 0.48 to explore the spa- +tial profile of this gas and the mass dependence of the +profile. Specifically, we have combined the CGM2 and +CASBaH H I measurement and constructed a catalog +of 7244 absorber-galaxy pairs around 28 QSO sightlines +(6589 absorber-galaxy pairs when we restrict our galaxy +sample to galaxies with 8 < log M⋆/M⊙ < 10.5). The +CGM2 survey has better sampling of galaxies at low im- +pact parameter while CASBaH samples galaxies out to +20 cMpc. This allows us to characterize the H I profile +via the covering fraction as a tracer of the gas. +1. By modeling the covering fraction as a power-law +with a mass dependent length scale, we find good +agreement with previous studies, such as T14, of +our clustering amplitude and power law slope pa- +rameters. +2. In Section 3.1, we find the clustering scale has a +mass dependence with a power-law slope of β2h = +0.08 ± 0.03. +3. We compare the slope of our absorber-galaxy bias +to the galaxy-dark matter bias of Tinker et al. +(2010). The absorber-galaxy bias is a steeper func- +tion of galaxy mass than the galaxy-dark matter +bias. However, this difference is only significant at +a sub-2σ level. +4. We model the data with an exclusionary two- +component model where we adopt an inner-CGM +Gaussian profile to describe the data at smaller +impact parameters and the customary two-halo +single power-law model at larger impact param- +eters. This model faithfully reproduces the data +for galaxies M⋆ > 108M⊙. + +15 +5. The two component model allows us to calculate +the crossover radius, Rcross(M⋆), where the mod- +els are equal. Rcross(M⋆) represents a soft upper +estimate of the furthest impact parameter needed +to optimally fit the inner CGM component. We +then use Rcross as an estimate of the extent of the +CGM and find Rcross(M⋆) ≈ 2 ± 0.6Rvir for galax- +ies 108 ≤ M⋆/M⊙ ≤ 1010.5. Additionally, we find +excellent agreement between Rcross(M⋆) and the +splashback radius, Rsp for galaxies in this mass +range. +7. ACKNOWLEDGMENTS +MCW, KT, and JKW acknowledge support for this +work from NSF-AST 1812521, NSF-CAREER 2044303, +the Research Corporation for Science Advancement, +grant ID number 26842. Support for the CASBaH HST +programs HST-GO-11741 and HST-GO-13846 was pro- +vided through grants from the Space Telescope Science +Institute under NASA contract NAS5-26555. +Support for the CASBaH HST programs HST-GO- +11741 and HST-GO-13846 was provided through grants +from the Space Telescope Science Institute under NASA +contract NAS5-26555. +The CGM2 Survey would not have been possible with- +out the substantial contributions from a dedicated group +of UW undergraduate Student Quasar Absorption Di- +agnosticians, the Werk SQuAD, with over 50 individ- +ual undergraduate research assistants since 2016. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Werk,1 Todd M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Tripp,2 Joseph N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Burchett,3, 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Xavier Prochaska,3, 5, 6 Nicolas Tejos,7 Nicolas Lehner,8 Rongmon Bordoloi,9 John M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' O’Meara,10 Jason Tumlinson,11 and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Christopher Howk12 1University of Washington, Department of Astronomy, Seattle, WA 98195, USA 2Department of Astronomy, University of Massachusetts, 710 North Pleasant Street, Amherst, MA 01003-9305, USA 3University of California, Santa Cruz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 1156 High St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=', Santa Cruz, CA 95064, USA 4Department of Astronomy , New Mexico State University, PO Box 30001, MSC 4500, Las Cruces, NM 88001 5Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU) The University of Tokyo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan 6Division of Science, National Astronomical Observatory of Japan,2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 7Instituto de F´ısica, Pontificia Universidad Cat´olica de Valpara´ıso, Casilla 4059, Valpara´ıso, Chile 8Department of Physics and Astronomy, University of Notre Dame, Notre Dame, IN 46556 9North Carolina State University, Department of Physics, Raleigh, NC 27695-8202 10W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Keck Observatory, 65-1120 Mamalahoa Hwy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=', Kamuela, HI 96743, USA 11Space Telescope Science Institute, Baltimore, MD, USA 12Department of Physics and Astronomy, University of Notre Dame, Notre Dame, IN 46556, USA ABSTRACT We combine datasets from the CGM2 and CASBaH surveys to model a transition point, Rcross, between circumgalactic and intergalactic media (CGM and IGM, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In total, our data consist of 7244 galaxies at z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 with precisely measured spectroscopic redshifts, all having impact parameters of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='01 − 20 comoving Mpc from 28 QSO sightlines with high-resolution UV spectra that cover H I Lyα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Our best-fitting model is an exclusionary two-component model that combines a 3D absorber-galaxy cross correlation function with a simple Gaussian profile at inner radii to represent the CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' By design, this model gives rise to a determination of Rcross as a function of galaxy stellar mass, which can be interpreted as the boundary between the CGM and IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' For galaxies with 108 ≤ M⋆/M⊙ ≤ 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5, we find that Rcross(M⋆) ≈ 2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='6Rvir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Additionally, we find excellent agreement between Rcross(M⋆) and the theoretically-determined splashback radius for galaxies in this mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Overall, our results favor models of galaxy evolution at z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 that distribute T ≈ 104K gas to distances beyond the virial radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' INTRODUCTION The formation and evolution of galaxies involves a complex interplay between gravitational collapse of gas from the intergalactic medium (IGM), galaxy mergers, and feedback due to stellar evolution and active galactic nuclei (AGN) that drive gaseous outflows and change the ionization state of the galaxies’ gaseous halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Together, these processes drive the “cosmic baryon cycle” which takes place largely in the region of a galaxy referred to as the circumgalactic medium (CGM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Indeed, under- standing the CGM is critical for developing a complete theory of galaxy evolution, as highlighted by the recent decadal survey (National Acadamy of Sciences 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Corresponding author: Matthew C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Wilde mwilde@uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='edu In particular, the extent of the gaseous CGM relative to the extent of the dark matter halo is a subject of great interest for models that aim to reproduce the properties of gaseous halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The existence of the CGM, first predicted by Bahcall & Spitzer (1969), was initially revealed by detection of Mg II and H I absorption at large projected distances (R⊥ > 20 kpc) from L∗ galaxies (Bergeron 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Morris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Bergeron & Boiss´e 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Lanzetta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2005), and subsequently traced via higher- energy metal-line transitions such as Si III, C IV and O VI that are observed to correlate with galaxies and their global properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Tripp & Savage 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Tripp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Tumlinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Werk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 Rvir of L ∼ L* galaxies, the metal line incidence is found to be 60 − 90 % for a range of ionized metal species (Werk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Con- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='02718v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='GA] 6 Jan 2023 2 versely, Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2022) find an 80% chance of find- ing a massive galaxy nearby to any high-metallicity ab- sorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The CGM of M⋆ > 108 M⊙ galaxies is now well- established to be metal-enriched (Liang & Chen 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Bordoloi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2022), and to extend to at least 1 Rvir, and very likely beyond it (Wakker & Savage 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Burchett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Finn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Wilde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Borthakur 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Generally, hydrodynamical simulations of galaxy evo- lution, which exhibit complex interactions between gravitational collapse from the cosmological large scale structure and subsequent feedback from supernovae and AGN-driven winds that heat and enrich the CGM and IGM (EAGLE, Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' IllustrisTNG, Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' SIMBA, Dav´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' and CAMELS, Villaescusa-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2022), are con- sistent with the range of observations of the CGM in absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Yet these models still rely on simplistic implementations of the “sub-grid” physics in order to model entire galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Ford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Hummels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2013), and physical properties of the CGM are dependent on the simulation resolution (Hummels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Peeples et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' More sensitive observations of the CGM, including the ability to detect the diffuse gas in emission, are needed both to break degenera- cies in these models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=', between heating and cooling mechanisms, and to develop a flexible parametric model of the CGM (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The two-point correlation function between H I ab- sorption along QSO sightlines and galaxies has proven to be an essential tool to understand the connection of galaxies to the IGM (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Morris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Ryan-Weber 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Tejos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The primary advan- tages of leveraging the clustering of these two entities over one-to-one association analyses is that it provides results for large scales (1-10 Mpc) as well as the rel- atively smaller scales where the baryonic processes as- sociated with the CGM play out, and the correlation function statistically characterizes absorber-galaxy rela- tionships when multiple galaxies are close to the sight- line and a one-to-one assignment is ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Since H I traces both enriched material from galaxies as well as primordial accretion from the IGM, observations of the CGM, IGM, and galaxies in the same volume are fundamental to both testing the predictions of galaxy evolution models and providing a means to differentiate between them (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Fumagalli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Oppenheimer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Stinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Ford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Hum- mels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Butsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Understanding the physical profile and size of the CGM sheds light on the non-linear processes of galaxy formation: on what spatial scale(s) do virialization, ac- cretion, and feedback transform these galactic atmo- spheres?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Astronomers have long used some version of the virial radius as an estimator for the size of galaxy halos, but this estimate is somewhat arbitrary and is based on the distibution of unobservable dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' By observing the radial gas profile around galaxies out to large scales, we can effectively map the gaseous halo, which in turn constrains the physics of galaxy-scale feed- back processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Observationally determining the galac- tic atmosphere’s extent has additional implications for constraining galaxy evolution and assembly models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' For example, the galaxy baryon and metal budgets require a scale to integrate the total mass (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Peeples et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Werk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Furthermore, the gaseous halo likely plays an important role in the quenching of dwarf satel- lite galaxies as they become stripped by ram-pressure in a low-density CGM (Putman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2021), and it is useful to constrain where this occurs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=', the extent of the CGM, and how this depends on central galaxy mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The presence of H I absorption beyond the virial ra- dius is now widely accepted for a range of galaxy stellar masses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Tejos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2012, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Wilde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Bouma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Borthakur 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In Wilde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2021) (Paper I) we found an em- pirical relation between galaxy stellar mass and the ex- tent of the CGM as indicated by H I covering fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' For galaxies with stellar masses 108 < M⋆/M⊙ < 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5, we found that the CGM extends to two times the virial radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In this paper, we focus on the functional forms of the mass dependence of the H I-traced CGM using a power-law model similar to the 2-halo correlation func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We also investigate other two-component models that differentiate the CGM from the IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We combine the CGM2 Survey, which focuses on sightlines at low galaxy impact parameters (< 1 Mpc), with the COS Absorption Survey of Baryon Harbors (CASBaH) that probes larger spatial scales (< 20 Mpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In doing so, we greatly increase the absorber-galaxy sample from 543 spectroscopically-confirmed absorber-galaxy pairs to 7244 pairs spanning 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='003 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Our goal is to provide the most reliable constraints to date on the spatial extent of the CGM as traced by H I absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The paper is structured as follows: In Section 2, we briefly review each of the galaxy-absorber surveys and discuss their combined properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In Section 3, we in- troduce two models of the H I-galaxy correlation func- tions and cover our main results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We com- pare our results with simulations and previous results and discuss their implications for galaxy evolution mod- els in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Finally, we summarize our results in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' DATA - COMBINING CGM2 AND CASBAH Both surveys feature far-ultraviolet spectroscopy of QSOs with HST, using both the Cosmic Origins Spectro- graph (COS, Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2012) and the Space Telescope Imaging Spectrograph (STIS, Woodgate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 1998), and deep, ground-based optical spectroscopy of foreground galaxies in the QSO fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' CASBaH is well suited to the study of the interface between the CGM and the IGM, at scales ≳ 1 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' CGM2 provides a relatively more complete mapping of the inner CGM at scales ≲ 1 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' By combining CGM2 and CASBaH data, we leverage the strengths of each survey, as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Figure (1) shows the distributions of galaxy stellar masses and impact parameters versus redshift from both surveys out to z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Together, the surveys allow us to probe the CGM as it transitions into the IGM for a large sample of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' CGM2 The CGM2 survey, first presented in Wilde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2021), includes precise spectroscopic redshifts and bulk galaxy properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' stellar masses, M∗, and star formation rates, SFR) from a combination of Gemini GMOS spectra and deep, broadband photometry for ∼1000 galaxies in the foreground of 22 QSOs, each with S/N ≈10 HST/COS G130M+G160M spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' By matching galaxy and absorber redshifts in ±500 km s−1 windows, the CGM2 survey is ultimately a large col- lection of measurements pertaining to the CGM of z < 1 galaxies over a wide range of stellar masses, 108 ≲ M⋆/M⊙ ≲ 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The data acquisition and analysis are explained in detail in Wilde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Here we present a brief overview of the survey data relevant to the present analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The CGM2 galaxy spectra were obtained using Gemini-GMOS spectrographs on the twin Gemini North and South telescopes (Hook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Gimeno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Galaxy redshifts were inferred from the template fitting code, Redrock1 (v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='14) and manually inspected with VETRR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The typical statistical uncertainly of our redshifts is σz ∼ 50-100 km s−1 (z ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='00016-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='00030).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Photometry of the CGM2 galaxy catalog was obtained from the Gemini-GMOS pre-imaging in g and i bands as well as all available bands from DESI Legacy Imag- ing Surveys Data Release 8 (DR8) (Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2019), WISE (Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2013), Pan-STARRS Data Release 2 (Chambers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2016), and SDSS DR14 (Abolfathi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='com/desihub/redrock 2 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='com/mattcwilde/vetrr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='6 z 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 log R [cMpc] CASBaH CGM2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='6 z 7 8 9 10 11 12 logM [M ] CASBaH CGM2 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Top: Distribution of the combined CGM2 (blue dots) and CASBaH (purple dots) data sets in both logarith- mic impact parameter, and redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The data are roughly uniform in redshift space but we can see the relative contri- butions of the data sets in impact parameter space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' CGM2 is highly concentrated at lower impact parameters while CAS- BaH explores much greater impact parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Bottom: Galaxy stellar mass distribution as a function of redshift for the two data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 4 The 22 QSOs included in the CGM2 survey have HST/COS spectra selected from the COS-Halos (GO11598, GO13033;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Tumlinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2013) and COS-Dwarfs (GO12248;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Bordoloi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2014) surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In general, the CGM2 QSO targets have zQSO > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='6 and avail- able HST imaging, which permits detailed analysis of absorption-hosting galaxies with z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' All COS spec- tra include both the G130M and G160M gratings, and have a S/N ≃ 8 − 12 per resolution element (FWHM ≃ 16-18 km s−1) or better over 1150-1800 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The COS data and their reduction are presented in detail in Tum- linson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2013) and Bordoloi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2014) and follows the same method used by Tripp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2011), Meiring et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2011), Tumlinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2011) and Thom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' CASBaH The CASBaH program was designed to take advan- tage of the multitude of resonance transitions at rest- frame wavelengths < 912 ˚A to probe the physical condi- tions, metallicity, and physics of the multiphase CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' A wide variety of elements and ionization stages have res- onance lines only at λ < 912 ˚A (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=', Verner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 1994), so observations of this wavelength range provide new diagnostics and precise constraints using banks of adjacent ions such as N i through N v, O i through O vi, and Ne ii through Ne viii (see Tripp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2011, for examples of lines detected by CASBaH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The Ne viii 770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='4, 780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='3 ˚A doublet has received particular attention as a probe of warm-hot gas at ≈ 105 − 106 K (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=', Sav- age et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Burchett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Wijers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In many contexts such as the Milky Way interstellar medium, these lines are inaccessible because they are blocked by the H i Lyman limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' CASBaH overcomes this limitation by observing QSO absorbers with suffi- cient redshift to bring the lines into the observable band of HST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The motivation and design of the CASBaH program is summarized in section 1 of Haislmaier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2021), and the CASBaH galaxy redshift survey is presented in Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Briefly, CASBaH obtained both HST/COS and HST/STIS spectra of nine QSOs at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='92 < zQSO < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='48, with two primary selection crite- ria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' First, since some of the most important target lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=', Ne viii) are weak, the QSOs were required to be UV-bright so that good signal-to-noise and sensitivity to weak lines would be attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Second, the targets were required to have zQSO > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='9 to provide a total redshift path that is sufficient to accumulate a statistically use- ful sample of absorbers of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' No considerations were given to known foreground galaxies or absorbers, so the targets were not selected in a way that would fa- vor particular types of foreground absorbers or galaxies, except that sightlines with known black Lyman limits at λob > 1150 ˚A were excluded to avoid using HST time on sightlines that would not contribute useful pathlengths to the samples (see Burchett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The CASBaH UV spectra were reduced in the same way as the CGM2 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The CASBaH galaxy-redshift survey (Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2019) measured thousands of redshifts in the fields of seven of the CASBaH QSOs using the Keck DEIMOS and MMT Hectospec spectrographs, with typical red- shift uncertainties of ≈ 30 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The survey used a wedding-cake strategy with the Hectospec covering galaxies in the ≈ 1◦ fields centered on the QSOs and the DEIMOS survey providing a deeper survey with a smaller field of view (81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 arcmin2) (see Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Using the CASBaH galaxy database, supple- mented with data from public surveys such SDSS, we selected a sample of 6701 galaxies with spectroscopic redshifts z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='481 and comoving impact parameters less than 13 cMpc, appropriate for the H I analysis pre- sented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Synergy of CGM2 + CASBaH The CASBaH and CGM2 surveys have complemen- tary designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' On the one hand, CGM2 is built on COS-Halos and thus favors at least one L∗ galaxy close to the sightline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' CGM2 also covers a smaller FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' On the other hand, CASBaH is a blind survey that covers a larger FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Consequently, CASBaH provides more information about galaxies and large-scale struc- tures at larger impact parameters, but as a blind sur- vey, it is cross-section weighted in favor of galaxies at larger impact parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Also, since CASBaH avoided sightlines with black Lyman limits in the HST band (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=', at λob ≥ 1150 ˚A), it will not include galaxies at zgal > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='26 that harbor absorbers with N(H i) ≳ 1017 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Thus, CGM2 probes the inner CGM including higher N(H i) absorbers, while CASBaH complements CGM2 by adding very large samples of galaxies and structures at larger distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Galaxy Properties To estimate the galaxy properties for both surveys, we used CIGALE (Noll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Boquien et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2019) to fit the spectral energy distribution (SED) and retrieve stellar mass and star formation rates (SFR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We used the Bruzual & Charlot (2003) stellar population mod- els, assuming a Chabrier (2003) initial mass function (IMF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We chose a grid of metallicities ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='001-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5Z⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' A delayed star formation history (SFH) model was employed with an exponential burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The 5 e-folding time of the main stellar population models ranged from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='1-8 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We varied the age of the oldest stars in the galaxy from 2-12 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We included an op- tional late burst with an e-folding time of 50 Myr and an age of 20 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The burst mass fraction varied from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='0 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='1 to turn this feature on or off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Nebular emis- sion and reprocessed dust models (Dale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2014) were also included with the default values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The dust models have slopes ranging from 1−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 and the nebular models include no active galactic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We employed the Calzetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (1994) dust attenua- tion law, but we also included a “bump” in the UV (see discussion in Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2019) at 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 nm with a FWHM of 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='6 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The bump amplitude is set at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='3 and the power law slope is -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='13 (Lo Faro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We varied the color excess of the stellar continuum from the young population, E(B-V), from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='12-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Finally, we used a reduction factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='44 to the color excess for the old population compared to the young stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' CIGALE then provides us with Bayesian estimates for the stellar mass and SFR for each galaxy in the com- bined catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In order to calculate the virial radius we used the abundance matching method of Moster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2013) with the modifications used in Burchett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We adopt the convention of using Rvir = R200m, the radius within which the average mass density is 200 times the mean matter density of the universe, as the virial radius (Rvir) of a galaxy halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Combining the CGM2 and CASBaH Surveys In order to combine the surveys, we modified both catalogs to ensure the same matching criteria between galaxies and absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In the original CGM2 survey, we measured the 2σ upper limit on absorption within δv = ±30 km s−1 of the galaxies redshift using the nor- malized error of the quasar flux when no absorption sys- tem was found within our |δv| < 500 km s−1 window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In order to match the CASBaH survey, we adjusted this to a 3σ upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' This did not change our results in a meaningful way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The original CASBaH survey used a ve- locity window of |δv| < 400 km s−1 to match the galaxies to absorption systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We adjusted the window for this work to |δv| < 500 km s−1 to match the CGM2 survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' As in Paper I, we restrict our H I measurements to those less than z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='481 since at this redshift, the Lyman-α line redshifts out of the G160 grating band, and thus we are only sensitive to higher order transitions at higher redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Having made these two small changes to each survey, both could be combined to give us a total survey that in- cludes 7244 galaxies spanning ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='01−8 comoving Mpc in impact parameter around 28 QSO sightlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The distributions of impact parameter, redshift, and stellar mass are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In this paper, we will fo- cus on galaxies with 8 < log M⋆/M⊙ < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5, a stellar mass range with good coverage in both surveys, which trims our galaxy sample to 6136 galaxies from CASBaH and 453 galaxies from CGM2 for a total sample of 6589 absorber-galaxy pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The number of absorber-galaxy pairs is summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' MODELING ABSORBER-GALAXY CLUSTERING We model the CGM using an absorber-galaxy cross- correlation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' This technique is based on model- ing the covering fraction, fc, as a binomial probability distribution of detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' To ensure high completeness in the absorber sample, based on the S/N of the data, we require a total column density NHI ≥ 1014 cm−2 to consider the sightline to have a “detection”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Likewise, a non-detection is the case where we do not detect gas above this threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The models used here are based on the models employed in Paper I, which was inspired by the model developed by Hennawi & Prochaska (2007) and Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' A more detailed explana- tion can be found in those three papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In Paper I, we found a mass dependence of the extent of the CGM based on dividing the data into three mass bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In this work, we wish to quantify the mass dependence of the clustering as well as determine the redshift dependence given our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Single Power-Law Model The single power-law model consists of two terms: the base rate of detection due to the random incidence of ab- sorbers greater than this threshold and an excess above this base rate due to the clustering of galaxy-absorber pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Much like Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2019), we define the 3D absorber-galaxy cross-correlation function, ξag(r) as ξag(r) = � r r0 �−γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (1) To model the galaxy mass dependence of the cluster- ing, we add a new mass dependence to the clustering scale, r0, r0,m(m) = r0 �M⋆ M0 �β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2) As before, we examine the projected 2-D correlation function, which is obtained by integrating the 3-D cor- relation function over the line of sight χ⊥(r) = 1 ∆r∥ � r∥ ξag( � r2 ∥ + r2 ⊥ )dr∥, (3) 6 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Number of Absorber-Galay Pairs Survey 107−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='3M∗/M⊙ 108−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5M∗/M⊙ 108−9M∗/M⊙ 109−10M∗/M⊙ 1010−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5M∗/M⊙ (1) (2) (3) (4) (5) (6) CGM2 543 453 103 271 79 CASBaH 6701 6136 1265 3545 1326 Total 7244 6589 1368 3816 1405 Note—Summary of absorber-galaxy pairs used in this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (1) The number of absorber-galaxy pairs in each survey and total of the combined surveys;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2) the number of absorber-galaxy pairs in the entire mass range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (3) the mass range used to perfom the model fitting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (4, 5, 6) the number of absorber-galaxy pairs within each mass bin used for model verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Corner plots showing the posterior parameter probabilities for the parameters in the single power-law clustering model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We find a non-zero, positive mass dependence term in the two-halo absorber-galaxy clustering, β2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' ro = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='60+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='05 B2h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='08+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='03 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='74+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='24 Co = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='14+2:22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='14 Co 2h ro7 where r∥ is the line-of-sight distance, r⊥ is the transverse distance, and ∆r∥ is the size of the redshift window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' For simplicity of notation, r is equivalent to r⊥ in the following analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In the following definitions, we label the single power law clustering terms “2-halo,” as the galaxy clustering method we adopt here describes the clustering of sepa- rate dark matter halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' This approach distinguishes the “two-halo” only method from the two-component model we develop later in this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In order to model fc, we assume that the number of detected absorbers above the column-density threshold has a Poisson distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We consider two cases: (1) one or more absorbers detected, and (2) the case where no absorbers are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In this framework the prob- ability of seeing no absorbers is P miss = λ0 exp(−λ) 0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (4) where we denote the rate of incidence (see below) as λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The probability of finding one or more absorbers is just the complement of Equation 4, fc = 1 − P miss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (5) We model the rate of absorber incidence as the pro- jected correlation function, the 2-halo term, as the ex- cess over the probability of intersecting an absorber with NHI > 1014 cm−2 in the redshift window, λ = (1 + χ2h ⊥ ) ⟨dN/dz⟩δz, (6) where ⟨dN/dz⟩ is the base rate of detection due to the random incidence of absorbers greater than this thresh- old and deltaz is the line-of-sight redshift window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In addition to parameterizing the mass dependence as in Equation (2), we also parameterize the redshift dependence of ⟨dN/dz⟩ as follows: dN(NHI ≥ N14 HI, z) dz = C0(1 + z)α, (7) where N 14 HI denotes absorbers with column densities of 1014 cm−2, C0 is the random rate of incidence at z = 0, and δz is the redshift window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We adopt a redshift window to be ±500 km s−1 in velocity units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Thus, we have a rate of incidence of the form λ = (1 + [χ2h ⊥ (r, m|r2h 0 , γ2h, β2h)]) ⟨dN(z|C0, α)/dz⟩ δz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (8) Finally, we construct the likelihood function, L = � i P hit(ri, zi, mi|θ) � j P miss(rj, zj, mj|θ), (9) where θ = [r2h 0 , γ2h, β2h, C0, α].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In constructing our Bayesian model, we must choose priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' For the single power law parameters, we chose the priors based on the results of cross-correlation anal- ysis by Tejos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2014) except for our new mass de- pendent term, β2h, which was motivated by physical arguments: r2h 0 ∼ N(µ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='2, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='3), r2h 0 > 0 γ2h ∼ N(µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='7, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='1), γ2h > 0 β2h > 0, where N is the normal distribution with mean µ and variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The priors for the redshift dependence were chosen based on the findings in Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2021): C0 ∼ Lognormal(µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='25, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='11) , C0 > 0 α ∼ N(µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='97, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='87) , −3 < α < 3 We note that we chose to use the more recent results of Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2021) in modeling the redshift evolution instead of that from Danforth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2016), as were used in Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' As in Paper I, we apply the Bayesian Markov Chain Monte Carlo (MCMC) sampler emcee (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2013) to generate samples from the posterior prob- ability distribution function to estimate the parameters of interest and their distributions, using Equation (9) and the priors described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In Figure 2, we show the posterior distributions of our single power-law model with M0 = 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' These were fit only to data with 8 < log M⋆/M⊙ < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5, as above this range there is a change in the virial radius due to the M⋆−Mhalo relation from abundance matching (Moster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Below this mass range we find a very flat covering fraction profile, which does not show a clustering signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Two-component Models The single power-law model used in galaxy-galaxy clustering and adapted above to model the galaxy- absorber clustering makes no assumption of a CGM or overlapping (in projection) gaseous halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' However, the existence of the CGM is now well-established (Tumlin- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In particular, the trends of ionized metal species with impact parameter around L* and sub-L* galaxies from z = 0 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 distinctly show that metal-enriched gaseous atmospheres are a fundamental component of galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Werk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Lehner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Bordoloi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Borthakur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 8 Rudie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In the following section, we therefore assume the existence of the CGM and use a simple Gaus- sian profile to model the excess clustering signal due to the presence of the CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In addition, we investigated several other functional forms of the CGM component, which we describe in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We find that the particular functional form of this component has little impact on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The Gaussian CGM Two-Component Model We now add a third term to the detection rate: a Gaussian 1-halo component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The detection rate now consists of a baseline random incidence rate, an enhance- ment due to large-scale absorber-galaxy clustering, and an additional enhancement due to the CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We em- ploy an exclusion model where the contribution from the 2-halo term terminates at the distance it reaches the 1-halo component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' This scheme, shown in Figure 3, also allows us to determine a natural estimate of the extent of the CGM: the crossing point of the 1-and 2- halo components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' More explicitly, within some radius, the galaxy has a CGM that we define as the gas of that galaxy and any other satellite galaxies within its halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Our formalism then defines the Rcross where this CGM component exceeds the 2-halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 10 1 100 R [cMpc] 100 101 102 103 104 3D Correlation Function Rcross G(r)1h (r)2h Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' A schematic depiction of our two-component ex- clusion model and the determination of Rcross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The 2-halo component cuts off interior to Rcross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The model is similar to that single power-law we intro- duced before with a few key differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We introduce a Gaussian one-halo term defined as: G(r)1h = Ae−(r/σ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (10) Where the two models intersect, Rcross, we can solve for σ as σ = � 1 2 R2cross ln(A) + γ ln(Rcross/r0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (11) It should be noted that Rcross here is the 3-D distance and not the projected distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In order to characterize the mass dependence of Rcross we define Rcross = Rcross,0 �M⋆ M0 �β1h , (12) where Rcross,0 is the 1-halo term extent for a galaxy at the fixed pivot mass M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The galaxy mass dependence of σ includes contributions from the mass dependencies of Rcross and r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' This parameterization allows us to compare the mass dependence of the 1-halo term, β1h with that of the 2- halo term, β2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In order to solve for the projected clustering signal, ξ, we first make some definitions to ease the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We use s = r∥ in the remainder of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The integration is performed over different portions of the line of sight distance, s, corresponding to the 1 and 2- halo components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We define the line of sight crossing point scross as scross = � max(R2cross − r2 ⊥, 0), (13) and we can then integrate Equation 10 to seval = min(scross, smax), where smax is the maximum interval we wish to integrate over, which in our case is [−500, 500] km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Thus we have χ(r⊥) ∝ 2 � seval 0 G(r⊥, s)1hds + 2 � smax seval ξ(r⊥, s)2hds (14) where the factor of 2 comes from the fact that both components are symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Here we integrate the one- halo component over the more nearby regime out to seval and only integrate the 2-halo term beyond seval out to the maximum line of sight distance, thus excluding the regimes in which the models do not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' For the two- component model, we choose fairly weak priors on un- known parameters based on physical arguments while following the same priors as described above for the pa- rameters in the single power-law model: β1h > −3 A > 0 Rcross > 0 9 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Posterior probabilities for the parameters in the two-component clustering model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We again recover a non-zero, positive mass dependence term in the two-halo absorber-galaxy clustering, β2h but find an even stronger one-halo CGM clustering mass dependence β1h ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We can then follow the same MCMC fitting proce- dure described above to determine the posteriors for the parameters in this model as well as the crossing ra- dius, Rcross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' These are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' As before, we only fit data with 8 < log M⋆/M⊙ < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 and use M0 = 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Other Two-Component Models While the single power-law clustering model does an adequate job reproducing the data on large spatial scales, its contribution is insufficient at R⊥ ≲ 200 kpc as can be seen in Figure 5 (pink curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Furthermore, the primary goal of our study is to find the boundary between the CGM and IGM, and thus including a CGM component is essential for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We explored several candidate functional forms for this CGM com- ponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We first investigated a two-component model where each component is represented by a power law, inspired by the 1-halo and 2-halo terms that are used to model the clustering of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The 3D and projected forms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='07+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='03 22h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='0 519.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='06 ross 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='07 S 公 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 00 QQ Q: Rcross B2h Co Blh10 102 103 104 R [ckpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='0 Covering Fraction logM /M =8 9 G+ Rcross Observed 102 103 104 R [ckpc] logM /M =9 10 102 103 104 R [ckpc] logM /M =10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Comparison of our two models to the empirical covering fraction as a function of impact parameter in comoving kpc in mass bins of 108−9M⊙, 109−10M⊙ and 1010−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The data are shown in black with 1σ error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The single power-law model is shown in pink while the two-component model is shown in purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The vertical dotted line denotes Rcross in each mass bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Both models recreate the covering fraction of the data in all mass bins except for the lowest mass bin where the clustering signal disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The two-component model provides a better match to the data for galaxies of M⋆ > 109M⊙ at the lowest impact parameters where the single power law model underestimates the covering fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' of the two absorber-galaxy correlation functions are given by Equations 1 and 3, respectively, and the two- component correlation function is the sum of these parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We also considered a model where the two-component correlation function is, in 3D, the maximum of the two power laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' This is similar to our chosen model, but with an inner power law rather than an inner Gaussian profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' To rise above the outer power law component at small radii, the inner power law has to be steeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In practice, the two power law indices turned out to be similar, yield- ing essentially the same result as a single power law fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' This outcome is not unexpected: the enhancement in the incidence rate or surface density of gas near galaxies often does not resemble a steepening power law at small radii (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Lan 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In those studies, the enhancement is better described by a function that declines gradually (compared to a power law) at small radii and quickly at large radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The top-hat function, which has amplitude A inside a bound- ary and amplitude 0 outside the boundary, is an extreme example of this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Our adopted Gaussian profile al- lows a smoother transition between the CGM-like and outer components of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' However, we note that a fit to the data combining a inner 3D top-hat with an outer power law yields an Rcross(M∗) that is effectively indistinguishable from the one that emerges from the Gaussian component model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Model Comparison In addition to comparing the two models to each other, Figure 5 compares the models to the empirical covering fraction as a function of impact parameter and mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The data are shown in black with 1σ error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The sin- gle power-law model is shown in pink while the two com- ponent model is shown in purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Both models recreate the covering fractions in all mass bins at all values of R⊥ except for one data point in the log M∗/M⊙ = 9 − 10 bin at R⊥ ≈ 200 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Moreover, the two models make different predictions at low R⊥ except for in the lowest mass bin (log M∗/M⊙ < 9) where there is no discernible excess above the clustering signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' This does not pre- clude the presence of a CGM around these galaxies, but rather suggests that we require more data at lower R⊥ for galaxies with log M∗/M⊙ < 9 to be able to constrain Rcross at these masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The two-halo only model under-predicts the observed signal for galaxies at intermediate masses (log M∗/M⊙ = 9−10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The two component model does better for galax- ies of M⋆ = 109−10M⊙ at the lowest impact parame- ters where the single power law model underestimates the covering fraction, although not significantly so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' For Rcross < 300 kpc, one detects 52 H I systems where 46 systems are predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Assuming Poisson statistics, the two-halo only model is consistent with the data at 1σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Analogous to the one-halo term of galaxy- galaxy clustering, the data themselves do not require an enhanced covering fraction of H I absorption that we identify as the CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We find the 1-halo component has a stronger clus- tering mass dependence, β1h ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='07, than the two-halo term, β2h ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We also find the 2-halo clustering terms in each model to be internally consistent with each other as seen in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' RESULTS 11 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Comparison of the two-halo 3D cross correlation posteriors between the two-component model (r0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='99+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='24 cMpc, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='07) and the single power-law model (r0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='58+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='24 cMpc, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The two models are consistent with each other within the 1σ limits and have a power-law slope consistent with the absorber-galaxy 3D cross correlation found in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Tejos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2014) of γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Clustering Mass Dependence As seen in Figure 2, we find the clustering parameters to be r0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='3 cMpc, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' r0 and γ are consistent with those found in Tejos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2014) who find r0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='1 cMpc and γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We also find a mass dependence of the absorber-galaxy clustering of β2h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='07+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We find the the two component model better fits the data as can be seen in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Specifically, the two component model better matches the covering fraction for galaxies of M⋆ > 109−10M⊙ at the lower impact parameters where the single power law model underes- timates the covering fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In addition, we find the two-component model reproduces the mass dependence of the 2-halo clustering term, β2h ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='07 while also pro- ducing a stronger mass dependence of the 1-halo clus- tering term, β1h ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Physically-Motivated Extent of the CGM As mentioned above, using the two-component model produces an estimate of Rcross, a natural metric for the extent of the CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' This 3-D distance demarcates where the contribution to the clustering begins to be domi- nated by the CGM above the expected two-halo clus- tering due to isolated galaxy halos traced by H I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Rcross 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='06 ro Y B2h12 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' A comparison of Rcross with the virial radius (Rvir, grey filled region) as well as the splashback radius (Rsplash, pink shaded region) of the galaxy sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The filled regions in Rvir and Rsplash denote the redshift range for the galaxies in our sample (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='1 ≲ z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The filled blue region represents the 1σ limits of the distribution in Rcross while the blue line denotes the median of this distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The black crosses correspond to the values published in Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The vertical dotted lines denote the mass range of 8 < log(M⋆/M⊙) < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 to which we limited the fitting in our MCMC analysis in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' can be viewed as the maximum radius to which an en- hancement from the CGM could extend without over- predicting the data at large radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In Figure 7, we see Rcross (blue) compared with the spread in virial radii of the galaxy sample (grey filled region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The filled blue region represents the 1σ limits of the distribution in Rcross while the blue line denotes the median of this distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We find Rcross is ∼ 2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='6Rvir for galaxies in the range 8 < log(M⋆/M⊙) < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The black crosses correspond to the values published in Paper I defined as the extent where there is 50%chance to see H I absorption above 1014 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The vertical dotted lines denote the mass range of 8 < log(M⋆/M⊙) < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 that was used in our MCMC analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Above this range, we see a change in the relation of the virial radius with stellar mass, and below this mass range, we find little to no correlation between absorbers and galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We also calculated the splashback radius, Rsp, us- ing the method from Diemer (2018) and encoded in the COLOSSUS3 package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' This radius denotes the location at which particles reach the apocenter of their first orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We find excellent agreement of Rcross with the results in Paper I and Rcross neatly matches the splashback radius 3 https://bdiemer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='bitbucket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='io/colossus/ for galaxies in this mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We discuss these results in more detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' DISCUSSION Both of the models we investigate do an adequate job of recreating the cross correlation signal at all impact parameters and masses 108 < M⋆ < 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5M⊙ as seen in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' It is not entirely clear that the single power law model has any physically-consistent meaning, how- ever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Effectively, it would seem to signify that every time one measures H I absorption at the same redshift as a particular galaxy (|∆v| < 500 km s−1), the absorp- tion is always due to another galaxy’s CGM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Note, we would conclude this for all galaxies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' each has no CGM and only neighbors with a CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' This is clearly impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The two-halo-only model for the CGM effec- tively breaks down when the galaxies lie within the halo under consideration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' when they “mix.” We cannot and do not try to distinguish between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' How- ever, our formalism does allow one to identify the outer extent of this “mixing.” The two-component model asserts that galaxies with M⋆ > 108M⊙ have a CGM, an assumption that is moti- vated by previous survey results (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Werk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Additionally, this model is able to better recreate the data – from the combined datasets of CGM2 + CAS- BaH, which together represent the largest sample of galaxies with confirmed spectroscopic redshifts in the foregrounds of UV-bright QSOs with high-resolution ab- sorption spectroscopy – both at smaller impact param- eters and at M⋆ > 109 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The much larger number of galaxies at larger impact parameters drives the fit of the models to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' There is, however, a > 1σ inconsistency between the two-halo only model and the data at R⊥ ∼ 200 and for both models at R⊥ ∼ 600 in the logM⋆ = 9 − 10M⊙ mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The latter inconsistency may be due to cosmic variance or the assumption that the absorber- galaxy measurements are independent and are not cor- related, which would increase the scale of the error bars at R⊥ ∼ 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Comparing the mass dependence of the single and two-component models Our galaxy sample includes a large number of galaxies at low (< 500 kpc) impact parameters which allows us to better model the regime in which the two-halo galaxy clustering becomes dominated by the signal of galaxies that inhabit the same dark matter halo, the one-halo term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' By separating these two terms in the manner pre- sented here, we can disentangle the large scale clustering as well as the contribution of the CGM to the 3D corre- lation of absorbers and galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Rvir 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Rlm Wilde+21 [ckpc] R 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 7 8 6 10 11 log10(M*)13 Our analysis finds nearly identical terms for the mass dependence of the clustering at large scales, β2h as well as the contribution of absorbers at random, C0 and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We do find a stronger mass dependence in the one-halo term, β1h than at larger scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' This can be seen in Figure 5 where the correlation steepens in higher mass bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Absorber-Galaxy Bias Our covering fraction analyses provide an estimate of the galaxy-absorber correlation function, ξag (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Here, we test if the mass dependence of ξag outside the CGM is consistent with absorption systems and galax- ies simply being two independent tracers of the same underlying dark matter distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Assuming both tracers have linear bias, ξag should be equal to babgξDM, where ba and bg are the absorber and galaxy bias, respec- tively, and ξDM is the dark matter 3D correlation func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Following Tinker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2010) (hereafter, T10), we assume the dark matter correlation function can be described by a power-law function of radius with index γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We fix the power-law index in the ξag deter- mined by fitting a single power-law to the data to this same value, with which it is consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' With the above assumptions, ξag = (r/r0(M))−γ = babgξDM(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The radial dependence cancels, leaving the proportionality r0(M)γ ∝ babg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We show a scaled r0(M)γ in Figure 8 along with the galaxy bias as a function of stellar mass from T10 and implemented in the COLOSSUS package (Diemer 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' If ba is constant and the assumptions stated above hold, r0(M)γ should have the same mass dependence as galaxy bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' While there is a visually apparent differ- ence between the galaxy bias and the best-fit r0(M)γ, this difference is not significant at a 2σ level and so is merely suggestive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' If the difference is real, it could be a consequence of the H I mass per dark matter mass be- ing a function of overdensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Up to the overdensities at which Mstar = 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 M⊙ galaxies tend to be found, this function would be increasing: H I would be less common in low density regions than in higher density filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' This behavior would be consistent with theoretical ex- pectations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=', Hui & Gnedin 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Schaye 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Dav´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2010) and observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=', Rudie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Burchett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Comparison to Previous Work One of the key aspects of this analysis is determining the mass dependence of the extent of the NHI > 1014 cm−2 for which our model provides a direct metric, Rcross(M⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We compare our resulting Rcross(M⋆) to the method and results from Paper I in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 Log10M * /M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='2 Scaled bias = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='07 + / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='03, = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='62 Tinker2010 galaxy bias Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' A comparison of the slopes of the relative bias as a function of mass derived from our analysis (orange) com- pared to Tinker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2010) (T10, black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The dashed lines correspond to the ranges spanned by the 1σ limits in in β2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The relative bias, r0(M) ∝ (M⋆/M0)γβ, are normalized to the value of T10 at log M⋆/M⊙ = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We find a steeper mass dependence than T10, but the significance of the dif- ference is less than 2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' result of Paper I, R14 CGM, which are based only on the CGM2 survey are shown as black crosses in the mass bins they span in that paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We also compare the method used in that paper to determine R14 CGM, the radius at which the probability of detecting NHI > 1014 cm−2 is > 50%, calculated with the two-component model us- ing the combined CGM2 + CASBaH surveys and find it to be consistent within 1σ with our newer model for Rcross(M⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We find that our mass dependent estimate of the extent of the CGM, Rcross(M⋆) corroborates the findings of Paper I that the NHI > 1014 cm−2 extends to approximately twice the virial radius (∼ 2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='6Rvir).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' One of the main strengths of the CGM2+ CASBaH sample is the large number of galaxies at small projected separations (<1 Mpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' This allows us to investigate the smaller scale regime in more detail within the context of similar studies such as Tejos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2014) (hereafter, T14) who uses a single power law model to measure the two-point correlation between H I and galaxies above NHI > 1014 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In this work they break up their measurements into SF vs non-SF samples while we do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Our sample however is dominated by the more common SF galaxies and we will compare our results to their SF sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Comparing our cross-correlation results with T14, we find good agreement between the results in T14, rT14 0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='2 Mpc, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='1 and the results from both models presented here, r0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='99+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='24 Mpc, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='62±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='07) and the single power-law model (r0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='58+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='24 Mpc, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We find 14 a mass dependence of this cross-correlation, however as parameterized by β2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Our results are slightly in tension with Momose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2021) who find galaxies in the 109−10M⊙ range domi- nate their H I-galaxy cross correlation signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We find the largest mass bin sample to have the most elevated covering fractions at low impact parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Physical Extent of Galaxy Halos Astronomers often use the viral radius as a means to describe the characteristic size of galaxy halos and it is convenient to compare this to the extent of the gaseous galactic atmosphere as we have done here and in Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The virial radius is typically defined in terms of the spherical overdensity mass definition which is based on the radius which encloses an overdensity of 200 times the critical or mean density, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=', R200c and R200m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Because the mean and critical densities are decreasing over cos- mic time, this can lead to a pseudo-evolution as pointed out in Diemer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In addition, subhalos show evidence of being stripped outside the virial radius of clusters (Behroozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' An alternative physically motivated halo scale is the splashback radius, Rsp (Diemer & Kravtsov 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Ad- hikari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' More et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' This radius effec- tively distinguishes infalling material from matter orbit- ing in the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We compare our results to the splash- back radius in Figure 7 and find that our estimate of the extent of the H I CGM, Rcross, neatly aligns with Rsp over the mass range 108 < M⋆/M⊙ < 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' This result implies that Rsp is a better approximation of the CGM extent than the more commonly used viral radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' O’Neil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2021) compared Rsp as estimated from dark matter and gas profiles in the IllustrisTNG simula- tions and found that the gas Rsp is consistently smaller than the dark matter Rsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' However, they were look- ing at much more massive halos Mhalo > 1013 in which shocks dominate the gas distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Nonetheless, the fact that Rcross ≈ Rsp at the mass ranges considered here (Mhalo 1010−12M⊙) is intriguing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The halo mass accretion rate generally sets whether Rsp exceeds Rvir;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' a rapid accretion rate will impact the growth of the grav- itational potential well, leading to Rsp < Rvir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' If the location of Rcross reflects the extent of orbiting gas in a halo, then our observational results imply a halo mass accretion rate that is slow enough to keep the apocenters of orbiting structures at large radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Another way of defining the extent of the CGM is to use the boundary of the pressure-supported CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' For galaxies with halo masses ≳ 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5M⊙ (M⋆ ≈ 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='8M⊙), this pressure support comes from fact that the gas that has fallen into the gravitational potential well is virially shocked and cannot cool within a Hubble time (Binney 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Rees & Ostriker 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Silk 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' For the galax- ies in our survey, which are predominately below this halo mass, however, the gas would rapidly cool and thus this pressure support might come from galactic winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Fielding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2017) and Lochhaas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2018) show that supernovae winds with reasonable mass loading ef- ficiencies could shock the gas to distances past the virial radius and account for the survival of cool gas at these large radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Using a more comprehensive model of the multiphase CGM, Fielding & Bryan (2022) show that SF in the galactic disk can slow cooling and accretion as part of a global preventive self-regulation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In addition, the winds can transport cold clouds to large radii, consistent with these constraints from our com- bined survey data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' SUMMARY Herein, we have examined the associations of galax- ies with Lyα absorption z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='48 to explore the spa- tial profile of this gas and the mass dependence of the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Specifically, we have combined the CGM2 and CASBaH H I measurement and constructed a catalog of 7244 absorber-galaxy pairs around 28 QSO sightlines (6589 absorber-galaxy pairs when we restrict our galaxy sample to galaxies with 8 < log M⋆/M⊙ < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The CGM2 survey has better sampling of galaxies at low im- pact parameter while CASBaH samples galaxies out to 20 cMpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' This allows us to characterize the H I profile via the covering fraction as a tracer of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' By modeling the covering fraction as a power-law with a mass dependent length scale, we find good agreement with previous studies, such as T14, of our clustering amplitude and power law slope pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='1, we find the clustering scale has a mass dependence with a power-law slope of β2h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We compare the slope of our absorber-galaxy bias to the galaxy-dark matter bias of Tinker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The absorber-galaxy bias is a steeper func- tion of galaxy mass than the galaxy-dark matter bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' However, this difference is only significant at a sub-2σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We model the data with an exclusionary two- component model where we adopt an inner-CGM Gaussian profile to describe the data at smaller impact parameters and the customary two-halo single power-law model at larger impact param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' This model faithfully reproduces the data for galaxies M⋆ > 108M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The two component model allows us to calculate the crossover radius, Rcross(M⋆), where the mod- els are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Rcross(M⋆) represents a soft upper estimate of the furthest impact parameter needed to optimally fit the inner CGM component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' We then use Rcross as an estimate of the extent of the CGM and find Rcross(M⋆) ≈ 2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='6Rvir for galax- ies 108 ≤ M⋆/M⊙ ≤ 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Additionally, we find excellent agreement between Rcross(M⋆) and the splashback radius, Rsp for galaxies in this mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' ACKNOWLEDGMENTS MCW, KT, and JKW acknowledge support for this work from NSF-AST 1812521, NSF-CAREER 2044303, the Research Corporation for Science Advancement, grant ID number 26842.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Support for the CASBaH HST programs HST-GO-11741 and HST-GO-13846 was pro- vided through grants from the Space Telescope Science Institute under NASA contract NAS5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' Support for the CASBaH HST programs HST-GO- 11741 and HST-GO-13846 was provided through grants from the Space Telescope Science Institute under NASA contract NAS5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The CGM2 Survey would not have been possible with- out the substantial contributions from a dedicated group of UW undergraduate Student Quasar Absorption Di- agnosticians, the Werk SQuAD, with over 50 individ- ual undergraduate research assistants since 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQf3AIE/content/2301.02718v1.pdf'} +page_content=' The SQuAD confirmed all auto-fitted galaxy spectroscopic redshifts by eye, identified absorption systems along ev- ery quasar 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LATEX twocolumn style in AASTeX631 +Spatio-temporal characterization of Cassiopeia A +Yuto Ichinohe +1 and Toshiki Sato +1 +1Rikkyo University +3-34-1 Nishi-Ikebukuro, Toshima-ku +Tokyo 171-8501, Japan +ABSTRACT +Analyzing the X-ray data of supernova remnants (SNRs) are among the most challenging task in +the current X-ray astronomy because SNRs are both spatially extended and variable over time. We +developed the strategy to track the time-series properties of all the parts constituting a diffuse structure +by introducing the free-form image registration technique based on B-spline, and demonstrated the +methodology using the Chandra data of Cassiopeia A. We successfully extracted the spatial distribution +map of the time variability of continuum luminosity. To our knowledge, this is the first comprehensive +characterization of such a dynamic diffuse target both in spatial and temporal viewpoints. We found +that each of the four clusters derived by applying k-means algorithm to the extracted light curves +has a clear physical meaning distinct from other clusters, which shows that our method is not a mere +technique for automation but capable of capturing the underlying physics. +Keywords: X-ray astronomy (1810) — Astronomy data analysis (1858) — Supernova remnants (1667) +1. INTRODUCTION +Through X-ray observations, several physical quanti- +ties can be obtained, such as positions, lightcurves, en- +ergy spectra, and polarizations. In many cases, however, +only a low-dimensional slice of the complete dataset +is essential. +For example, the data of celestial point +sources are essentially two-dimensional; one only needs +to play with the spectra and lightcurves (energy E and +time t) because the spatial dimensions can be ignored +— they have no observable spatial substructures by def- +inition, and their locations are usually unchanged. Ef- +fectively three-dimensional data (two spatial dimensions +(x, y) and energy E) are obtained by the observations +of the objects that are spatially extended but stable in +human timescale such as galaxy clusters. As supernova +remnants (SNRs) are spatially extended and variable +over human timescale, the data obtained through the +X-ray observation of SNRs are truly four-dimensional +(x, y, t and E). In this regard, analyzing SNR data is +among the most challenging task in the current X-ray +astronomical data analyses. +When one wants to characterize the properties of a sin- +gle diffuse system from a spatially comprehensive view- +point (e.g., making the spatial distribution map of el- +emental abundances), typically a two-step strategy has +been taken; (i) first, defining multiple regions so that +they cover the entire system, and then (ii) performing +the same analysis for all the regions. As both the steps +can usually be automated, there have been many studies +in this line (e.g., Ichinohe et al. 2015, 2017, 2019, 2021). +On the other hand, when one wants to characterize +the time-series properties of the specific structure in the +object (temporally comprehensive viewpoint; e.g., ex- +tracting the time variability of the spectral indices of a +hotspot), one needs to define the regions in all the time +frames corresponding to that structure. This is not a +difficult task when the object is stationary (e.g. distant +point sources) because all the regions should be identical +among observations. Even investigating the time-series +properties of all the regions (comprehensive both spa- +tially and temporally) in a stationary diffuse target (e.g. +clusters of galaxies) is, at least conceptually, as easy as +performing it for a single feature. +However, if the structures are moving or changing +their shapes in the object, the analysis that is compre- +hensive in both spatial and temporal viewpoint (e.g., +tracking the time variability of every part in a dynamic +diffuse object) is, in turn, a very complex task for the +following reasons. Firstly, even for a single distinct fea- +ture, defining the corresponding regions in all the time +frames is usually not straightforward and requires hu- +arXiv:2301.02026v1 [astro-ph.HE] 5 Jan 2023 + +ID2 +Y. Ichinohe and T. Sato +man efforts to make them consistent and appropriate +through the time frames. The situation is even worse +when a region segmentation algorithm is employed to +divide the field of view into multiple regions; as the algo- +rithm is operated on each image independently, it is not +guaranteed that the number of the regions, the topology +of the regions (i.e., how the regions are connected to each +other), and what fraction of the system each region rep- +resents, are same between any two images; Even if the +algorithm guarantees all the things described in the pre- +vious sentence, associating the regions in two different +images is nontrivial in itself. +Indeed, such time-series analyses have been performed +only on the prominent features e.g. +(e.g., Uchiyama +et al. 2007; Uchiyama & Aharonian 2008; Patnaude & +Fesen 2009, 2014; Matsuda et al. 2020, 2022). This indi- +cates that we might have missed important things hap- +pening in the regions that have not been analyzed yet. In +order not to miss intriguing phenomena in the available +data, the method that is able to capture the properties +of the entire system in an unbiased manner is required. +To improve this situation and not to leave significant +amount of data unexplored, we have developed the strat- +egy to track the time-series properties of all the parts +constituting a dynamic diffuse structure. The key idea +is to find the segmentation of a given image that is same +in the physical sense as a given segmentation of the ref- +erence image. We implement the strategy and demon- +strate it using the multiple Chandra images of the su- +pernova remnant Cassiopeia A (hereafter Cas A). +X-ray emissions of Cas A are known to be very com- +plicated, where a mixture of thermal and non-thermal +emissions and their temporal variations are observed in +the monitoring observations since the launch of Chandra +in 1999 (e.g., Hughes et al. 2000; Patnaude et al. 2011; +Patnaude & Fesen 2014; Hwang & Laming 2012; Sato +et al. 2017). X-ray emitting structures in the remnant +show different time variations while changing its posi- +tion, making it difficult to know what causes the time +variability in each component. Therefore, this remnant +would be the best target for demonstrating our new tech- +nique. The outline of this paper is as follows. In Sec- +tion 2, we explain the concept of our new strategy to +track the time-series properties of the entire diffuse sys- +tem. In Section 3, we present how we implement the +strategy, and demonstrate its effectiveness by applying +it to the Chandra data of Cas A. We discuss the results +in Section 4 and present the conclusion in Section 5. +2. SPATIO-TEMPORAL CHARACTERIZATION +The main part of the algorithm consists of two steps; +(i) finding the transformation that morphs a given image +so that it matches the reference image best, based on the +B-spline registration algorithm (Lee et al. 1996, 1997); +(ii) using the transformation, inverse-transforming the +image segmentation in the reference image’s coordinates +into the corresponding one in the given image’s coordi- +nates. +2.1. Free-form image registration using B-spline +Image registration is the task of finding the transfor- +mation that maps any point in an image to the cor- +responding point in another image. Image registration +methods are well studied in the medical field where com- +paring two images of the same object taken in different +situations is essential (e.g., two MRI images taken be- +fore and after the injection of a contrast agent; see e.g., +Rueckert et al. 1999; Mattes et al. 2003). +Rigid transformation is the simplest option. When the +image is two-dimensional, the transformation only has +three degrees of freedom corresponding to a rotation and +two translations. Affine transformation is more general +and has extra three degrees of freedom that describe the +scaling and shearing. Although these methods are sim- +ple and easy to implement, they have the critical short- +coming that they can capture only the global motion. +It is often the case that the substructures in an astro- +physical object changes non-uniformly across the field- +of-view; for example, in an SNR, nonthermal filamentary +shells tend to move outward from the center of explo- +sion, while inner thermal substructures sometimes move +inward. Moreover, often the velocities of the substruc- +tures are different due to the difference in the actual +three-dimensional distance from the center as well as +the surrounding environments. +In such a complex case, simple rigid transformation +is severely insufficient and a more flexible transforma- +tion method is necessary. The main requirements to the +method are; (1) it should be able to capture the local +non-uniformity of the motions across the field of view, +and at the same time, that (2) it can express the smooth- +ness of the motion field. The latter point is important +because although each substructure moves rather inde- +pendently, the motion of a pixel would be similar to its +neighboring pixels as astrophysical objects such as SNRs +evolve in time mostly continuously. +There have been several methods to realize such free- +form image registration. For example, the combination +of feature detection (e.g., Lowe 1999; Rublee et al. 2011) +and feature matching is often used to find correspond- +ing key points in the two given images. However, X-ray +images are usually dominated by the Poisson noises and +image binning or smoothing is necessary to avoid detect- +ing image fluctuations as feature points. This worsens + +Spatio-temporal characterization of Cassiopeia A +3 +the image resolution and thus is a significant drawback +especially when the image is high-resolution such as the +one taken with Chandra. +Another option is optical flow (e.g., Lucas & Kanade +1981), the algorithm developed for motion tracking in +computer vision. However, optical flow is designed to +take into account only the local information, namely, +each pixel is independently assumed to be moving to +a certain direction. Therefore, although this can be a +choice when the motion of a local independent structure +is focused on (e.g., Sato et al. 2018), this is not the best +when the global consistency of the whole system such as +smoothness of the motion field should be considered. +B-spline registration (Lee et al. 1996, 1997) is an im- +age registration algorithm that has been applied to the +motion analysis of medical images. The basic concept +of the method is to express the deformation field of the +image using a finite number of discrete control points. +The control points are arranged in a latticed pattern +and each one represents the local motion of its neighbor- +ing coordinates. The motion field of the points between +the control points are interpolated using cubic B-spline +functions; +Tlocal(x, y) = +3 +� +k=0 +3 +� +l=0 +Bk(s)Bl(t)φi+k,j+l, +(1) +where i = ⌊x⌋ − 1, j = ⌊y⌋ − 1, s = x − ⌊x⌋, t = y − ⌊y⌋ +and Bk represents the kth basis function of the B-spline +B0(t) = (1 − t)3/6 +B1(t) = (3t3 − 6t2 + 4)/6 +B2(t) = (−3t3 + 3t2 + 3t + 1)/6 +B3(t) = t3/6, +where 0 ≤ t < 1. As the B-Spline representation of the +motion field is continuous, differentiable, and bijective, +this method is suitable for feature tracking of smoothly +moving objects such as supernova remnants. +We want to identify the best transformation within +this modeling. +However, a caveat is that the defi- +nition of the best transformation is somewhat vague. +It is straightforward to determine the best one when +the ground truth deformation exists; that is, the best +transformation should be the one that approximates the +ground truth best within its expressive power. However, +in the actual case, there is no ground truth deformation +that transforms one observed image frame into another. +Instead, there are just motions of astrophysical struc- +tures – some move coherently, others independently –, +and we want to express these collective motions by a +parameterized transformation. +Therefore, in this work, we define the best transforma- +tion as the one that yields the most similar transformed +image to the reference image, i.e., the one that mini- +mizes a certain similarity metric predetermined by the +analyzer of the data. In order to find the best transfor- +mation, one only needs to minimize the similarity met- +ric, which measures the difference between the reference +image and the other image after deformation. As this +transformation is parameterized with a relatively small +number of free parameters, i.e., the motion vector asso- +ciated with each of the control points, it is relatively easy +to find the best parameters with common optimization +algorithms. +In some problematic cases, it is possible that the al- +gorithm associates points in different images that are +independent1. It should be noted that this method im- +plicitly assumes the smallest possible motions. We ex- +pect that this assumption applies to most astrophysical +applications, including the present work. +2.2. Region transformation +Once the best transformation is found, the reference +image and the other image after deformation are similar +to each other. It is thus expected that the region seg- +mentation in the reference image generated by a physical +motivation can also be used as a physically-motivated re- +gion segmentation of the other image after deformation. +Therefore, the regions in the other image’s original coor- +dinates can be obtained by simply inverse-transforming +the regions in the reference image. The resulting region +segmentation should be physically motivated as is the +one in the reference image. +3. DEMONSTRATION +3.1. Datasets +The Advanced CCD Imaging Spectrometer (ACIS) +of Chandra has observed Cas A multiple times since +its launch in 1999 (e.g., Hughes et al. 2000; Hwang +et al. 2000, 2004; Uchiyama & Aharonian 2008; Pat- +naude et al. 2011; Patnaude & Fesen 2014; Sato et al. +2017, 2018) and here we used the archival data from +2000 to 2019. The ObsIDs used for this work are sum- +marized in Table 1. We reprocessed the archival level +1 event lists produced by the Chandra pipeline in the +standard manner using the CIAO software package (ver- +sion 4.12) and the CALDB (version 4.9.2.1) to apply the +1 For example, when three points are arranged in a triangle and +all the points are rotating clockwise at the angular speed of 119◦ +per frame, the optimizer is likely to halt by finding 1◦ per frame +anticlockwise motion because the optimization step starts from +zero motion. + +4 +Y. Ichinohe and T. Sato +Year +ObsID +Exp. (ks) +Obs. Date +2000 +114 +49.9 +2000 Jan 30 +2002 +1952 +49.6 +2002 Feb 06 +2004 +4634 +148.6 +2004 Apr 28 +4635 +135.0 +2004 May 01 +4636 +143.5 +2004 Apr 20 +4637 +163.5 +2004 Apr 22 +4638 +164.5 +2004 Apr 14 +4639 +79.0 +2004 Apr 25 +5196 +49.5 +2004 Feb 08 +5319 +42.3 +2004 Apr 18 +5320 +54.4 +2004 May 05 +2007 +9117 +24.8 +2007 Dec 05 +9773 +24.8 +2007 Dec 08 +2009 +10935 +23.3 +2009 Nov 02 +12020 +22.4 +2009 Nov 03 +2010 +10936 +32.2 +2010 Oct 31 +13177 +17.2 +2010 Nov 02 +2012 +14229 +49.1 +2012 May 15 +2013 +14480 +48.8 +2013 May 20 +2014 +14481 +48.4 +2014 May 12 +2015 +14482 +49.4 +2015 Apr 30 +2016 +18344 +25.8 +2016 Oct 21 +19903 +24.7 +2016 Oct 20 +2017 +19604 +49.5 +2017 May 16 +2018 +19605 +49.4 +2018 May 15 +2019 +19606 +49.4 +2019 May 13 +Table 1. Chandra observations +appropriate gain maps and the latest calibration prod- +ucts. +Cas A is observed as a complex mixture of thermal +and non-thermal X-ray radiations. +For simplicity, we +generated one image in the 4.2–6.0 keV band for each +observation, resulting in fourteen images in total. +In +this energy band, both thermal bremsstrahlung and non- +thermal (synchrotron) radiation are present, and these +featureless continuum radiations are dominant. Avoid- +ing emission lines from various elements reduces infor- +mation on specific elements (i.e., information on the el- +emental abundance). This allows us to focus only on +thermal and non-thermal variations. +3.2. Implementation +We implemented the cubic B-spline registration using +SimpleITK toolkit (Lowekamp et al. 2013; Beare et al. +2018; Yaniv et al. 2018). We employed an 8×8 mesh for +the control points in the transform domain. Each control +point has two control parameters corresponding to the +two components of the motion field vector there in the +two-dimensional image, and the cubic B-spline interpo- +lation requires three extra control points per dimension +for the interpolation of the domain close to the image +boundaries. These result in (8 + 3) × (8 + 3) × 2 = 242 +parameters to optimize. We employed the L-BFGS-B +algorithm implemented in SimpleITK for the metric op- +timization (Liu & Nocedal 1989; Byrd et al. 1995; Zhu +et al. 1997). +For the optimization metric, the corre- +lation of two images is used because the luminosity of +Cas A is known to be gradually decreasing (e.g., Pat- +naude et al. 2011), and simpler, intensity-based metrics +such as mean-squared error are not suitable. +In this work, we use the image obtained in the 2004 +observation as the reference image because it is of the +highest quality. We preprocessed all the fourteen images +with a Gaussian filter of the kernel size σ = 4 px prior to +the deformation. For each of thirteen images other than +the reference, the corresponding transformation was de- +rived by comparing it with the reference image. +Although all the images are directly compared to the +2004 image for deriving the transformations (one-shot +strategy) in the present implementation, it should be +noted that another strategy exists; the two-shot strat- +egy, in which the transformation between every two suc- +cessive frames is first computed, and the appropriate +transformations are composed to reproduce the trans- +formations equivalent to the one-shot counterparts. We +checked both strategies and found that both yield almost +the same results, while some of the structures show more +residual motions when the two-shot strategy is taken. +We think this is due to the larger uncertainties in the +determination of the transformation when using worse- +quality images. There are at least two competing factors +that cause larger errors in the resulting transformations; +(1) larger time intervals and (2) much accumulation of +the errors. In the present case, we think that the lat- +ter effect dominates so that the one-shot strategy works +better. While the 2004 image has an exposure time of +∼1 Ms, other images have exposure times of only ∼50 ks +(Table 1). In the two-shot strategy, most of the transfor- +mations are computed using these worse-quality images, +which would result in larger errors for each transforma- +tion. In the present case, the motions of the structures +are relatively smooth and slow, and we thus think the +accumulation of the errors has a larger negative impact +on the overall uncertainties compared to the larger time +intervals. If, for example, the motions of the structures +are fast, the time interval is very large, and/or the qual- +ity of all the images is very high (or low), the former fac- +tor could dominate. In such cases, the two-shot strategy +would work better. + +Spatio-temporal characterization of Cassiopeia A +5 +Relative deviation +Relative deviation (Transformed) +Transformation field +original points +transformed points +4 +3 +2 +1 +0 +1 +2 +3 +4 +(F2019 +F2004)/F2004 +4 +3 +2 +1 +0 +1 +2 +3 +4 +(F2019 +F2004)/F2004 +Figure 1. Left and middle: relative deviation images computed by using (X − Y )/Y with X and Y being the 2019 and 2004 +(reference) images, respectively. Left: created using the original images (without any transformations). Middle: the 2019 image +is transformed to match the 2004 image best. Right: visualization of the transformation field. The coordinates in the 2019 +image represented by the black points are transformed to the coordinates represented by the red points. Note that, in the left +and middle panels, most of the pixels outside the remnant show negative deviations because the 2004 image contains far fewer +zero-count pixels due to the long exposure time (∼1 Ms, see also Table 1). +The left panel in Fig. 1 shows the relative deviation +image computed by using (X − Y )/Y , where X and Y +is the images obtained in the 2019 and 2004 (reference) +observations, respectively. The image shows clear fila- +mentary structures aligned in a circular shape, reflecting +the non-thermal filaments in Cas A, which are actually +moving outwards. The image in the middle panel is com- +puted by the same formula, in which the 2019 image de- +formed using B-spline so as to match the 2004 image best +is used as X, instead of the original 2019 image. The +disappearance of the filaments in all directions clearly +shows that the 2019 image is successfully deformed by +the B-spline transformation. The right panel shows a vi- +sualization of the transformation field. The coordinates +in the 2019 image represented by the black points are +transformed to the ones represented by the red points. It +can be observed that in addition to the dominant radial +motions of the filaments, the non-uniformity of their am- +plitudes and directions are captured, demonstrating the +effectiveness of this method on the registration problem +of SNR images. +To divide the field of view into subregions, we used +the contour binning algorithm (Sanders 2006). We ap- +plied the algorithm to the reference image with the pa- +rameters sn=50, constrainval=2.5, and smoothsn=15. +The region segmentation in the reference image’s coordi- +nates was converted into the ones in all the other images’ +original coordinates by the respective inverse transfor- +mations. +Once the regions are obtained in all the images, one +can easily obtain the time-series property associated to +any features; one only needs to refer to the region ex- +pressed in the coordinates corresponding to the image of +interest without being bothered by, e.g., the consistency +of the regions among different time frames. +For the +demonstration purpose, we simply extracted the contin- +uum X-ray flux in each region for all the time frames. As +a result, ∼2000 light curves, each of which corresponds +to a certain morphological structure in Cas A, were ob- +tained. +Using the simulated image frames, we tested +the validity of the pipeline and conservatively estimated +the systematic uncertainties in deriving the light curve +values at ∼ 10%. See Appendix A for the details. +For the current implementation, most of the calcula- +tions take less than a few seconds using an Intel Xeon +E5-1620 v4 CPU (3.50 GHz, 8-core). The processes that +can take longer time are the image registration and the +inverse transformation of the region map into the target +coordinates. +With the image dimensions of 921×921, +the typical time the former process takes ranges from +about ten seconds to several tens of seconds per pair of +images depending on the registration parameters. The +latter takes ∼20 seconds per target frame. +3.3. Unsupervised clustering +To characterize the thousands of light curves, we +adopted the k-means clustering algorithm (Macqueen +1967). +k-means is one of the unsupervised machine +learning algorithms to perform the partition of N ob- +servations contained in a given dataset into k clusters. +After successful classification, each element belongs to +one of the k clusters whose center is the nearest to the +element among the k centers. The center of each clus- +ter is calculated by the mean of the elements belonging +to the cluster, which are expected to be similar to each +other. We ran k-means with k = 4 with the Euclidean + +6 +Y. Ichinohe and T. Sato +metric (see Appendix B for the details regarding the +choice of k). Before applying k-means, each of the light +curves was normalized using the respective 2004 value. +Fig. 2 shows the result of k-means. The average light +curve (black dashed line) of each cluster represents the +rough trend of the light curves belonging to the cluster. +Namely, the X-ray flux of the regions belonging to the +clusters 0, 1, 2, and 3 is increasing, mildly decreasing, +quickly decreasing, and staying constant, respectively. +3.4. Visualization +The resulting product of all the above procedures is a +set of light curves, grouped by time variability. As each +light curve represents a certain part in the object, one +can convert this information into the spatial distribu- +tion map of X-ray time variability. The top right panel +of Fig. 3 shows the spatial distribution map of the clus- +tering results. The middle and bottom rows of Fig. 3 +show the Cas A images filtered by the cluster member- +ships. The components whose flux are increasing (clus- +ter 0; middle left) or rapidly decreasing (cluster 2; bot- +tom left) are distributed in clumpy morphology. On the +other hand, the components whose flux are gradually +decreasing (cluster 1; middle right) or constant (cluster +3; bottom right) are distributed in diffuse morphology. +Although both clumpy, the distribution of the bright- +ening (cluster 0) and rapidly dimming (cluster 2) com- +ponents are different; the former is rather localized in +the northwestern direction, while the latter is rather dis- +tributed uniformly. The bright compact clumps tend to +belong to the cluster 2, while the cluster 0 contains fila- +mentary structures of moderate luminosity . Regarding +the diffuse components, the gradually dimming compo- +nent (cluster 1) is apparently brighter than the constant +luminosity component (cluster 3). The regions covered +by the cluster 1 extend to the radii of non-thermal shells, +while the cluster 3 only covers the region inside the +shells. +4. DISCUSSION +In the previous section, we have shown the concept of +the combination of the B-spline registration and the k- +means clustering algorithm in characterizing the evolv- +ing diffuse structures, and demonstrated the effective- +ness of the concept using the data of Cas A. To our +knowledge, this is the first comprehensive characteriza- +tion of such a dynamic diffuse target both in spatial and +temporal viewpoints. In this section, we show that the +method is not a mere tool for automation by demon- +strating that the clusters thus obtained are actually sci- +entifically interpretable. We also discuss the advantages, +caveats and future prospects of our method. +4.1. Physical interpretation of each cluster +We have identified four clusters showing different time +variations. +These can be broadly classified into two +types of variations – thermal or non-thermal emissions. +Here, we discuss the origin of the flux variations using +the typical timescale of each cluster. +Fig. 4 show the spatial distribution maps of the ratio +of the flux in 2019 and 2000, plotted for each of the clus- +ters. We can see that the regions with flux increase and +decrease between 2000 and 2019 were characterized well. +Interestingly, the different rates of variation can also be +visualized (e.g., between the clusters 1 (deep blue) and +2 (light blue)), suggesting that different physics is in the +background. +Generally, non-thermal emissions in SNRs are ob- +served as small-scale filamentary structures (e.g., Bamba +et al. 2003, 2005). This is because the non-thermal X- +rays from the accelerated cosmic-ray electrons are emit- +ted from small regions that are compressed by shock +waves. +Therefore, the small-scale knotty/filamentary +structures seen in the clusters 0 and 2 would probably +be associated with the non-thermal phenomena. In ad- +dition, these clusters show fast variability with ∼ +3.5 +% yr−1 (e-folding time τinc ∼ 17 yr) for the cluster 0 and +≳ −4 % yr−1 (e-folding time τdec ∼ 49 yr) for the clus- +ter 2, which could be related to rapid acceleration and +cooling in an amplified magnetic field (e.g., Uchiyama +et al. 2007; Uchiyama & Aharonian 2008; Patnaude & +Fesen 2009). +The synchrotron cooling timescale τsyn +and the acceleration timescale τacc, which are e-folding +timescales, can be estimated as +τsyn ∼ 0.7 B−1.5 +mG ϵ−0.5 +5keV year +(2) +τacc ∼ 0.8 η B−1.5 +mG ϵ0.5 +5keVV −2 +sh,5 year, +(3) +where BmG, Vsh,5, ϵ5keV and η are the magnetic field +in units of 1 mG, the shock velocity in units of 5,000 +km s−1, the photon energy in units of 5 keV, and the +Bohm factor, respectively. We found that the amplified +magnetic field of B ∼ 0.12 mG and the Bohm factor of +η ∼ 3 reproduce the observational timescales well, which +agrees with the previous estimations (e.g., Sato et al. +2018; Tsuji et al. 2021). Thus, we conclude that most +of the clusters 0 and 2 are related to the non-thermal +phenomena. +We note that our method is also useful for identify- +ing peculiar structures, such as inward-moving shocks in +this remnant (Sato et al. 2018). These structures can be +interpreted as signatures of the interaction of the rem- +nant with an asymmetric dense circumstellar shell that +occurred between ∼180 and ∼240 yr after the super- +nova event (Vink et al. 2022; Orlando et al. 2022), thus +it would be important for understanding the progeni- + +Spatio-temporal characterization of Cassiopeia A +7 +0 +2 +4 +6 +8 +10 +12 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Cluster 0 +83 regions +0 +2 +4 +6 +8 +10 +12 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +Cluster 1 +1235 regions +0 +2 +4 +6 +8 +10 +12 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +Cluster 2 +412 regions +0 +2 +4 +6 +8 +10 +12 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +Cluster 3 +587 regions +Figure 2. The results of k-means (k = 4). Each panel corresponds to a cluster. In each panel, the colored lines exemplify the +light curves belonging to the cluster, and the black dashed line and the gray band correspond to the average light curve and +standard deviation, respectively. +tor’s activity. We found that these structures, classified +as clusters 0 and 2 in the southern and western regions, +move differently from most of the structures that move +outward, but were successfully tracked by our method +(see the rightmost panel of Fig. 1 and the movie corre- +sponding to Fig. 3). While there are also several bright +small-scale structures classified as clusters 0 and 2 in the +northwestern reverse-shock region, they differ from the +inward-moving structures in that they are moving out- +ward. This suggests that there is not enough material in +the northwest direction to drive the shock wave inward, +and may support the recently proposed asymmetric cir- +cumstellar shell for Cas A. In addition to these struc- +tures, our method identified a forward-shock filament +that is getting brighter in the northeastern edge of the +remnant (see the top left panel of Fig. 4). It has been +suggested that diffusion in this filament appears to be +occurring near the Bohm limit (Stage et al. 2006), which +implies efficient cosmic-ray acceleration at the location. +Although the interpretation of the cause of the flux vari- +ation is outside the scope of this paper, our analysis in- +dicates that the flux increase in this filament reported +in Patnaude & Fesen (2009) has continued since then. +Diffuse components that are gradually dimming with +a rate of ∼ −1.4 % yr−1 were classified into the clus- +ter 1, which could be related to the thermal emission +of Cas A. In particular, the regions classified into this +cluster appear to be concentrated in the northern and +southeastern reverse shock regions where the thermal +emission is dominant (e.g., DeLaney et al. 2004; Helder +& Vink 2008). Assuming that this cluster represents the +thermal radiation, we here discuss the time evolution +of the thermal component in the remnant. The ther- +mal emissions in young supernova remnants are thought +to decrease due to adiabatic expansion. In Sato et al. +(2017), the decay rate of the thermal X-rays in 4.2–6.0 +keV by the adiabatic expansion in Cas A was estimated +to be −1.36 (m/0.66)(t/340 yr)−1 % yr−1, where m and +t is the expansion index and the age of the remnant, +respectively. This rate explains well the temporal varia- +tion of the cluster 1, thus we conclude that this cluster +would be related to the thermal emission. On the other +hand, we note that the non-thermal emissions dimming +with a longer timescale (i.e., cooling at a less amplified +magnetic field) would also be classified into this cluster. +Faint regions with less time variability were classified +into the cluster 3. +We consider that acceleration (or +heating) and cooling are balanced in these regions. Most +of the regions in this cluster are located at the south- +western reverse shock regions and the forward shock re- +gions, where the non-thermal emission is dominant (e.g., +DeLaney et al. 2004; Helder & Vink 2008; Grefenstette +et al. 2015). If cosmic-ray electrons were regularly accel- +erated and cooled there, this would explain the constant +luminosity. In addition, materials are constantly being +heated in the vicinity of the shock wave, which may off- +set the decay of the thermal component due to adiabatic +expansion. These balanced components could have been +classified as the cluster 3. +4.2. Advantages, caveats and future prospects +As shown in the previous section, we found that each +of the cluster represents a clear physical meaning dis- +tinct from other clusters. This means that the physical +motivation in each cluster is aligned and thus that our +methodology can extract the physics underlying a dy- +namic diffuse object in an organized manner. Notably, +all the processes, i.e., the region segmentation, image +registration, and clustering, were performed automat- +ically. +This is already a huge advantage that lead to +the first comprehensive characterization of Cas A, which +have been practically impossible to do manually. Such +an automated extraction of physics would become more +important when the future powerful missions such as +Athena are in operation. +Most of the future missions, including Athena, will +have worse angular resolutions than Chandra. Although +worse angular resolutions cause worse image quality, we + +8 +Y. Ichinohe and T. Sato +Figure 3. Visualization of the results. Top left: the Cas A X-ray images in the original coordinates. Top, second from left: +the same images in the transformed coordinates. Top, third from left: the spatial distribution map of the clustering results in +the 2004 coordinates. Top, right: the same maps inverse-transformed onto the target frames’ coordinates. Middle row: the +Cas A X-ray images filtered by the clusters, shown in the transformed coordinates. The leftmost, second, third, and rightmost +panels show the regions in Cas A, whose light curves are classified as clusters 0, 1, 2, and 3, respectively. Bottom row: the same +images as in the middle row, shown in the original coordinates. The contours represent the X-ray surface brightness at 2004. +The animation (14 frames, corresponding to the years shown in Table 1) shows how the overall morphology of Cas A evolves +with time and how this morphological evolution is seemingly suppressed by our method. + +2004 (original coordinates) +2004 (transformed coordinates) +Cluster label map (2004 coordinates) +Cluster label map (inverse-transformed) +Cluster o (transformed coordinates) +Cluster 1 (transformed coordinates) +Cluster 2 (transformed coordinates) +Cluster 3 (transformed coordinates) +Cluster o (original coordinates) +Cluster 1 (original coordinates) +Cluster 2 (original coordinates) +Cluster 3 (original coordinates)Spatio-temporal characterization of Cassiopeia A +9 +Cluster 0 +Cluster 1 +Cluster 2 +Cluster 3 +2 +1 +0 +1 +2 +log(F2019/F2000) +Figure 4. The spatial distribution maps of the ratio of the flux in 2019 (F2019) and 2000 (F2000). In each panel, only the +component classified as the respective cluster is shown. +think the registration should still work if the object ap- +pears to change its shape under that quality. If the angu- +lar resolution is so poor that actually moving structures +do not appear to be moving, our method is just un- +necessary, and one can simply take identical sets of the +regions for all the time frames. In general, when the an- +gular resolution is poor, one needs to be careful in deter- +mining the regions to take the PSF (point spread func- +tion) blending effect into consideration properly, and it +is probably required to tune the hyperparameters of the +tessellation algorithm. +That said, we emphasize that +this is the issue in the tessellation algorithm but not in +the registration algorithm presented in this paper. +The concept of our method is fairly simple — using +image registration technique to convert dynamic objects +into static ones. Therefore, at least conceptually, our +method can be applied to a wide range of other dy- +namic diffuse targets, regardless of the instrument (e.g., + +10 +Y. Ichinohe and T. Sato +Chandra or XMM-Newton; it does not even have to be +X-ray instruments). That said, there are cases that our +method would not work properly. For example, when +the neighboring image frames are not very similar, im- +age registration would fail. This happens, for example, +when the structures are moving too fast compared to +the time separation of two images or when the motion +is chaotic. Low quality of images (low statistical counts, +high noise, or existence of instrumental artefacts) would +of course cause the fail of the method. When the con- +trol points are too sparse compared to the spatial scale +of the motion, this method might also fail. The cases +include the motions whose scale is substantially sharper +than the mesh grid, and the motions which are highly +non-uniform within the neighrboring of a single control +point. This behavior is also observed in our result; some +filamentary structures close to the center show residual +motion after transformation (see the movie correspond- +ing to Fig. 3). Although employing denser control points +should resolve the problem, this leads to the increase of +the number of parameters to optimize. Therefore careful +trade-off study may be required. +In this demonstration, we used only the total luminos- +ity of the continuum (4.2–6.0 keV) band for both image +transformation and the subsequent clustering. Although +we leave advanced analysis for future works, we note +that it would be easy to extend this method to incor- +porate more information contained in the dataset. For +example, by extracting lightcurves from multiple energy +bands, it would be possible to characterize the time evo- +lution of the object from the time evolution of spatial +distribution of metals. By using high-resolution spec- +troscopic data using future instruments, one can trace +the evolution of line-of-sight velocity of moving struc- +tures to constrain 3D shock dynamics. Ultimate goal +would be using the entire spectral information. Char- +acterizing a huge amount of complex spectra is itself a +significant task and several progresses have been made +(e.g., Iwasaki et al. 2019). We think that combining our +method with such methods based on data science would +facilitate the analysis of future high-quality astronomi- +cal datasets. +5. CONCLUSIONS +We have developed the strategy to track the time- +series properties of all the parts constituting a diffuse +structure by introducing the free-form image registra- +tion technique based on B-spline. +We have demon- +strated the methodology using the Chandra data of +Cas A by extracting the spatial distribution map of the +time variability of continuum luminosity. To our knowl- +edge, this is the first comprehensive characterization of +such a dynamic diffuse target both in spatial and tem- +poral viewpoints. +We have found that each of the four clusters derived +by applying k-means algorithm to the extracted light +curves has a clear physical meaning distinct from other +clusters, which shows that our method is not a mere +technique for automation but capable of capturing the +underlying physics. +This work was partially supported by the Grants-in- +Aid for Scientific Research by the Japan Society for +the Promotion of Science with KAKENHI Grant Nos. +18H05458, 19K14749, 20K14524, and 20K20527. +APPENDIX +A. TEST OF THE PIPELINE +In this section, we present the results of the exper- +iment to test the validity of the presented algorithm +and to estimate the systematic uncertainties in the light +curve reconstruction. +We simulated 50 movies consisting of four 50 px×50 px +mock image frames and randomly generated four struc- +tures per movie. Each structure is either a ‘clump’ or +a ‘filament’. For a given frame number i (i = 0, 1, 2, 3), +the shape of a clump is computed by a two-dimensional +Gaussian with the center location, normalization, and +radius calculated by (x0 +i∆x,y0 +i∆y), n0 +i∆n, and +σ, respectively, using the values presented in Table 2. +The shape of a filament is a line segment smeared by +a two-dimensional Gaussian. The center location, rota- +tion, normalization, Gaussian radius, and line length are +calculated by (x0 + i∆x,y0 + i∆y), θ0 + i∆θ n0 + i∆n, +σ, and l, respectively, using the values presented in Ta- +ble 2. The left two columns of Fig. 5 show examples of +the first frames of the movies and the corresponding last +frames. +Using the pixel values corresponding to each structure, +we defined three regions for each structure by selecting +the pixels whose value is larger than 0.3, 0.1, and 0.01 +times the maximum. Larger threshold values correspond +to smaller region sizes, as exemplified in the right two +columns of Fig. 5. We found that sometimes those ran- +domly generated structures overlap, which results in the +regions not being properly determined. To avoid such + +Spatio-temporal characterization of Cassiopeia A +11 +situations, we only kept the random trials that did not +cause the overlap of regions. +Using the region maps, we followed the same proce- +dure presented in Section 2 to transform the region maps +into the coordinates of target frames and extracted the +sum of the pixel values corresponding to the transformed +regions. +Fig. 6 shows the ratios of the reconstructed +normalization values to the ground truth normalization +values. The errors correspond to the standard deviation +of the derived values. +The ratios are consistent with unity, suggesting that +the light curves are recovered correctly by our strategy. +We found that the scatter of the reconstructed values +increases with frame numbers, which can be naturally +interpreted as that it is easier to perform image regis- +tration between two similar frames than two rather dif- +ferent frames. We also found that the scatters are larger +for higher threshold values (i.e., smaller regions). This +is because the errors in the deformation cause the mis- +alignment of structures and the corresponding regions. +The systematic errors introduced by this effect should +be larger for smaller regions because the relative error +of the transformation field to the region size becomes +larger. +It is difficult to quantitatively estimate how much er- +ror is caused by how the structures in different frames +differ because other factors, such as the distance and +flux ratio to the surrounding structures, would also af- +fect the predictions. However, considering that our sim- +ulation includes the structures whose motion scales are +as large as ∼ 1/10 of the frame size (e.g., the filaments in +Fig. 5), which is way larger than the structure motions +existing in Cas A, we think the overall systematic errors +due to the reconstruction accuracy is sufficiently below +the error bars presented in Fig. 6, i.e., a conservative +baseline value is ∼ 10%. We expect that such a level of +error does not significantly affect the k-means algorithm +(which resulted in four clusters of rather distinct trends, +see Fig 2) and the following discussions. +B. NOTE ON THE K-MEANS +HYPERPARAMETERS +In this work, we ran k-means with k = 4 with the Eu- +clidean metric. In determining the number of clusters, +we checked the performance metrics of the k-means al- +gorithm, such as the silhouette score (Rousseeuw 1987), +the elbow method, and the Bayesian information crite- +rion (BIC, Schwarz 1978; Pelleg & Moore 2000). +Al- +though all the methods suggested the optimal number +of around k = 3 − 5, we found it difficult to choose one. +We think this is because the light curve properties are +relatively smoothly distributed, and there are no clear +boundaries between the components. 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Softw., 23, 550–560, +doi: 10.1145/279232.279236 + diff --git a/WNA0T4oBgHgl3EQfE_93/content/tmp_files/load_file.txt b/WNA0T4oBgHgl3EQfE_93/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..549b92cb45d70e34bc7f4cf44c3f0ce1d3e62d7e --- /dev/null +++ b/WNA0T4oBgHgl3EQfE_93/content/tmp_files/load_file.txt @@ -0,0 +1,887 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf,len=886 +page_content='Draft version January 6, 2023 Typeset using LATEX twocolumn style in AASTeX631 Spatio-temporal characterization of Cassiopeia A Yuto Ichinohe 1 and Toshiki Sato 1 1Rikkyo University 3-34-1 Nishi-Ikebukuro, Toshima-ku Tokyo 171-8501, Japan ABSTRACT Analyzing the X-ray data of supernova remnants (SNRs) are among the most challenging task in the current X-ray astronomy because SNRs are both spatially extended and variable over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We developed the strategy to track the time-series properties of all the parts constituting a diffuse structure by introducing the free-form image registration technique based on B-spline, and demonstrated the methodology using the Chandra data of Cassiopeia A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We successfully extracted the spatial distribution map of the time variability of continuum luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' To our knowledge, this is the first comprehensive characterization of such a dynamic diffuse target both in spatial and temporal viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We found that each of the four clusters derived by applying k-means algorithm to the extracted light curves has a clear physical meaning distinct from other clusters, which shows that our method is not a mere technique for automation but capable of capturing the underlying physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Keywords: X-ray astronomy (1810) — Astronomy data analysis (1858) — Supernova remnants (1667) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' INTRODUCTION Through X-ray observations, several physical quanti- ties can be obtained, such as positions, lightcurves, en- ergy spectra, and polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In many cases, however, only a low-dimensional slice of the complete dataset is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' For example, the data of celestial point sources are essentially two-dimensional;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' one only needs to play with the spectra and lightcurves (energy E and time t) because the spatial dimensions can be ignored — they have no observable spatial substructures by def- inition, and their locations are usually unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Ef- fectively three-dimensional data (two spatial dimensions (x, y) and energy E) are obtained by the observations of the objects that are spatially extended but stable in human timescale such as galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' As supernova remnants (SNRs) are spatially extended and variable over human timescale, the data obtained through the X-ray observation of SNRs are truly four-dimensional (x, y, t and E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In this regard, analyzing SNR data is among the most challenging task in the current X-ray astronomical data analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' When one wants to characterize the properties of a sin- gle diffuse system from a spatially comprehensive view- point (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', making the spatial distribution map of el- emental abundances), typically a two-step strategy has been taken;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' (i) first, defining multiple regions so that they cover the entire system, and then (ii) performing the same analysis for all the regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' As both the steps can usually be automated, there have been many studies in this line (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', Ichinohe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2015, 2017, 2019, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' On the other hand, when one wants to characterize the time-series properties of the specific structure in the object (temporally comprehensive viewpoint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', ex- tracting the time variability of the spectral indices of a hotspot), one needs to define the regions in all the time frames corresponding to that structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' This is not a difficult task when the object is stationary (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' distant point sources) because all the regions should be identical among observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Even investigating the time-series properties of all the regions (comprehensive both spa- tially and temporally) in a stationary diffuse target (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' clusters of galaxies) is, at least conceptually, as easy as performing it for a single feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' However, if the structures are moving or changing their shapes in the object, the analysis that is compre- hensive in both spatial and temporal viewpoint (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', tracking the time variability of every part in a dynamic diffuse object) is, in turn, a very complex task for the following reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Firstly, even for a single distinct fea- ture, defining the corresponding regions in all the time frames is usually not straightforward and requires hu- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='02026v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='HE] 5 Jan 2023 ID2 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Ichinohe and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Sato man efforts to make them consistent and appropriate through the time frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The situation is even worse when a region segmentation algorithm is employed to divide the field of view into multiple regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' as the algo- rithm is operated on each image independently, it is not guaranteed that the number of the regions, the topology of the regions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', how the regions are connected to each other), and what fraction of the system each region rep- resents, are same between any two images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Even if the algorithm guarantees all the things described in the pre- vious sentence, associating the regions in two different images is nontrivial in itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Indeed, such time-series analyses have been performed only on the prominent features e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', Uchiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Uchiyama & Aharonian 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Patnaude & Fesen 2009, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Matsuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2020, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' This indi- cates that we might have missed important things hap- pening in the regions that have not been analyzed yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In order not to miss intriguing phenomena in the available data, the method that is able to capture the properties of the entire system in an unbiased manner is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' To improve this situation and not to leave significant amount of data unexplored, we have developed the strat- egy to track the time-series properties of all the parts constituting a dynamic diffuse structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The key idea is to find the segmentation of a given image that is same in the physical sense as a given segmentation of the ref- erence image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We implement the strategy and demon- strate it using the multiple Chandra images of the su- pernova remnant Cassiopeia A (hereafter Cas A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' X-ray emissions of Cas A are known to be very com- plicated, where a mixture of thermal and non-thermal emissions and their temporal variations are observed in the monitoring observations since the launch of Chandra in 1999 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Patnaude et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Patnaude & Fesen 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Hwang & Laming 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Sato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' X-ray emitting structures in the remnant show different time variations while changing its posi- tion, making it difficult to know what causes the time variability in each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Therefore, this remnant would be the best target for demonstrating our new tech- nique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The outline of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In Sec- tion 2, we explain the concept of our new strategy to track the time-series properties of the entire diffuse sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In Section 3, we present how we implement the strategy, and demonstrate its effectiveness by applying it to the Chandra data of Cas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We discuss the results in Section 4 and present the conclusion in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' SPATIO-TEMPORAL CHARACTERIZATION The main part of the algorithm consists of two steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' (i) finding the transformation that morphs a given image so that it matches the reference image best, based on the B-spline registration algorithm (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 1996, 1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' (ii) using the transformation, inverse-transforming the image segmentation in the reference image’s coordinates into the corresponding one in the given image’s coordi- nates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Free-form image registration using B-spline Image registration is the task of finding the transfor- mation that maps any point in an image to the cor- responding point in another image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Image registration methods are well studied in the medical field where com- paring two images of the same object taken in different situations is essential (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', two MRI images taken be- fore and after the injection of a contrast agent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', Rueckert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Mattes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Rigid transformation is the simplest option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' When the image is two-dimensional, the transformation only has three degrees of freedom corresponding to a rotation and two translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Affine transformation is more general and has extra three degrees of freedom that describe the scaling and shearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Although these methods are sim- ple and easy to implement, they have the critical short- coming that they can capture only the global motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' It is often the case that the substructures in an astro- physical object changes non-uniformly across the field- of-view;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' for example, in an SNR, nonthermal filamentary shells tend to move outward from the center of explo- sion, while inner thermal substructures sometimes move inward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Moreover, often the velocities of the substruc- tures are different due to the difference in the actual three-dimensional distance from the center as well as the surrounding environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In such a complex case, simple rigid transformation is severely insufficient and a more flexible transforma- tion method is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The main requirements to the method are;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' (1) it should be able to capture the local non-uniformity of the motions across the field of view, and at the same time, that (2) it can express the smooth- ness of the motion field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The latter point is important because although each substructure moves rather inde- pendently, the motion of a pixel would be similar to its neighboring pixels as astrophysical objects such as SNRs evolve in time mostly continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' There have been several methods to realize such free- form image registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' For example, the combination of feature detection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', Lowe 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Rublee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2011) and feature matching is often used to find correspond- ing key points in the two given images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' However, X-ray images are usually dominated by the Poisson noises and image binning or smoothing is necessary to avoid detect- ing image fluctuations as feature points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' This worsens Spatio-temporal characterization of Cassiopeia A 3 the image resolution and thus is a significant drawback especially when the image is high-resolution such as the one taken with Chandra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Another option is optical flow (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', Lucas & Kanade 1981), the algorithm developed for motion tracking in computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' However, optical flow is designed to take into account only the local information, namely, each pixel is independently assumed to be moving to a certain direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Therefore, although this can be a choice when the motion of a local independent structure is focused on (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', Sato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2018), this is not the best when the global consistency of the whole system such as smoothness of the motion field should be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' B-spline registration (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 1996, 1997) is an im- age registration algorithm that has been applied to the motion analysis of medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The basic concept of the method is to express the deformation field of the image using a finite number of discrete control points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The control points are arranged in a latticed pattern and each one represents the local motion of its neighbor- ing coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The motion field of the points between the control points are interpolated using cubic B-spline functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Tlocal(x, y) = 3 � k=0 3 � l=0 Bk(s)Bl(t)φi+k,j+l, (1) where i = ⌊x⌋ − 1, j = ⌊y⌋ − 1, s = x − ⌊x⌋, t = y − ⌊y⌋ and Bk represents the kth basis function of the B-spline B0(t) = (1 − t)3/6 B1(t) = (3t3 − 6t2 + 4)/6 B2(t) = (−3t3 + 3t2 + 3t + 1)/6 B3(t) = t3/6, where 0 ≤ t < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' As the B-Spline representation of the motion field is continuous, differentiable, and bijective, this method is suitable for feature tracking of smoothly moving objects such as supernova remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We want to identify the best transformation within this modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' However, a caveat is that the defi- nition of the best transformation is somewhat vague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' It is straightforward to determine the best one when the ground truth deformation exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' that is, the best transformation should be the one that approximates the ground truth best within its expressive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' However, in the actual case, there is no ground truth deformation that transforms one observed image frame into another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Instead, there are just motions of astrophysical struc- tures – some move coherently, others independently –, and we want to express these collective motions by a parameterized transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Therefore, in this work, we define the best transforma- tion as the one that yields the most similar transformed image to the reference image, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', the one that mini- mizes a certain similarity metric predetermined by the analyzer of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In order to find the best transfor- mation, one only needs to minimize the similarity met- ric, which measures the difference between the reference image and the other image after deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' As this transformation is parameterized with a relatively small number of free parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', the motion vector asso- ciated with each of the control points, it is relatively easy to find the best parameters with common optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In some problematic cases, it is possible that the al- gorithm associates points in different images that are independent1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' It should be noted that this method im- plicitly assumes the smallest possible motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We ex- pect that this assumption applies to most astrophysical applications, including the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Region transformation Once the best transformation is found, the reference image and the other image after deformation are similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' It is thus expected that the region seg- mentation in the reference image generated by a physical motivation can also be used as a physically-motivated re- gion segmentation of the other image after deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Therefore, the regions in the other image’s original coor- dinates can be obtained by simply inverse-transforming the regions in the reference image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The resulting region segmentation should be physically motivated as is the one in the reference image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' DEMONSTRATION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Datasets The Advanced CCD Imaging Spectrometer (ACIS) of Chandra has observed Cas A multiple times since its launch in 1999 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Hwang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2000, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Uchiyama & Aharonian 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Pat- naude et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Patnaude & Fesen 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Sato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2017, 2018) and here we used the archival data from 2000 to 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The ObsIDs used for this work are sum- marized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We reprocessed the archival level 1 event lists produced by the Chandra pipeline in the standard manner using the CIAO software package (ver- sion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='12) and the CALDB (version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='1) to apply the 1 For example, when three points are arranged in a triangle and all the points are rotating clockwise at the angular speed of 119◦ per frame, the optimizer is likely to halt by finding 1◦ per frame anticlockwise motion because the optimization step starts from zero motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 4 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Ichinohe and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Sato Year ObsID Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' (ks) Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Date 2000 114 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='9 2000 Jan 30 2002 1952 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='6 2002 Feb 06 2004 4634 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='6 2004 Apr 28 4635 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0 2004 May 01 4636 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5 2004 Apr 20 4637 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5 2004 Apr 22 4638 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5 2004 Apr 14 4639 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0 2004 Apr 25 5196 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5 2004 Feb 08 5319 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='3 2004 Apr 18 5320 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='4 2004 May 05 2007 9117 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='8 2007 Dec 05 9773 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='8 2007 Dec 08 2009 10935 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='3 2009 Nov 02 12020 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='4 2009 Nov 03 2010 10936 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='2 2010 Oct 31 13177 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='2 2010 Nov 02 2012 14229 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='1 2012 May 15 2013 14480 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='8 2013 May 20 2014 14481 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='4 2014 May 12 2015 14482 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='4 2015 Apr 30 2016 18344 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='8 2016 Oct 21 19903 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='7 2016 Oct 20 2017 19604 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5 2017 May 16 2018 19605 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='4 2018 May 15 2019 19606 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='4 2019 May 13 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Chandra observations appropriate gain maps and the latest calibration prod- ucts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Cas A is observed as a complex mixture of thermal and non-thermal X-ray radiations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' For simplicity, we generated one image in the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='2–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0 keV band for each observation, resulting in fourteen images in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In this energy band, both thermal bremsstrahlung and non- thermal (synchrotron) radiation are present, and these featureless continuum radiations are dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Avoid- ing emission lines from various elements reduces infor- mation on specific elements (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', information on the el- emental abundance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' This allows us to focus only on thermal and non-thermal variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Implementation We implemented the cubic B-spline registration using SimpleITK toolkit (Lowekamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Beare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Yaniv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We employed an 8×8 mesh for the control points in the transform domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Each control point has two control parameters corresponding to the two components of the motion field vector there in the two-dimensional image, and the cubic B-spline interpo- lation requires three extra control points per dimension for the interpolation of the domain close to the image boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' These result in (8 + 3) × (8 + 3) × 2 = 242 parameters to optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We employed the L-BFGS-B algorithm implemented in SimpleITK for the metric op- timization (Liu & Nocedal 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Byrd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' For the optimization metric, the corre- lation of two images is used because the luminosity of Cas A is known to be gradually decreasing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', Pat- naude et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2011), and simpler, intensity-based metrics such as mean-squared error are not suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In this work, we use the image obtained in the 2004 observation as the reference image because it is of the highest quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We preprocessed all the fourteen images with a Gaussian filter of the kernel size σ = 4 px prior to the deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' For each of thirteen images other than the reference, the corresponding transformation was de- rived by comparing it with the reference image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Although all the images are directly compared to the 2004 image for deriving the transformations (one-shot strategy) in the present implementation, it should be noted that another strategy exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' the two-shot strat- egy, in which the transformation between every two suc- cessive frames is first computed, and the appropriate transformations are composed to reproduce the trans- formations equivalent to the one-shot counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We checked both strategies and found that both yield almost the same results, while some of the structures show more residual motions when the two-shot strategy is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We think this is due to the larger uncertainties in the determination of the transformation when using worse- quality images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' There are at least two competing factors that cause larger errors in the resulting transformations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' (1) larger time intervals and (2) much accumulation of the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In the present case, we think that the lat- ter effect dominates so that the one-shot strategy works better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' While the 2004 image has an exposure time of ∼1 Ms, other images have exposure times of only ∼50 ks (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In the two-shot strategy, most of the transfor- mations are computed using these worse-quality images, which would result in larger errors for each transforma- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In the present case, the motions of the structures are relatively smooth and slow, and we thus think the accumulation of the errors has a larger negative impact on the overall uncertainties compared to the larger time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' If, for example, the motions of the structures are fast, the time interval is very large, and/or the qual- ity of all the images is very high (or low), the former fac- tor could dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In such cases, the two-shot strategy would work better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Spatio-temporal characterization of Cassiopeia A 5 Relative deviation Relative deviation (Transformed) Transformation field original points transformed points 4 3 2 1 0 1 2 3 4 (F2019 F2004)/F2004 4 3 2 1 0 1 2 3 4 (F2019 F2004)/F2004 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Left and middle: relative deviation images computed by using (X − Y )/Y with X and Y being the 2019 and 2004 (reference) images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Left: created using the original images (without any transformations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Middle: the 2019 image is transformed to match the 2004 image best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Right: visualization of the transformation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The coordinates in the 2019 image represented by the black points are transformed to the coordinates represented by the red points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Note that, in the left and middle panels, most of the pixels outside the remnant show negative deviations because the 2004 image contains far fewer zero-count pixels due to the long exposure time (∼1 Ms, see also Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The left panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 1 shows the relative deviation image computed by using (X − Y )/Y , where X and Y is the images obtained in the 2019 and 2004 (reference) observations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The image shows clear fila- mentary structures aligned in a circular shape, reflecting the non-thermal filaments in Cas A, which are actually moving outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The image in the middle panel is com- puted by the same formula, in which the 2019 image de- formed using B-spline so as to match the 2004 image best is used as X, instead of the original 2019 image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The disappearance of the filaments in all directions clearly shows that the 2019 image is successfully deformed by the B-spline transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The right panel shows a vi- sualization of the transformation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The coordinates in the 2019 image represented by the black points are transformed to the ones represented by the red points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' It can be observed that in addition to the dominant radial motions of the filaments, the non-uniformity of their am- plitudes and directions are captured, demonstrating the effectiveness of this method on the registration problem of SNR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' To divide the field of view into subregions, we used the contour binning algorithm (Sanders 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We ap- plied the algorithm to the reference image with the pa- rameters sn=50, constrainval=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5, and smoothsn=15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The region segmentation in the reference image’s coordi- nates was converted into the ones in all the other images’ original coordinates by the respective inverse transfor- mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Once the regions are obtained in all the images, one can easily obtain the time-series property associated to any features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' one only needs to refer to the region ex- pressed in the coordinates corresponding to the image of interest without being bothered by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', the consistency of the regions among different time frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' For the demonstration purpose, we simply extracted the contin- uum X-ray flux in each region for all the time frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' As a result, ∼2000 light curves, each of which corresponds to a certain morphological structure in Cas A, were ob- tained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Using the simulated image frames, we tested the validity of the pipeline and conservatively estimated the systematic uncertainties in deriving the light curve values at ∼ 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' See Appendix A for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' For the current implementation, most of the calcula- tions take less than a few seconds using an Intel Xeon E5-1620 v4 CPU (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='50 GHz, 8-core).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The processes that can take longer time are the image registration and the inverse transformation of the region map into the target coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' With the image dimensions of 921×921, the typical time the former process takes ranges from about ten seconds to several tens of seconds per pair of images depending on the registration parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The latter takes ∼20 seconds per target frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Unsupervised clustering To characterize the thousands of light curves, we adopted the k-means clustering algorithm (Macqueen 1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' k-means is one of the unsupervised machine learning algorithms to perform the partition of N ob- servations contained in a given dataset into k clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' After successful classification, each element belongs to one of the k clusters whose center is the nearest to the element among the k centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The center of each clus- ter is calculated by the mean of the elements belonging to the cluster, which are expected to be similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We ran k-means with k = 4 with the Euclidean 6 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Ichinohe and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Sato metric (see Appendix B for the details regarding the choice of k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Before applying k-means, each of the light curves was normalized using the respective 2004 value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2 shows the result of k-means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The average light curve (black dashed line) of each cluster represents the rough trend of the light curves belonging to the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Namely, the X-ray flux of the regions belonging to the clusters 0, 1, 2, and 3 is increasing, mildly decreasing, quickly decreasing, and staying constant, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Visualization The resulting product of all the above procedures is a set of light curves, grouped by time variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' As each light curve represents a certain part in the object, one can convert this information into the spatial distribu- tion map of X-ray time variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The top right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 3 shows the spatial distribution map of the clus- tering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The middle and bottom rows of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 3 show the Cas A images filtered by the cluster member- ships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The components whose flux are increasing (clus- ter 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' middle left) or rapidly decreasing (cluster 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' bot- tom left) are distributed in clumpy morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' On the other hand, the components whose flux are gradually decreasing (cluster 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' middle right) or constant (cluster 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' bottom right) are distributed in diffuse morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Although both clumpy, the distribution of the bright- ening (cluster 0) and rapidly dimming (cluster 2) com- ponents are different;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' the former is rather localized in the northwestern direction, while the latter is rather dis- tributed uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The bright compact clumps tend to belong to the cluster 2, while the cluster 0 contains fila- mentary structures of moderate luminosity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Regarding the diffuse components, the gradually dimming compo- nent (cluster 1) is apparently brighter than the constant luminosity component (cluster 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The regions covered by the cluster 1 extend to the radii of non-thermal shells, while the cluster 3 only covers the region inside the shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' DISCUSSION In the previous section, we have shown the concept of the combination of the B-spline registration and the k- means clustering algorithm in characterizing the evolv- ing diffuse structures, and demonstrated the effective- ness of the concept using the data of Cas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' To our knowledge, this is the first comprehensive characteriza- tion of such a dynamic diffuse target both in spatial and temporal viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In this section, we show that the method is not a mere tool for automation by demon- strating that the clusters thus obtained are actually sci- entifically interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We also discuss the advantages, caveats and future prospects of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Physical interpretation of each cluster We have identified four clusters showing different time variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' These can be broadly classified into two types of variations – thermal or non-thermal emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Here, we discuss the origin of the flux variations using the typical timescale of each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 4 show the spatial distribution maps of the ratio of the flux in 2019 and 2000, plotted for each of the clus- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We can see that the regions with flux increase and decrease between 2000 and 2019 were characterized well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Interestingly, the different rates of variation can also be visualized (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', between the clusters 1 (deep blue) and 2 (light blue)), suggesting that different physics is in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Generally, non-thermal emissions in SNRs are ob- served as small-scale filamentary structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', Bamba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2003, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' This is because the non-thermal X- rays from the accelerated cosmic-ray electrons are emit- ted from small regions that are compressed by shock waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Therefore, the small-scale knotty/filamentary structures seen in the clusters 0 and 2 would probably be associated with the non-thermal phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In ad- dition, these clusters show fast variability with ∼ +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5 % yr−1 (e-folding time τinc ∼ 17 yr) for the cluster 0 and ≳ −4 % yr−1 (e-folding time τdec ∼ 49 yr) for the clus- ter 2, which could be related to rapid acceleration and cooling in an amplified magnetic field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', Uchiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Uchiyama & Aharonian 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Patnaude & Fesen 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The synchrotron cooling timescale τsyn and the acceleration timescale τacc, which are e-folding timescales, can be estimated as τsyn ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='7 B−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5 mG ϵ−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5 5keV year (2) τacc ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='8 η B−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5 mG ϵ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5 5keVV −2 sh,5 year, (3) where BmG, Vsh,5, ϵ5keV and η are the magnetic field in units of 1 mG, the shock velocity in units of 5,000 km s−1, the photon energy in units of 5 keV, and the Bohm factor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We found that the amplified magnetic field of B ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='12 mG and the Bohm factor of η ∼ 3 reproduce the observational timescales well, which agrees with the previous estimations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', Sato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Tsuji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Thus, we conclude that most of the clusters 0 and 2 are related to the non-thermal phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We note that our method is also useful for identify- ing peculiar structures, such as inward-moving shocks in this remnant (Sato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' These structures can be interpreted as signatures of the interaction of the rem- nant with an asymmetric dense circumstellar shell that occurred between ∼180 and ∼240 yr after the super- nova event (Vink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Orlando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2022), thus it would be important for understanding the progeni- Spatio-temporal characterization of Cassiopeia A 7 0 2 4 6 8 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0 Cluster 0 83 regions 0 2 4 6 8 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='6 Cluster 1 1235 regions 0 2 4 6 8 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='4 Cluster 2 412 regions 0 2 4 6 8 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0 Cluster 3 587 regions Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The results of k-means (k = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Each panel corresponds to a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In each panel, the colored lines exemplify the light curves belonging to the cluster, and the black dashed line and the gray band correspond to the average light curve and standard deviation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' tor’s activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We found that these structures, classified as clusters 0 and 2 in the southern and western regions, move differently from most of the structures that move outward, but were successfully tracked by our method (see the rightmost panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 1 and the movie corre- sponding to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' While there are also several bright small-scale structures classified as clusters 0 and 2 in the northwestern reverse-shock region, they differ from the inward-moving structures in that they are moving out- ward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' This suggests that there is not enough material in the northwest direction to drive the shock wave inward, and may support the recently proposed asymmetric cir- cumstellar shell for Cas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In addition to these struc- tures, our method identified a forward-shock filament that is getting brighter in the northeastern edge of the remnant (see the top left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' It has been suggested that diffusion in this filament appears to be occurring near the Bohm limit (Stage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2006), which implies efficient cosmic-ray acceleration at the location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Although the interpretation of the cause of the flux vari- ation is outside the scope of this paper, our analysis in- dicates that the flux increase in this filament reported in Patnaude & Fesen (2009) has continued since then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Diffuse components that are gradually dimming with a rate of ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='4 % yr−1 were classified into the clus- ter 1, which could be related to the thermal emission of Cas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In particular, the regions classified into this cluster appear to be concentrated in the northern and southeastern reverse shock regions where the thermal emission is dominant (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', DeLaney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Helder & Vink 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Assuming that this cluster represents the thermal radiation, we here discuss the time evolution of the thermal component in the remnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The ther- mal emissions in young supernova remnants are thought to decrease due to adiabatic expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In Sato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' (2017), the decay rate of the thermal X-rays in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='2–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0 keV by the adiabatic expansion in Cas A was estimated to be −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='36 (m/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='66)(t/340 yr)−1 % yr−1, where m and t is the expansion index and the age of the remnant, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' This rate explains well the temporal varia- tion of the cluster 1, thus we conclude that this cluster would be related to the thermal emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' On the other hand, we note that the non-thermal emissions dimming with a longer timescale (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', cooling at a less amplified magnetic field) would also be classified into this cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Faint regions with less time variability were classified into the cluster 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We consider that acceleration (or heating) and cooling are balanced in these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Most of the regions in this cluster are located at the south- western reverse shock regions and the forward shock re- gions, where the non-thermal emission is dominant (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', DeLaney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Helder & Vink 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Grefenstette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' If cosmic-ray electrons were regularly accel- erated and cooled there, this would explain the constant luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In addition, materials are constantly being heated in the vicinity of the shock wave, which may off- set the decay of the thermal component due to adiabatic expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' These balanced components could have been classified as the cluster 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Advantages, caveats and future prospects As shown in the previous section, we found that each of the cluster represents a clear physical meaning dis- tinct from other clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' This means that the physical motivation in each cluster is aligned and thus that our methodology can extract the physics underlying a dy- namic diffuse object in an organized manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Notably, all the processes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', the region segmentation, image registration, and clustering, were performed automat- ically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' This is already a huge advantage that lead to the first comprehensive characterization of Cas A, which have been practically impossible to do manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Such an automated extraction of physics would become more important when the future powerful missions such as Athena are in operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Most of the future missions, including Athena, will have worse angular resolutions than Chandra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Although worse angular resolutions cause worse image quality, we 8 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Ichinohe and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Sato Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Visualization of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Top left: the Cas A X-ray images in the original coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Top, second from left: the same images in the transformed coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Top, third from left: the spatial distribution map of the clustering results in the 2004 coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Top, right: the same maps inverse-transformed onto the target frames’ coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Middle row: the Cas A X-ray images filtered by the clusters, shown in the transformed coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The leftmost, second, third, and rightmost panels show the regions in Cas A, whose light curves are classified as clusters 0, 1, 2, and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Bottom row: the same images as in the middle row, shown in the original coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The contours represent the X-ray surface brightness at 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The animation (14 frames, corresponding to the years shown in Table 1) shows how the overall morphology of Cas A evolves with time and how this morphological evolution is seemingly suppressed by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='2004 (original coordinates) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='2004 (transformed coordinates) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='Cluster label map (2004 coordinates) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='Cluster label map (inverse-transformed) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='Cluster o (transformed coordinates) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='Cluster 1 (transformed coordinates) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='Cluster 2 (transformed coordinates) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='Cluster 3 (transformed coordinates) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='Cluster o (original coordinates) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='Cluster 1 (original coordinates) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='Cluster 2 (original coordinates) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='Cluster 3 (original coordinates)Spatio-temporal characterization of Cassiopeia A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='Cluster 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='Cluster 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='Cluster 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='Cluster 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='log(F2019/F2000) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The spatial distribution maps of the ratio of the flux in 2019 (F2019) and 2000 (F2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In each panel, only the component classified as the respective cluster is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' think the registration should still work if the object ap- pears to change its shape under that quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' If the angu- lar resolution is so poor that actually moving structures do not appear to be moving, our method is just un- necessary, and one can simply take identical sets of the regions for all the time frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In general, when the an- gular resolution is poor, one needs to be careful in deter- mining the regions to take the PSF (point spread func- tion) blending effect into consideration properly, and it is probably required to tune the hyperparameters of the tessellation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' That said, we emphasize that this is the issue in the tessellation algorithm but not in the registration algorithm presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The concept of our method is fairly simple — using image registration technique to convert dynamic objects into static ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Therefore, at least conceptually, our method can be applied to a wide range of other dy- namic diffuse targets, regardless of the instrument (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', 10 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Ichinohe and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Sato Chandra or XMM-Newton;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' it does not even have to be X-ray instruments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' That said, there are cases that our method would not work properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' For example, when the neighboring image frames are not very similar, im- age registration would fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' This happens, for example, when the structures are moving too fast compared to the time separation of two images or when the motion is chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Low quality of images (low statistical counts, high noise, or existence of instrumental artefacts) would of course cause the fail of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' When the con- trol points are too sparse compared to the spatial scale of the motion, this method might also fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The cases include the motions whose scale is substantially sharper than the mesh grid, and the motions which are highly non-uniform within the neighrboring of a single control point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' This behavior is also observed in our result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' some filamentary structures close to the center show residual motion after transformation (see the movie correspond- ing to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Although employing denser control points should resolve the problem, this leads to the increase of the number of parameters to optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Therefore careful trade-off study may be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In this demonstration, we used only the total luminos- ity of the continuum (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='2–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0 keV) band for both image transformation and the subsequent clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Although we leave advanced analysis for future works, we note that it would be easy to extend this method to incor- porate more information contained in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' For example, by extracting lightcurves from multiple energy bands, it would be possible to characterize the time evo- lution of the object from the time evolution of spatial distribution of metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' By using high-resolution spec- troscopic data using future instruments, one can trace the evolution of line-of-sight velocity of moving struc- tures to constrain 3D shock dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Ultimate goal would be using the entire spectral information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Char- acterizing a huge amount of complex spectra is itself a significant task and several progresses have been made (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', Iwasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We think that combining our method with such methods based on data science would facilitate the analysis of future high-quality astronomi- cal datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' CONCLUSIONS We have developed the strategy to track the time- series properties of all the parts constituting a diffuse structure by introducing the free-form image registra- tion technique based on B-spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We have demon- strated the methodology using the Chandra data of Cas A by extracting the spatial distribution map of the time variability of continuum luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' To our knowl- edge, this is the first comprehensive characterization of such a dynamic diffuse target both in spatial and tem- poral viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We have found that each of the four clusters derived by applying k-means algorithm to the extracted light curves has a clear physical meaning distinct from other clusters, which shows that our method is not a mere technique for automation but capable of capturing the underlying physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' This work was partially supported by the Grants-in- Aid for Scientific Research by the Japan Society for the Promotion of Science with KAKENHI Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 18H05458, 19K14749, 20K14524, and 20K20527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' TEST OF THE PIPELINE In this section, we present the results of the exper- iment to test the validity of the presented algorithm and to estimate the systematic uncertainties in the light curve reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We simulated 50 movies consisting of four 50 px×50 px mock image frames and randomly generated four struc- tures per movie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Each structure is either a ‘clump’ or a ‘filament’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' For a given frame number i (i = 0, 1, 2, 3), the shape of a clump is computed by a two-dimensional Gaussian with the center location, normalization, and radius calculated by (x0 +i∆x,y0 +i∆y), n0 +i∆n, and σ, respectively, using the values presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The shape of a filament is a line segment smeared by a two-dimensional Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The center location, rota- tion, normalization, Gaussian radius, and line length are calculated by (x0 + i∆x,y0 + i∆y), θ0 + i∆θ n0 + i∆n, σ, and l, respectively, using the values presented in Ta- ble 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The left two columns of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 5 show examples of the first frames of the movies and the corresponding last frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Using the pixel values corresponding to each structure, we defined three regions for each structure by selecting the pixels whose value is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='1, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='01 times the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Larger threshold values correspond to smaller region sizes, as exemplified in the right two columns of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We found that sometimes those ran- domly generated structures overlap, which results in the regions not being properly determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' To avoid such Spatio-temporal characterization of Cassiopeia A 11 situations, we only kept the random trials that did not cause the overlap of regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Using the region maps, we followed the same proce- dure presented in Section 2 to transform the region maps into the coordinates of target frames and extracted the sum of the pixel values corresponding to the transformed regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 6 shows the ratios of the reconstructed normalization values to the ground truth normalization values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The errors correspond to the standard deviation of the derived values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The ratios are consistent with unity, suggesting that the light curves are recovered correctly by our strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We found that the scatter of the reconstructed values increases with frame numbers, which can be naturally interpreted as that it is easier to perform image regis- tration between two similar frames than two rather dif- ferent frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We also found that the scatters are larger for higher threshold values (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', smaller regions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' This is because the errors in the deformation cause the mis- alignment of structures and the corresponding regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The systematic errors introduced by this effect should be larger for smaller regions because the relative error of the transformation field to the region size becomes larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' It is difficult to quantitatively estimate how much er- ror is caused by how the structures in different frames differ because other factors, such as the distance and flux ratio to the surrounding structures, would also af- fect the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' However, considering that our sim- ulation includes the structures whose motion scales are as large as ∼ 1/10 of the frame size (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', the filaments in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 5), which is way larger than the structure motions existing in Cas A, we think the overall systematic errors due to the reconstruction accuracy is sufficiently below the error bars presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 6, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', a conservative baseline value is ∼ 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We expect that such a level of error does not significantly affect the k-means algorithm (which resulted in four clusters of rather distinct trends, see Fig 2) and the following discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' NOTE ON THE K-MEANS HYPERPARAMETERS In this work, we ran k-means with k = 4 with the Eu- clidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' In determining the number of clusters, we checked the performance metrics of the k-means al- gorithm, such as the silhouette score (Rousseeuw 1987), the elbow method, and the Bayesian information crite- rion (BIC, Schwarz 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Pelleg & Moore 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Al- though all the methods suggested the optimal number of around k = 3 − 5, we found it difficult to choose one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' We think this is because the light curve properties are relatively smoothly distributed, and there are no clear boundaries between the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Therefore, we ar- bitrarily chose k = 4 because we found that k = 5 yields a cluster containing only ∼30 samples, and that k = 3 failed to separate the brightening component from the constant luminosity component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' For a situation like this, k-means is probably not the optimal algorithm, and other algorithms that are capa- ble of characterizing such a smoothly distributed dataset would be more suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Exploring the k-means hy- perparameters (which include not only the number of clusters, but also e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', the distance measure and how to calculate the center) and/or the optimal algorithm is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' REFERENCES Bamba, A.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Sato Parameter Distribution (clump) Distribution (filament) x0, y0 Uniform(5, 45) Uniform(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5, 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='4) ∆x, ∆y (frame−1) Uniform(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0) Uniform(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0) θ0 (deg) N/A Uniform(0, 360) ∆θ (deg frame−1) N/A Uniform(-10, 10) n0 Uniform(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5) Uniform(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5) ∆n (frame−1) Uniform(-n0/6, n0) Uniform(-n0/6, n0) σ Uniform (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5) Uniform (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='5) l N/A Uniform(5, 15) Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Parameter distribution to generate random structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Trial 0 Initial frame (i=0) Last frame (i=3) Binmap (thr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='3) Binmap (thr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='01) Trial 1 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Examples of the first and last frames of randomly generated mock movies (left two columns) and the corresponding region maps with the threshold values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='01 (right two columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' The first frames are presented in the left column, and the corresponding last frames in the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Top row: three clumps and a filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Bottom row: a clump and three filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' 1 2 3 Frame number i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='1 Flux relative to i = 0 Reconstructed / Ground Truth Clump, thr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='3 Clump, thr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='1 Clump, thr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='01 Filament, thr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='3 Filament, thr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='1 Filament, thr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='01 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' ˆni,recon/ˆni,GT, the ratio of the reconstructed nor- malization value (ˆni,recon) to the ground truth normalization value (ˆni,GT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=' Before calculating the ratio, all the values are normalized by the corresponding normalization value at the first frame (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNA0T4oBgHgl3EQfE_93/content/2301.02026v1.pdf'} +page_content=', ˆni ≡ ni/n0).' metadata={'source': 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b/WNE0T4oBgHgl3EQfVwAY/content/tmp_files/2301.02267v1.pdf.txt @@ -0,0 +1,1925 @@ +Identifying modified theories of gravity using binary black-hole ringdowns +Costantino Pacilio∗ +Dipartimento di Fisica “G. Occhialini”, Universit´a degli Studi di +Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy and +INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy +Swetha Bhagwat† +Institute for Gravitational Wave Astronomy & School of Physics and Astronomy, +University of Birmingham, Edgbaston, Birmingham B15 2TT, UK +(Dated: January 9, 2023) +Black-hole spectroscopy, that is, measuring the characteristic frequencies and damping times of +different modes in a black-hole ringdown, is a powerful probe for testing deviations from the general +theory of relativity (GR). In this work, we present a comprehensive study on its ability to identify +deviations from the spectrum of a Kerr black hole in GR. Specifically, we investigate the performance +of black hole spectroscopy on a diverse set of theoretically motivated as well as phenomenologically +modified spectra. We find that while the signal-to-noise ratio ρRD in the ringdown required to iden- +tify a modification to the GR Kerr black hole spectrum depends on the details of the modifications, +a modification that introduces ∼ 1% shift in the fundamental mode frequencies can typically be dis- +tinguished with ρRD ∈ [150, 500]. This range of ρRD is feasible with the next-generation detectors, +showing a promising science case for black hole spectroscopy. +I. +INTRODUCTION +Gravitational waves (GWs) with characteristic fre- +quencies and damping times are radiated as the distorted +black hole (BH) formed during a binary BH merger re- +laxes into its final stable state. This signal is called the +ringdown and comprises of a linear superposition of the +spectral modes of the BH, known as the quasi-normal +modes (QNMs). +We can obtain the QNMs by solving +the BH perturbation equations [1–3] and the ringdown +signal can be used to validate dynamics in linear strong +field regime. The QNM spectra of a perturbed BH in the +general theory of relativity (GR) are obtained by solving +Teukolsky’s equation [4] and under the Kerr hypothesis +[5, 6] i.e., that the remnant BH in binary BH coalescence +relaxes to a Kerr BH. +BH spectroscopy [7], defined as measuring QNM spec- +tra from ringdown signals, allows to put forth consistency +tests of the joint hypotheses that +1. the asymptotic equilibrium state of the remnant is +described by the Kerr metric a.k.a., the Kerr hy- +pothesis and +2. the dynamics of the perturbed Kerr BH is governed +by Teukolsky’s equation, i.e., (linearized) GR dy- +namics. +Further, BH spectroscopy can observationally validate +the no-hair theorem obeyed by BHs in GR; it demands +that all aspects of a Kerr spacetime, including its QNM +spectrum, be fully characterized by just two parameters. +∗ costantino.pacilio@unimib.it +† sbhagwat@star.sr.bham.ac.uk +Often and most naturally, the two parameters are chosen +as the mass Mf and spin χf of the BH. If more than two +QNM parameters are measured, it can be verified that +different pairs of QNM parameters solve for the same +Mf and χf. This allows us to perform a null test of the +no-hair theorem. +There has been much focus in the literature on the fea- +sibility of measuring the subdominant QNM modes and +performing null tests to validate the underlying theory +of gravity as GR with BH spectroscopy [8–20]. In this +study, we concentrate on a complementary aspect and +investigate the ability of BH spectroscopy to identify de- +viations from GR when the spectrum is not described by +the GR Kerr QNMs. We perform a comprehensive study +utilizing the publicly available QNM spectra in various +modified theories as well as two phenomenological modi- +fications and assess the performance of BH spectroscopy +to differentiate them from a GR Kerr BH spectra. We +then investigate the signal-to-noise ratio ρRD in the ring- +down at which different modified theories of gravity can +be distinguished from GR using BH spectroscopy. We +find that the required ρRD depends on the details of the +QNM spectra in a given theory and on their degeneracies +with the GR Kerr BH spectra in the mass-spin space. +However, at a broad level, we observe that ρRD ≥ 150 +is required to confidently identify modified theories that +produce ≤ 1% deviation in the dominant mode from GR +using BH spectroscopy (c.f., [8, 9] for a detailed study on +expected ρRD for measurability of QNM parameters with +the next-generation GW detectors). +The remainder of this paper is organized as follows. In +Section II we outline the conceptual structure adopted +for this study to test the no-hair theorem using BH spec- +troscopy. +In Section III, we detail the modified QNM +spectra used in this study. Then, in Section IV we sum- +marize the setup and implementation used to perform +arXiv:2301.02267v1 [gr-qc] 5 Jan 2023 + +2 +this study. This is followed by the results in Section V +and a discussion of their implications in Section VI. +II. +TESTING THE NO-HAIR HYPOTHESIS +WITH BH SPECTROSCOPY +The ringdown waveform observed at asymptotic infin- +ity can be approximated as a linear superposition of a +countably infinite set of (complex) QNMs with +ωlmn = 2πflmn − i/τlmn. +(1) +Here flmn and τlmn are the characteristic frequencies and +damping times of the spectral modes. +(l, m, n) index +the mode’s angular, azimuthal and overtone numbers. +As in any perturbation theory, the excitation amplitude +of the modes depend on the initial perturbation condi- +tions; for a binary BH merger these are set largely during +the plunge-merger phase. A quasi-circular merger excites +(2, 2, 0) dominantly, and depending on the initial binary +BH’s mass ratio and spins, the most prominent subdom- +inant angular modes can be {(3, 3, 0), (2, 1, 0), (4, 4, 0)} +[21–28]. +To outline our setup, let us consider a ringdown where +more than two QNM parameters are measurable, and +a case where we have identified its QNM indices. The +minimum ρRD required for this has been investigated in +studies such as [24, 29–31]. If the underlying theory of +gravity is GR and if the Kerr hypothesis holds, we can +invert any pair of QNM parameters, preferably the fre- +quency and damping time of the dominant mode, to infer +the mass and spin of the BH – +{f220, τ220} → {M Kerr +f +, χKerr +f +}. +(2) +From this mass and spin estimate, we can compute the +full set of QNM spectra of the Kerr BH. Let f (infer) +lmn +be +the inferred subdominant mode frequency +{M Kerr +f +, χKerr +f +} → f (infer) +lmn +. +(3) +Here we use the superscript (infer) to differentiate +f (infer) +lmn +from f (meas) +lmn +which are the frequencies mea- +sured from the ringdown signal. While a similar argu- +ment holds for QNM damping times, subdominant mode +damping times are poorly measured [8, 22, 32] and there- +fore, we focus on tests using solely the subdominant mode +frequencies. +A null test can be performed by checking if the rel- +ative difference between the inferred and the measured +quantity is compatible with zero. We define the relative +difference as +δflmn = f (meas) +lmn +− f (infer) +lmn +f (infer) +lmn +. +(4) +We can infer steps (2)-(4) through a convenient +reparametrization of the waveform during the parame- +ter estimation. We briefly summarize this and point the +reader to a detailed treatment in [33]. +A generic modified QNM spectrum can be phenomeno- +logically written as +flmn = f Kerr +lmn (Mf, χf)(1 + δflmn) , +(5a) +τlmn = τ Kerr +lmn (Mf, χf)(1 + δτlmn) , +(5b) +where {Mf, χf} are the true values of the final mass +and spin, and {δflmn, δτlmn} are the relative shifts of +the spectrum w.r.t. the QNM spectra of a GR BH. +{δflmn, δτlmn} can be non-trivial functions of {Mf, χf} +and of the physical parameters of the modified theory +such as the additional coupling constants or charges. At +this stage of setting up the formalism, we do not differen- +tiate between a modification to the underlying theory of +gravity and a modification in the nature of the compact +object. +Now, notice that {δf220, δτ220} are redundant param- +eters because we can always find a pair { ˜ +Mf, ˜χf} of ef- +fective final mass and spin that satisfy +f220 = f Kerr +220 ( ˜ +Mf, ˜χf) , +(6a) +τ220 = τ Kerr +220 ( ˜ +Mf, ˜χf) . +(6b) +The subdominant modes can be re-expressed as +flmn = f Kerr +lmn ( ˜ +Mf, ˜χf)(1 + ˜δflmn) , +(7a) +τlmn = τ Kerr +lmn ( ˜ +Mf, ˜χf)(1 + ˜δτlmn) , +(7b) +for (lmn) +̸= +(220). +Further, +the effective shifts +{˜δflmn, ˜δτlmn} satisfy +f Kerr +lmn (Mf, χf) (1 + δflmn) = f Kerr +lmn ( ˜ +Mf, ˜χf)(1 + ˜δflmn) , +(8a) +τ Kerr +lmn (Mf, χf) (1 + δτlmn) = τ Kerr +lmn ( ˜ +Mf, ˜χf)(1 + ˜δτlmn) . +(8b) +For the QNM spectrum of a Kerr BH in GR, +{δflmn, δτlmn} vanish; therefore { ˜ +Mf, ˜ +χf} = {Mf, χf} +and {˜δflmn, ˜δτlmn} vanish. We set up our framework to +identify departure from the GR Kerr BH QNM spectrum +by constraining the effective shifts away from zero. +Note that the mass and spin appearing in Eq.s (2)-(3) +are not the true values {Mf, χf} but rather the effec- +tive values { ˜ +Mf, ˜χf}. We emphasise that we can only +measure the effective final mass and spin, and not the +true values corresponding to the BHs. While developing +a framework for observational test, the effective (mea- +sured) parameters deviations ˜δ’s are the instrumental +variables 1. Similarly, the magnitudes of the δ’s are not +1 Note also that if one is not interested in testing the no-hair the- + +3 +directly accessible in BH spectroscopy and we can only +estimate ˜δ’s. In Section III, we inspect modified QNM +spectra and show that ˜δ can be significantly different +from δ. +III. +MODIFIED QNM SPECTRA +In this study, we quantify the ability of BH spec- +troscopy to constrain ˜δflmn away from zero for various +class of modifications to the Kerr BH spectrum (c.f., [35– +38] for other works on BH spectroscopy in the context of +modified theories of gravity). Given the lack of a best- +candidate theory for modified gravity and the fact that +QNM spectra in modified theories are available in a very +few theories, of which even fewer theories have QNMs +computed at a beyond-leading order in BH spins [39, 40], +we consider both publicly available modified spectra and +phenomenologically modified spectra. +The spectra are chosen to encompass a variety of mod- +ifications to stress-test the ability of spectroscopy to dis- +tinguish them from a GR Kerr spectrum. We don’t con- +cern ourselves with the physical plausibility of these mod- +ifications. Below we describe the modifications to the GR +Kerr BH spectrum used in this study: +• EdGB: The Einstein-dilaton-Gauss-Bonnet theory +[41, 42] is a modified theory of gravity that intro- +duces a dilaton scalar field that is non-minimally +coupled to higher orders of the curvature, specifi- +cally to the Gauss-Bonnet invariant. The BHs in +EdGB have a scalar hair as they are endowed with +a monopole scalar charge. +However, this charge +is not an independent parameter but it is a “sec- +ondary hair” [43], i.e., it is completely determined +by the mass and spin of the BH and by the coupling +constants of the theory. The QNMs of EdGB BHs +at the next-to-leading order in the spin are derived +in [40]. The numerical approximations in [40] re- +strict the validity of the spectrum to ζEdGB ≲ 0.4, +where ζEdGB = αEdGB/M 2 +f and here αEdGB is the +coupling constant of the theory. +Further, the fi- +nal spins χf ≳ 0.3 can be potentially outside the +range of validity of the O(χ2 +f) expansion. To miti- +gate these effects, we consider the Pad´e resummed +version of the spectrum provided in [40]. Note that +the EdGB QNMs break isospectrality [44] between +axial and polar sectors due to the non-minimal cou- +pling of the scalar field. In this work, we choose to +focus on the spectrum corresponding to the polar +sector. +orem, the signals can be analyzed by assuming the Kerr BH +spectrum (i.e., +setting all δ’s to zero) and recovering posterior +estimates of the mass and spin. The posteriors so obtained will +generally differ from the posteriors of { ˜ +Mf, ˜χf}. This means that +{ ˜ +Mf, ˜χf} obtained here cannot be used to gauge the performance +of tests like the Inspiral-merger-ringdown test [34]. +• Kerr-Newman: +The Kerr-Newman spectrum for +GW perturbation is derived in [45, 46] at first order +in the final spin expansion (c.f., [47] for a pertur- +bative expansion in the electric charge). There is +strong numerical evidence that the Kerr-Newman +spectrum is isospectral, which is confirmed by the +full non-perturbative analysis in [39]. +Therefore, +unlike in the EdGB, there is no ambiguity in choos- +ing polar or axial sectors in its modified spectrum. +Note that a Kerr-Newman BH becomes extremal at +charge-to-mass ratio Q = (1 − χ2 +f)1/2 but we only +consider values of Q away from this limit. +• Horndeski: The Horndeski action gives a general +scalar-tensor gravity with second order equations +of motion [48]. The Horndeski field equations ad- +mit standard GR BH solutions under various con- +ditions [49]. +Linear perturbations around slowly +rotating Kerr BHs were studied in [50] for the sub- +class of Horndeski theories in which GW propagates +at the speed of light. They show that the equations +are reduced to a massive scalar perturbation with +an effective mass parameter µ. Although the spec- +trum does not correspond to perturbations in GW +sector, in principle, it can be sourced by the GW +sector, and we expect imprints of these frequencies +in the GW signals. In this work, we only look at the +QNMs in scalar sector presented in Eq.s (34)-(35) of +[50]. Note that this spectrum reduces to the scalar +perturbations of the Kerr BH in the limit µ → 0; +therefore, to augment our battery of modifications, +we linearly re-scale it to recover the gravitational +GR Kerr BH spectrum in the limit µ → 0 and pro- +mote it as yet another modified spectrum. We re- +mind that for this work, we are interested in study- +ing the performance of BH spectroscopy to distin- +guish a non-Kerr GR spectrum and do not aim to +put bounds on any given modified theory/spectrum +in particular. +• dCS: In dynamical Chern-Simons (dCS) theory +[51], a scalar field is non-minimally coupled to the +higher-curvature Pontryagin invariant, resulting in +a breakdown of parity symmetry. +Rotating BHs +in dCS have a secondary hair in the form of a +monopole scalar charge [52]. The QNM spectrum +of dCS BHs were computed in [53] at the leading +order in the spin and at second order in the non- +minimal coupling constant of the scalar field αdCS +(see also [54]). In the following we will use the di- +mensionless coupling ζdCS = α2 +dCS/M 4 +f . Here, we +are extrapolating the spectrum in [53] beyond the +small spin approximation but this is not a critical +concern for our study. Also, the non-minimal cou- +pling of the scalar field breaks isospectrality and +therefore, similar to EdGB we consider the polar +sector of the dCS spectrum. +• Delta: We generate an ad-hoc phenomenological + +4 +spectrum by modifying the frequencies of all modes +by a constant relative shift, flmn = f Kerr +lmn (1 + ∆). +We also choose to leave all damping times un- +changed. +• Delta220: We modify only the the frequency of the +dominant mode f220 = f Kerr +220 (1 + ∆220) and leave +all other mode frequencies and damping times un- +changed. +The above scenarios are distinct modifications to the +GR Kerr BH spectra where the no-hair hypothesis can +be violated. While Kerr-Newman BHs deviate from the +Kerr background due to the presence of a “primary hair” +(the electric charge), in the case of EdGB and dCS +BHs the background possesses a “secondary hair” (the +monopole scalar charge). Further in the Horndeski BHs +we consider here, the background coincides with Kerr. +Moreover, note that in all the theories here, the devia- +tions from GR appear also at the level of the field equa- +tions. +The Delta and Delta220 are simplistic ad-hoc unrealis- +tic modification schemes; in a realistic physical scenario, +we typically expect all or at least a subset of frequen- +cies and damping times to be modified and it is unlikely +that all modes are modified by the exact same amount. +We study them as they are simpler to implement and +to interpret the performance of BH spectroscopy and as +a benchmark. +It also allows us to check if any of the +realistic spectra can be approximated as simpler ad-hoc +modifications. +For the EdGB, Kerr-Newman, and dCS spectra, we +opt to impose consistency with the Kerr BH spectrum by +linear rescaling, similar to the Horndeski case described +above. If we take the limit of ζEdGB → 0 for the EdGB +spectrum in [40], we do not recover the GR Kerr BH +QNMs (which would be the case if the EdGB QNMs +could be computed non-perturbatively) because the spec- +trum is derived at O(χ2 +f). Additionally, each spectrum is +derived within its own set of approximations, and there- +fore, they return a different approximation to the GR +Kerr BH spectrum in the limit of the vanishing deviation +parameters. +We provide a procedure for imposing consistency +across the spectra by redefining the spectra. We illus- +trate our procedure on EdGB below – +f EdGB +lmn +(Mf, χf, ζEdGB) += f Kerr +lmn (Mf, χf) +� ˆf EdGB +lmn +(Mf, χf, ζEdGB) +ˆf EdGB +lmn +(Mf, χf, 0) +� +(9) +We enforce the GR Kerr BH spectrum in the limit +ζEdGB → 0. +Here, the hat denotes the expression of +EdGB spectrum presented in [40]. Then, we defined the +relative shifts as – +δflmn = +ˆf EdGB +lmn +(Mf, χf, ζEdGB) +ˆf EdGB +lmn +(Mf, χf, 0) +− 1 +(10) +and parametrized +f EdGB +lmn += f Kerr +lmn (1 + δflmn) . +(11) +The definition (10) preserves the values of the shifts given +by the traditional parametrization of the spectrum. We +also repeat the same procedure for the damping times. +Here, we re-emphasis that this procedure is only neces- +sary because the QNM spectra in these modified theories +are calculated perturbatively to a limited order in the fi- +nal spin of the BH. In the absence of the exact spectra, +we use Eq. (11) as a fiducial definition for all the modified +spectra considered here. +Finally, in Fig. 1 we plot the relative deviations δflmn +and effective deviations ˜δflmn in the QNM frequencies +for the modified spectra listed above. We remind that it +is the effective deviations ˜δflmn that are measured when +performing BH spectroscopy. Interestingly, we find that +for some spectra even when the actual spectrum has de- +viations at a percent level, the measurable effective spec- +trum deviates from GR Kerr BH at a much smaller sub- +percent level. This is particularly evident for the Delta +spectrum where all the true frequencies are shifted by +the same amount but the measurable deviations turn +out to be much smaller. We also see this in the Kerr- +Newman spectrum. For each spectrum and mass ratio, +we quantify these differences in Tab. I. We present the +values of the deviation parameters α such that the devi- +ation in the dominant mode frequency is at 1% level i.e., +|δf220| = 0.01. In Sec. V, using the values in Tab. I, we +study the ability of BH spectroscopy to constrain ˜δflmn +away from zero i.e., exclude the GR Kerr BH spectrum. +IV. +METHODS AND IMPLEMENTATIONS +We construct ringdowns that comprise of (l, m, n) ∈ +{(2, 2, 0), (3, 3, 0), (2, 1, 0)} +modes +with +the +modified +QNM spectra described in the previous Section 2. We +remind that the coupling parameters of the theory or +the extra charges that modify the QNM spectra are cho- +sen such that the dominant-mode frequency f220 differs +from the GR Kerr value f Kerr +220 +by 1%. This is a heuris- +tic choice but studying the effect of 1% deviation in f220 +is a reasonable goal from the data analysis perspective +as the next-generation detectors is expected to measure +f220 with sub-percent accuracy [8, 12, 60]. For Delta and +Delta220 modifications, the sign of the deviation is cho- +sen to be positive i.e., fractional frequency shift of +0.01. +We fix the final mass Mf = 70M⊙ in the detector +frame. +To compute ρRD, we consider events with ex- +trinsic parameters listed in Tab. II. These choices are +2 We do not consider overtones for simplicity as their measurablity +and physical interpretation is still debated even when assuming +that the underlying theory of gravity is GR [27, 30, 55–59]. + +5 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +EdGB +0.025 +0.020 +0.015 +0.010 +0.005 +0.000 +flmn +q = 1.4 +q = 3 +q = 5 +(2, 2, 0) +(3, 3, 0) +(2, 1, 0) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +EdGB +0.010 +0.005 +0.000 +0.005 +0.010 +0.015 +flmn +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +Q +0.0000 +0.0025 +0.0050 +0.0075 +0.0100 +0.0125 +0.0150 +0.0175 +0.0200 +flmn +q = 1.4 +q = 3 +q = 5 +(2, 2, 0) +(3, 3, 0) +(2, 1, 0) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +Q +0.0002 +0.0000 +0.0002 +0.0004 +0.0006 +0.0008 +0.0010 +0.0012 +0.0014 +flmn +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +0.200 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +flmn +q = 1.4 +q = 3 +q = 5 +(2, 2, 0) +(3, 3, 0) +(2, 1, 0) +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +0.200 +0.010 +0.005 +0.000 +0.005 +0.010 +0.015 +0.020 +flmn +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +dCS +0.012 +0.010 +0.008 +0.006 +0.004 +0.002 +0.000 +flmn +q = 1.4 +q = 3 +q = 5 +(2, 2, 0) +(3, 3, 0) +(2, 1, 0) +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +dCS +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +0.035 +0.040 +flmn +0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0200 +0.000 +0.001 +0.002 +0.003 +0.004 +0.005 +0.006 +0.007 +flmn +q = 1.4 +q = 3 +q = 5 +(3, 3, 0) +(2, 1, 0) +0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0200 +220 +0.0175 +0.0150 +0.0125 +0.0100 +0.0075 +0.0050 +0.0025 +0.0000 +flmn +q = 1.4 +q = 3 +q = 5 +(3, 3, 0) +(2, 1, 0) +FIG. 1. Rows 1-4: Relative deviations δflmn (left) and relative effective deviations ˜δflmn (right) from the GR Kerr BH spectrum, +as induced by the different modified spectra discusses in Sec. III: EdGB, Kerr-Newman, Horndeski and dCS, respectively. Last +row: Relative effective deviations ˜δflmn induces by the Delta spectrum (left) and the Delta220 spectrum (right). + +6 +Spectrum +α0.01 +˜ +Mf (M⊙) +˜χf +˜δf330 +˜δf210 +q = 1.4 , +χf = 0.67 +EdGB +0.28 +71.10 +0.68 −5.1 × 10−3 +9.7 × 10−3 +Kerr-Newman 0.25 +69.71 +0.68 −1.6 × 10−5 +6.6 × 10−5 +Horndeski +0.16 +70.38 +0.69 −4.3 × 10−3 +1.1 × 10−2 +dCS +0.055 +71.40 +0.68 +9.7 × 10−3 +1.3 × 10−2 +Delta +0.01 +69.76 +0.68 +4.2 × 10−4 +2.8 × 10−3 +Delta220 +0.01 +69.76 +0.68 −9.5 × 10−3 −7.1 × 10−3 +q = 3 , +χf = 0.54 +EdGB +0.31 +70.54 +0.54 −6.4 × 10−3 +4.8 × 10−3 +Kerr-Newman 0.26 +69.75 +0.55 −5.7 × 10−5 +4.5 × 10−4 +Horndeski +0.15 +70.57 +0.57 −4.3 × 10−3 +1.1 × 10−2 +dCS +0.058 +72.30 +0.54 +1.0 × 10−2 +1.8 × 10−2 +Delta +0.01 +69.82 +0.55 +3.6 × 10−4 +3.1 × 10−3 +Delta220 +0.01 +69.82 +0.55 −9.5 × 10−3 −6.8 × 10−3 +q = 5 , +χf = 0.42 +EdGB +0.34 +70.01 +0.40 −7.5 × 10−3 +5.2 × 10−5 +Kerr-Newman 0.27 +69.77 +0.43 −1.1 × 10−4 +8.2 × 10−4 +Horndeski +0.14 +70.71 +0.46 −4.3 × 10−3 +1.1 × 10−2 +dCS +0.063 +76.26 +0.55 +1.2 × 10−2 +4.2 × 10−2 +Delta +0.01 +69.08 +0.51 +1.9 × 10−3 +2.1 × 10−2 +Delta220 +0.01 +69.08 +0.51 −5.7 × 10−2 −4.0 × 10−2 +TABLE I. Values α0.01 of the deviation parameter α inducing a 1% shift in f220, alongside the effective measurable final mass +and spin { ˜ +Mf, ˜χf} as well as the effective shifts ˜δflmn in the frequencies of the subdominant modes, for the different spectra con- +sidered in this work. Here α is a collective name for the additional theory parameter. It represents {ζEdGB, Q, µ, ζdCS, ∆, ∆220} +depending on the spectrum. + +7 +compatible with the GW150914 event, with the only ex- +ception of the inclination angle; the posteriors distribu- +tion for the inclination angle of GW150914 prefers a face- +on/face-off orientation for which the subdominant modes +excitation is suppressed significantly. Instead, we use a +more optimal inclination angle for BH spectroscopy and +set ι = π/3. For each modified QNM scenario, we study +mass ratio q ∈ {1.4, 3, 5}. The final spin of the remnat +and the mode excitation amplitudes Almn are computed +from the q using the numerical fits provided in [61] and +[28], respectively, by assuming non-spinning progenitors. +The absolute amplitude scale A220 is set by Eq. (5) in +[22]3. In particular, χf ≈ {0.67, 0.54, 0.52} for the three +values of q chosen here. The phases φlmn do not signif- +icantly affect the recovery of the QNM frequencies and +damping times (c.f., [30, 32]) and therefore for this study, +we set it to 0. +The ringdown waveform is modeled as h = h+ − ih×, +with +h+ = +� +l,m>0,n +Almn −2Ylm,+ (ι)e−t/τlmn cos Φlmn , (12a) +h× = +� +l,m>0,n +Almn −2Ylm,× (ι)e−t/τlmn sin Φlmn , (12b) +where Φlmn = 2πflmnt + φlmn, Almn and φlmn are the +(real) excitation amplitudes and phases of the modes, +and the plus and cross spherical harmonics are defined +by4 +−2Ylm,+ (ι) = −2Ylm (ι, 0) + (−1)l +−2Ylm (ι, 0) , +(13a) +−2Ylm,+ (ι) = −2Ylm (ι, 0) − (−1)l +−2Ylm (ι, 0) . +(13b) +In +writing +the +expression +(12), +we +assume +a +non-precessing quasi-circular binary progenitor. +For +these systems equatorial reflection symmetries gives +Al−mneiφl−mn = (−1)lAlmne−iφlmn and we simultane- +ously sum over ±m using the symmetry relation ωl−mn = +−ω∗ +lmn +5. We assume a circularly polarized ringdown as +the numerical simulations favour it (c.f., [25] for a de- +tails). +We also use spherical harmonics instead of the +more natural spheroidal harmonics basis function for +3 Here we emphasize that depending on the modified theory, this +assumption may not hold to differing extents. Amplitudes are +governed dominantly by the plunge-merger dynamics and we typ- +ically need a fully numerical simulation to infer them. However, +we do not have numerical simulations for most of the theories +considered here; thus, we do not have knowledge of the ampli- +tudes of mode excitation in these theories. Instead, we use the +GR amplitudes as an approximate proxy for practicality. +4 A global phase in the definition of the spherical harmonics can be +reabsorbed in the definition of the phases φlmn of the ringdown +modes. +5 We use ωlmn to denote the prograde modes of the spectrum. The +QNM solutions also contain a set of retrograde modes, but in the +numerical simulations it is seen that these modes are not excited +significantly [24]. See also [56, 62] on the role of retrograde modes +in describing the ringdown waveform. +ringdown; this is a fairly standard approximation which +is known to introduces substantial errors only in high +spin limits [63, 64]. +To predict the statistical uncertainty in the measure- +ment of ˜δflmn, we employ a Fisher matrix formalism +following our work in [8]. +We use the power spectral +density (PSD) of the next-generation ground-based de- +tector – the Einstein Telescope, assuming a triangular +geometry, located in Sardinia [65] and operating at the +ET-D sensitivity [66]. However, note that all the quali- +tative statements and ball-park numbers reported in this +study should hold for any detector for a given ρRD. Our +study is not heavily influenced by the choice of the detec- +tor as long as the modifications in the spectra are cho- +sen to have 1% deviation of f220. Note that while the +results in this study will be approximately similar for +the next space-based detector LISA, LISA will be sen- +sitive to supermassive binary BH signals and therefore, +the theory-specific parameters i.e., coupling constants or +extra charges, that lead to 1% deviation of f220 will be +different from the choice made here. However, we expect +theory-parameters similar to the choice made here for the +Cosmic Explorer, as it is sensitive to stellar mass ranges +similar to the Einstein Telescope [11]. +For computing the Fisher covariance matrix, we pa- +rameterize the waveform as Eq.s (6)-(7), i.e., using ef- +fective parameters { ˜ +Mf, ˜χf, ˜δflmn, ˜δτlmn} instead of the +traditionally used {flmn, τlmn}. Next, for a choice of the- +ory parameters that produces a modified spectra such +that f220 deviates by 1% w.r.t. GR Kerr BH, we map the +true values {Mf, χf} to the measured values { ˜ +Mf, ˜χf} +using Eq.s (2) and (8). Note that once we compute the +values of { ˜ +Mf, ˜χf}, deviation parameters for all the sub- +dominant modes get fixed. We construct the covariance +matrix for a set of 4N parameters, where N is the number +of modes as – +θ = { ˜ +Mf, ˜χf, ˜δflmn, ˜δτlmn, log10 Almn, φlmn} . +(14) +For the dominant mode, we estimate { ˜ +Mf, ˜χf}, and not +{˜δf220, ˜δτ220}. +Next, we quantify the departure of the modified QNM +spectra from GR Kerr QNM spectra using QGR as a mea- +sure, where QGR is defined as the quantile at which the +posterior distribution of ˜δ excludes zero. Higher the value +of QGR, more confidently we can exclude the null hypoth- +esis that the ringdown contains QNMs corresponding to a +GR Kerr BH. QGR can either be defined mode-wise over +a marginalized 1-d probability distribution P(˜δflmn) or +defined on a joint probability distribution of all the mea- +sured modes. Here, we use QGR defined on the 1-d pos- +terior distributions corresponding to – a) ˜δf330 : Q330 +GR, +b) ˜δf210 : Q210 +GR and c) on a joint posterior distribution +of ˜δf330 and ˜δf210 : Q2D +GR. +In the Fisher matrix formalism, +by construction, +P(˜δflmn) is a Gaussian distribution centered at the true +value of ˜δflmn. From this, Qlmn +GR can be easily computed + +8 +ra +dec +ψ +ι +tGPS +dL(Mpc) +1.16 −1.19 1.12 π/3 1126259462.423 +403 +TABLE II. Right ascension, declination, polarization an- +gle, inclination angle, GPS time and luminosity distance for +GW150914 that were used to compute the SNR and the Fisher +covariance matrix. +as +Qlmn +GR = erf +� +p/ +√ +2 +� +, +(15) +where the p-value is given as +p = |˜δflmn| +σ +. +(16) +Further, Eq. (16) can be generalized to a 2-dimensional +case of the joint posterior P(˜δf330, ˜δf210). In the cases of +2-D posteriors – +Q2D +GR = 1 − exp +� +−1 +2⃗µT · Σ−1 · ⃗µ +� +(17) +where ⃗µ is the vector corresponding to the true values of +{˜δf330, ˜δf210} and Σ is the covariance matrix. +V. +RESULTS +Before turning our attention to the modified spectra, +we first look at the measurability of a GR Kerr BH spec- +trum with the next generation ground-based detectors. +We used ET-D PSD to illustrate the expected orders of +magnitudes and trends. +Specifically, we consider ring- +downs corresponding to q ∈ {1.4, 3, 5} with Mf = 70M⊙ +and the extrinsic parameters displayed in Tab. II. Tab. III +shows ρRD and the expected measurement uncertainties +on the deviation parameters σ(˜δflmn) — note that for a +GR Kerr BH spectrum, ˜δflmn = δflmn and we use tilde +here for consistency of notation. We report the normal- +ized uncertainties κ ≡ ρRDσ, where ρRD is the SNR in the +ringdown and σ is the expected statistical uncertainty in +the parameter computed with a Fisher information ma- +trix framework 6. Further, we confirm that the uncer- +tainty in the measurement of the deviations in damping +times σ(˜δτlmn) is large; therefore, we concentrate on the +subdominant mode frequencies. +To appreciate the quantitative results, we begin with il- +lustrating the performance of BH spectroscopy for a) the +best-case scenario, the dCS spectrum, and b) the worst- +case scenario, the Kerr-Newman spectrum in Fig. 2. For +6 Note, the relative excitation amplitudes Almn/A220 of the sub- +dominant modes increase monotonically with q for non-spinning +systems [21–23, 25, 28]. +However, ρRD decreases with q [22]. +Therefore, while spectroscopy with high q is favourable for fixed +ρRD, it need not be the case for fixed luminosity distance. +q +ρRD +κ = ρRD σ +˜ +Mf (M⊙) +˜χf +˜δf330 ˜δf210 ˜δτ220 ˜δτ330 +1.4 +87 +184.63 +3.68 +1.56 +7.78 22.43 46.14 +3 +74 +294.63 +7.43 +1.00 +3.27 +9.17 19.73 +5 +57 +436.37 +13.08 1.27 +3.38 +8.80 22.26 +TABLE III. ρRD and uncertainties σ over mass, spin and the +deviations parameters for a Kerr BH spectrum, as measured +by a triangular configuration of ET detector with ET-D PSD. +The ringdown corresponds to a BH whose detector-frame final +mass is Mf = 70M⊙ and the extrinsic system parameters are +enlisted in Tab. II. +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 +f +62.5 +65.0 +67.5 +70.0 +72.5 +75.0 +77.5 +80.0 +Mf [M +] +Kerr-Newman +f220 +220 +f330 +f210 +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 +0.65 +f +65 +70 +75 +80 +85 +90 +Mf [M +] +dCS +f220 +220 +f330 +f210 +FIG. 2. Projection of BH spectroscopy on the mass-spin plane +for the Kerr-Newman (top) and dCS (bottom) spectra. We +set ρRD = 300 and q = 5. The bands correspond to the 90% +credible intervals. +The black and red markers indicate the +measurable and true values for mass and spin, respectively. +both cases, we use ringdowns with ρRD = 300 and q = 5. +Specifically, we show the projections of the 90% credi- +bility bands of f220, τ220, f330 and f210 on the mass-spin +plane. The common region of intersection of f220 and τ220 +corresponds to the measurable mass and spin { ˜ +Mf, ˜χf} +estimate (indicated by a black marker), while the true +value of {Mf, χf} (indicated by a red marker) lies outside +the intersection region. For the Kerr-Newman spectrum, +all bands have a common intersection region, and there- +fore, BH spectroscopy fails to detect deviations from the +GR BH Kerr QNMs. On the contrary, for the dCS spec- + +9 +trum, there is no common intersection region and BH +spectroscopy can be used to identify that this spectrum +is incompatible with GR at the 90% confidence level. +Next, looking at the values of { ˜ +Mf, ˜χf} and of ˜δflmn +in Tab. I, we observe that { ˜ +Mf, ˜χf} does not differs sig- +nificantly from {Mf, χf} and ˜δflmn ≪ 1. Consequently, +the results reported in Tab. III for the GR Kerr BH spec- +tra can be expected to approximately hold for the mod- +ified QNM spectra too. This allows to derive a back-of- +the-envelope ρRD estimate required to distinguish vari- +ous modified spectra from the GR Kerr BH spectrum. +Further, we can use Eq.s (15)-(16) to find the ρRD nec- +essary to exclude 0 from P(˜δf330) at 90% confidence.For +instance, in the case of an EdGB spectrum, from (15) we +observe that Qlmn +GR = 0.9 corresponds to p ≈ 1.64. Invert- +ing (16) for ρRD = κ/σ and using the results in Tab.s I +and III, we get an approximate minimum value of ρRD +as – +ρ0.9 +RD ≈ +� +� +� +� +� +502 +(q = 1.4) +256 +(q = 3) +278 +(q = 5) +(18) +If we repeat this exercise for the dCS spectrum, we find +ρ0.9 +RD ≈ +� +� +� +� +� +264 +(q = 1.4) +164 +(q = 3) +174 +(q = 5) +(19) +Note that it is only for the sake of demonstrating a fast +and easy approximate estimation that we derive (18) us- +ing the values in Tab. III, which assumed a GR Kerr BH +spectrum. However, there is no fundamental obstruction +to being fully consistent by applying Eq.s (15) and (17) +using the covariance matrices from the modified spectra. +In Fig. 3, we depict marginalized 2-d posterior esti- +mates of ˜δf330 and ˜δf210 for all the modified ringdowns +studied. Here again, we use ringdowns with ρRD = 300 +to illustrate QGR as a measure to distinguish a modified +QNM spectrum from the GR Kerr BH spectrum. When +the posterior distributions are compatible with (0, 0) (in- +dicated with a black dot), the QNM spectra for the f330 +and f210 are compatible with the GR Kerr BH spectrum. +Note also that the shape of the contours changes with q +(and that the contours do not trivially shrink with chang- +ing in q). This foreshadows the non-trivial dependence +of QGR on q which will be further emphasised in fig 4. +We can identify the modifications to various level of con- +fidence – Q2D +GR = {0.33, 0.85, 0.97} for Horndeski, Q2D +GR = +{0.39, 0.91, 0.96} for EdGB, Q2D +GR = {0.85, 0.99, 1} for +dCS for q = {1.4, 3, 5} respectively. +We can contrast +this to the case of Kerr-Newman where we do not expect +to detect any deviations – we have Q2D +GR = O(10−6) for +q = 1.4, O(10−3) for q = 3 and O(10−2) for q = 5. We +will see in Fig. 4 that this is true even at very high ρRD. +Finally, in Fig. 4, we present our main results on ρRD +required to identify deviations from the GR Kerr QNM +spectrum for the various modification schemes. The first +two columns show QGR computed using a 1-d posterior +distribution of f330 and f210, respectively, and the right- +most column corresponds to the 2-d joint posterior distri- +bution on f330 and f210. Again, we study systems corre- +sponding to three mass ratios: in the first row, we study +the near-equal-mass scenario of q = 1.4; in the second +row, we study q = 3 and in the last row, we study q = 5. +Let us examine the ρRD at which QGR ≥ 0.9 for all +the modified QNMs considered here. We observe that +ρRD required to distinguish the modified QNMs from GR +Kerr BH spectrum depends acutely on the details of the +modifications. Modification of the spectra, such as that +predicted by Kerr-Newman and Delta, cannot be confi- +dently differentiated from the GR spectrum, even when +ρRD = 103. This happens because these modifications +can be approximated by the GR by suitably selecting +the values of ˜ +Mf and ˜χf. In other words, these modi- +fied spectra are highly degenerate with the GR Kerr BH +QNM in the mass-spin space. This can be quantitatively +seen in Fig. 1. +For instance, ˜δflmn for these modified +spectra is an order of magnitude smaller than ˜δflmn for +other modifications such as dCS or EdGB. In contrast to +this, modifications predicted by the EdGB and Horndeski +theories can be distinguished confidently from the GR at +a ρRD ∈ [250, 103]. Furthermore, the dCS or Delta220- +like QNM spectra could be confidently identified with an +even lower ρRD ∈ [250, 400]. +We highlight two non-trivial trends from Fig. 4 that +we observe. +• While all of them are valid measures to identify +modifications in a QNM spectrum, we find that the +ability to distinguish a modified theory from GR +using 1-d Q330 +GR, Q210 +GR and 2-d joint Q2D +GR are fairly +different and depends on theory. While modifica- +tions in the spectra predicted by dCS and Delta220 +benefit by using Q330 +GR, Kerr-Newmann and Delta +are more distinguishable using Q210 +GR. Furthermore, +certain types of modifications will only manifest in +a detectable fashion in one of the QGR measures. +For example, EdGB spectra with ρRD ≤ 103 cannot +be confidently distinguished if we used Q210 +GR while +it can be using Q330 +GR or Q2D +GR. +• The distinguishability of a given modified spectrum +does not have a monotonic behaviour with q. This +occurs because q dictates both the mode excita- +tion amplitudes and the final spin of the remnant +BH, and the interplay between the two produces +the non-monotonic trend we see here. For instance, +at a given ρRD, Q330 +GR for dCS increases monotoni- +cally with q, while Q330 +GR for EdGB and Horndeski +spectra seem to perform better for the case of q = 3 +than for q = 1.4 or 5. It is also worth noting that +the performance of Q330 +GR, Q210 +GR and Q2D +GR for differ- +ent theory can differ with q. + +10 +0.03 +0.02 +0.01 +0.00 +0.01 +f330 +0.10 +0.05 +0.00 +0.05 +0.10 +f210 +EdGB +q = 1.4 +q = 3 +q = 5 +0.02 +0.01 +0.00 +0.01 +0.02 +f330 +0.10 +0.05 +0.00 +0.05 +0.10 +f210 +Kerr-Newman +q = 1.4 +q = 3 +q = 5 +0.03 +0.02 +0.01 +0.00 +0.01 +f330 +0.10 +0.05 +0.00 +0.05 +0.10 +f210 +Horndeski +q = 1.4 +q = 3 +q = 5 +0.01 +0.00 +0.01 +0.02 +0.03 +f330 +0.10 +0.05 +0.00 +0.05 +0.10 +f210 +dCS +q = 1.4 +q = 3 +q = 5 +0.02 +0.01 +0.00 +0.01 +0.02 +f330 +0.10 +0.05 +0.00 +0.05 +0.10 +f210 +Delta +q = 1.4 +q = 3 +q = 5 +0.03 +0.02 +0.01 +0.00 +0.01 +f330 +0.10 +0.05 +0.00 +0.05 +0.10 +f210 +Delta220 +q = 1.4 +q = 3 +q = 5 +FIG. 3. Density plots of {˜δf330, ˜δf210} for the modified spectra considered in Sec. III and different mass ratios. Solid and dashed +lines indicate the 90% and 50% probability contours respectively. We assume a true injected mass Mf = 70M⊙ and vanishing +initial spins (see Tab. I for the values of the corresponding effective measurable masses and spins). We set to ρRD = 300 +to clearly display the different trends of the spectra in distinguishing deviations from GR. The null hypothesis under test is +denoted by a black marker. + +11 +101 +102 +103 +RD +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Q330 +GR +q = 1.4 +EdGB +Kerr-Newman +dCS +Horndeski +Delta +Delta220 +101 +102 +103 +RD +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Q210 +GR +q = 1.4 +EdGB +Kerr-Newman +dCS +Horndeski +Delta +Delta220 +101 +102 +103 +RD +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Q2D +GR +q = 1.4 +EdGB +Kerr-Newman +dCS +Horndeski +Delta +Delta220 +101 +102 +103 +RD +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Q330 +GR +q = 3 +EdGB +Kerr-Newman +dCS +Horndeski +Delta +Delta220 +101 +102 +103 +RD +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Q210 +GR +q = 3 +EdGB +Kerr-Newman +dCS +Horndeski +Delta +Delta220 +101 +102 +103 +RD +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Q2D +GR +q = 3 +EdGB +Kerr-Newman +dCS +Horndeski +Delta +Delta220 +101 +102 +103 +RD +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Q330 +GR +q = 5 +EdGB +Kerr-Newman +dCS +Horndeski +Delta +Delta220 +101 +102 +103 +RD +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Q210 +GR +q = 5 +EdGB +Kerr-Newman +dCS +Horndeski +Delta +Delta220 +101 +102 +103 +RD +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Q2D +GR +q = 5 +EdGB +Kerr-Newman +dCS +Horndeski +Delta +Delta220 +FIG. 4. Values of the GR quantile QGR as a function of the ρRD. We consider quantiles for the marginalised single parameter +posteriors P(˜δf330) (leftmost column) and P(˜δf210) (central column) and for the 2D joint posterior P(˜δf330, ˜δf210) (rightmost +column). The quantiles are computed using the exact expressions (15) and (17). + +12 +VI. +DISCUSSION AND CONCLUSION +We performed a comprehensive study on the ability of +BH spectroscopy to distinguish a modified spectra from a +GR Kerr BH QNMs. We studied theory-motivated mod- +ifications for QNM spectra that were publicly available +– specifically, EdGB, Kerr-Newman, Horndeski and dCS +theory, and for two phenomenologically modified QNM +spectra – Delta and Delta220. To investigate ρRD nec- +essary to distinguish these modified spectra from a GR +Kerr BH spectrum, we assessed the performance of BH +spectroscopy using a Fisher information matrix formal- +ism. The ringdowns with the modified spectra are gen- +erated such that in each case f220 deviates from the cor- +responding GR Kerr QNM by 1%. +First, we re-iterate that we can only measure the effec- +tive deviation parameters ˜δflmn and not the absolute de- +viation δflmn as the mass and the spin estimation can be +chosen suitably to compensate for the deviations of the +QNMs. Further, we show that in many theories ˜δflmn +significantly differs from δflmn. Therefore, to study the +ability BH spectroscopy to distinguish modified spectra, +the framework must be setup using the measurable effec- +tive deviation parameters ˜δflmn . +We found that ρRD necessary to distinguish a mod- +ified spectrum from the GR Kerr one depends on the +details of the modification and on the mass ratio. Fur- +ther, we used three different measures that quantify the +amount by which: a) the 1-d posterior estimate of ˜δf330 +excludes 0 a.k.a. Q330 +GR, b) the 1-d posterior estimate of +˜δf210 excludes 0 a.k.a. Q210 +GR and c) the 2-d joint posterior +estimation of ˜δf330−˜δf210 excludes (0,0) a.k.a. Q2D +GR. De- +pending on the spectrum and the magnitudes of ˜δflmn, +the performance of Q330 +GR, Q210 +GR and Q2D +GR can vary signif- +icantly. Roughly, we find that a ρRD ≥ 150 is required +to identify deviations from GR at a 90% credibility level. +This range of ρRD is attainable with the next generation +detectors and BH spectroscopy will be a powerful tool to +constrain GR as well as for identifying modifications to +Kerr BH spectrum. +In previous works [8, 9], we studied the measurability +of QNM parameters, that is, the statistical uncertainty +with which a QNM mode parameter can be estimated +from a signal to assess the landscape of BH spectroscopy +with a next-generation detector. However, we note here +that the subdominant QNM mode with the best mea- +surability may not always be the optimal mode for dis- +tinguishing a modified QNM spectrum from a GR Kerr +BH QNM spectrum. The best mode to identify a depar- +ture from GR Kerr spectrum depends on the interplay +between measurablity and the details of modification of +the spectra. For instance, measurablity of f330 is better +than f210 in a non-spinning binary BH ringdown. How- +ever, if the compact object is a Kerr-Newman BH instead +of a Kerr BH, f330 measured in the ringdown would be +compatible with the Kerr BH even for ringdown with +ρRD ∼ 103. +The modifications in the spectra in this +case can be observed predominantly by looking at f210. +Therefore, using a setup that uses information in all mea- +surable QNM mode parameters is optimal while looking +for departure from the GR Kerr spectrum in a ringdown +signal. Greater the number of QNM mode parameters +measured, greater the chances that we can distinguish a +modified QNM spectra from the GR Kerr spectra. +Furthermore, among the modified QNM spectra we +have studied, we see the theories separate out into two +groups : +• EdGB, Horndeski, dCS, Delta220: For these spec- +tra, a GR Kerr ringdown can be excluded at 90% +confidence level with a SNR ρ0.9 +RD ∈ [150, 500]. +The deviations manifest more prominently in the +(3, 3, 0) mode and the (2, 1, 0) mode offers a rela- +tively poor constraints. +• Kerr-Newman, Delta: here the deviations manifest +almost exclusively in the (2, 1, 0) mode, with Q210 +GR +returning the most performative no-hair test. How- +ever, the ρRD required to exclude 0 at 90% confi- +dence is so high that these deviations are indistin- +guishable, even at ρRD = 1000. +Finally, works such as this is essential for gauging the +potential of ringdown-based tests of gravity; but unfor- +tunately, we are restricted by publicly available modified +QMN spectra. While it is difficult to solve the QNM spec- +tra in modified theories of gravity, to adequately prepare +for test of GR with next-generation detectors including +designing the implementation of BH spectroscopy, we feel +that the field would benefit substantially from investing +effort in this direction. +ACKNOWLEDGMENTS +We thank L. Pierini for clarifications about the work +in [40]. We thank E. Berti, L. Gualtieri, A. Maselli and +P. Pani for fruitful discussions. C.P. is supported by Eu- +ropean Union’s H2020 ERC Starting Grant No. 945155– +GWmining and by Cariplo Foundation Grant No. 2021- +0555. 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Grav. 28, 094013 (2011), +arXiv:1012.0908 [gr-qc]. + diff --git a/WNE0T4oBgHgl3EQfVwAY/content/tmp_files/load_file.txt b/WNE0T4oBgHgl3EQfVwAY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..32ccaec25a0386aaf3788a82917fef6c9100bc9b --- /dev/null +++ b/WNE0T4oBgHgl3EQfVwAY/content/tmp_files/load_file.txt @@ -0,0 +1,1441 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf,len=1440 +page_content='Identifying modified theories of gravity using binary black-hole ringdowns Costantino Pacilio∗ Dipartimento di Fisica “G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Occhialini”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Universit´a degli Studi di Milano-Bicocca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Piazza della Scienza 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 20126 Milano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Italy and INFN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Sezione di Milano-Bicocca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Piazza della Scienza 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 20126 Milano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Italy Swetha Bhagwat† Institute for Gravitational Wave Astronomy & School of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' University of Birmingham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Edgbaston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Birmingham B15 2TT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' UK (Dated: January 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 2023) Black-hole spectroscopy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' measuring the characteristic frequencies and damping times of different modes in a black-hole ringdown,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' is a powerful probe for testing deviations from the general theory of relativity (GR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' In this work, we present a comprehensive study on its ability to identify deviations from the spectrum of a Kerr black hole in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Specifically, we investigate the performance of black hole spectroscopy on a diverse set of theoretically motivated as well as phenomenologically modified spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We find that while the signal-to-noise ratio ρRD in the ringdown required to iden- tify a modification to the GR Kerr black hole spectrum depends on the details of the modifications, a modification that introduces ∼ 1% shift in the fundamental mode frequencies can typically be dis- tinguished with ρRD ∈ [150, 500].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' This range of ρRD is feasible with the next-generation detectors, showing a promising science case for black hole spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' INTRODUCTION Gravitational waves (GWs) with characteristic fre- quencies and damping times are radiated as the distorted black hole (BH) formed during a binary BH merger re- laxes into its final stable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' This signal is called the ringdown and comprises of a linear superposition of the spectral modes of the BH, known as the quasi-normal modes (QNMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We can obtain the QNMs by solving the BH perturbation equations [1–3] and the ringdown signal can be used to validate dynamics in linear strong field regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The QNM spectra of a perturbed BH in the general theory of relativity (GR) are obtained by solving Teukolsky’s equation [4] and under the Kerr hypothesis [5, 6] i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=', that the remnant BH in binary BH coalescence relaxes to a Kerr BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' BH spectroscopy [7], defined as measuring QNM spec- tra from ringdown signals, allows to put forth consistency tests of the joint hypotheses that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' the asymptotic equilibrium state of the remnant is described by the Kerr metric a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=', the Kerr hy- pothesis and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' the dynamics of the perturbed Kerr BH is governed by Teukolsky’s equation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=', (linearized) GR dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Further, BH spectroscopy can observationally validate the no-hair theorem obeyed by BHs in GR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' it demands that all aspects of a Kerr spacetime, including its QNM spectrum, be fully characterized by just two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' ∗ costantino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='pacilio@unimib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='it † sbhagwat@star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='bham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='uk Often and most naturally, the two parameters are chosen as the mass Mf and spin χf of the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' If more than two QNM parameters are measured, it can be verified that different pairs of QNM parameters solve for the same Mf and χf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' This allows us to perform a null test of the no-hair theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' There has been much focus in the literature on the fea- sibility of measuring the subdominant QNM modes and performing null tests to validate the underlying theory of gravity as GR with BH spectroscopy [8–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' In this study, we concentrate on a complementary aspect and investigate the ability of BH spectroscopy to identify de- viations from GR when the spectrum is not described by the GR Kerr QNMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We perform a comprehensive study utilizing the publicly available QNM spectra in various modified theories as well as two phenomenological modi- fications and assess the performance of BH spectroscopy to differentiate them from a GR Kerr BH spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We then investigate the signal-to-noise ratio ρRD in the ring- down at which different modified theories of gravity can be distinguished from GR using BH spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We find that the required ρRD depends on the details of the QNM spectra in a given theory and on their degeneracies with the GR Kerr BH spectra in the mass-spin space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' However, at a broad level, we observe that ρRD ≥ 150 is required to confidently identify modified theories that produce ≤ 1% deviation in the dominant mode from GR using BH spectroscopy (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=', [8, 9] for a detailed study on expected ρRD for measurability of QNM parameters with the next-generation GW detectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' In Section II we outline the conceptual structure adopted for this study to test the no-hair theorem using BH spec- troscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' In Section III, we detail the modified QNM spectra used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Then, in Section IV we sum- marize the setup and implementation used to perform arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='02267v1 [gr-qc] 5 Jan 2023 2 this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' This is followed by the results in Section V and a discussion of their implications in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' TESTING THE NO-HAIR HYPOTHESIS WITH BH SPECTROSCOPY The ringdown waveform observed at asymptotic infin- ity can be approximated as a linear superposition of a countably infinite set of (complex) QNMs with ωlmn = 2πflmn − i/τlmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' (1) Here flmn and τlmn are the characteristic frequencies and damping times of the spectral modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' (l, m, n) index the mode’s angular, azimuthal and overtone numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' As in any perturbation theory, the excitation amplitude of the modes depend on the initial perturbation condi- tions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' for a binary BH merger these are set largely during the plunge-merger phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' A quasi-circular merger excites (2, 2, 0) dominantly, and depending on the initial binary BH’s mass ratio and spins, the most prominent subdom- inant angular modes can be {(3, 3, 0), (2, 1, 0), (4, 4, 0)} [21–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' To outline our setup, let us consider a ringdown where more than two QNM parameters are measurable, and a case where we have identified its QNM indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The minimum ρRD required for this has been investigated in studies such as [24, 29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' If the underlying theory of gravity is GR and if the Kerr hypothesis holds, we can invert any pair of QNM parameters, preferably the fre- quency and damping time of the dominant mode, to infer the mass and spin of the BH – {f220, τ220} → {M Kerr f , χKerr f }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' (2) From this mass and spin estimate, we can compute the full set of QNM spectra of the Kerr BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Let f (infer) lmn be the inferred subdominant mode frequency {M Kerr f , χKerr f } → f (infer) lmn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' (3) Here we use the superscript (infer) to differentiate f (infer) lmn from f (meas) lmn which are the frequencies mea- sured from the ringdown signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' While a similar argu- ment holds for QNM damping times, subdominant mode damping times are poorly measured [8, 22, 32] and there- fore, we focus on tests using solely the subdominant mode frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' A null test can be performed by checking if the rel- ative difference between the inferred and the measured quantity is compatible with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We define the relative difference as δflmn = f (meas) lmn − f (infer) lmn f (infer) lmn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' (4) We can infer steps (2)-(4) through a convenient reparametrization of the waveform during the parame- ter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We briefly summarize this and point the reader to a detailed treatment in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' A generic modified QNM spectrum can be phenomeno- logically written as flmn = f Kerr lmn (Mf, χf)(1 + δflmn) , (5a) τlmn = τ Kerr lmn (Mf, χf)(1 + δτlmn) , (5b) where {Mf, χf} are the true values of the final mass and spin, and {δflmn, δτlmn} are the relative shifts of the spectrum w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' the QNM spectra of a GR BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' {δflmn, δτlmn} can be non-trivial functions of {Mf, χf} and of the physical parameters of the modified theory such as the additional coupling constants or charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' At this stage of setting up the formalism, we do not differen- tiate between a modification to the underlying theory of gravity and a modification in the nature of the compact object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Now, notice that {δf220, δτ220} are redundant param- eters because we can always find a pair { ˜ Mf, ˜χf} of ef- fective final mass and spin that satisfy f220 = f Kerr 220 ( ˜ Mf, ˜χf) , (6a) τ220 = τ Kerr 220 ( ˜ Mf, ˜χf) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' (6b) The subdominant modes can be re-expressed as flmn = f Kerr lmn ( ˜ Mf, ˜χf)(1 + ˜δflmn) , (7a) τlmn = τ Kerr lmn ( ˜ Mf, ˜χf)(1 + ˜δτlmn) , (7b) for (lmn) ̸= (220).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Further, the effective shifts {˜δflmn, ˜δτlmn} satisfy f Kerr lmn (Mf, χf) (1 + δflmn) = f Kerr lmn ( ˜ Mf, ˜χf)(1 + ˜δflmn) , (8a) τ Kerr lmn (Mf, χf) (1 + δτlmn) = τ Kerr lmn ( ˜ Mf, ˜χf)(1 + ˜δτlmn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' (8b) For the QNM spectrum of a Kerr BH in GR, {δflmn, δτlmn} vanish;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' therefore { ˜ Mf, ˜ χf} = {Mf, χf} and {˜δflmn, ˜δτlmn} vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We set up our framework to identify departure from the GR Kerr BH QNM spectrum by constraining the effective shifts away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Note that the mass and spin appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='s (2)-(3) are not the true values {Mf, χf} but rather the effec- tive values { ˜ Mf, ˜χf}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We emphasise that we can only measure the effective final mass and spin, and not the true values corresponding to the BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' While developing a framework for observational test, the effective (mea- sured) parameters deviations ˜δ’s are the instrumental variables 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Similarly, the magnitudes of the δ’s are not 1 Note also that if one is not interested in testing the no-hair the- 3 directly accessible in BH spectroscopy and we can only estimate ˜δ’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' In Section III, we inspect modified QNM spectra and show that ˜δ can be significantly different from δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' MODIFIED QNM SPECTRA In this study, we quantify the ability of BH spec- troscopy to constrain ˜δflmn away from zero for various class of modifications to the Kerr BH spectrum (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=', [35– 38] for other works on BH spectroscopy in the context of modified theories of gravity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Given the lack of a best- candidate theory for modified gravity and the fact that QNM spectra in modified theories are available in a very few theories, of which even fewer theories have QNMs computed at a beyond-leading order in BH spins [39, 40], we consider both publicly available modified spectra and phenomenologically modified spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The spectra are chosen to encompass a variety of mod- ifications to stress-test the ability of spectroscopy to dis- tinguish them from a GR Kerr spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We don’t con- cern ourselves with the physical plausibility of these mod- ifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Below we describe the modifications to the GR Kerr BH spectrum used in this study: EdGB: The Einstein-dilaton-Gauss-Bonnet theory [41, 42] is a modified theory of gravity that intro- duces a dilaton scalar field that is non-minimally coupled to higher orders of the curvature, specifi- cally to the Gauss-Bonnet invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The BHs in EdGB have a scalar hair as they are endowed with a monopole scalar charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' However, this charge is not an independent parameter but it is a “sec- ondary hair” [43], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=', it is completely determined by the mass and spin of the BH and by the coupling constants of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The QNMs of EdGB BHs at the next-to-leading order in the spin are derived in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The numerical approximations in [40] re- strict the validity of the spectrum to ζEdGB ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4, where ζEdGB = αEdGB/M 2 f and here αEdGB is the coupling constant of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Further, the fi- nal spins χf ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='3 can be potentially outside the range of validity of the O(χ2 f) expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' To miti- gate these effects, we consider the Pad´e resummed version of the spectrum provided in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Note that the EdGB QNMs break isospectrality [44] between axial and polar sectors due to the non-minimal cou- pling of the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' In this work, we choose to focus on the spectrum corresponding to the polar sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' orem, the signals can be analyzed by assuming the Kerr BH spectrum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=', setting all δ’s to zero) and recovering posterior estimates of the mass and spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The posteriors so obtained will generally differ from the posteriors of { ˜ Mf, ˜χf}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' This means that { ˜ Mf, ˜χf} obtained here cannot be used to gauge the performance of tests like the Inspiral-merger-ringdown test [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Kerr-Newman: The Kerr-Newman spectrum for GW perturbation is derived in [45, 46] at first order in the final spin expansion (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=', [47] for a pertur- bative expansion in the electric charge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' There is strong numerical evidence that the Kerr-Newman spectrum is isospectral, which is confirmed by the full non-perturbative analysis in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Therefore, unlike in the EdGB, there is no ambiguity in choos- ing polar or axial sectors in its modified spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Note that a Kerr-Newman BH becomes extremal at charge-to-mass ratio Q = (1 − χ2 f)1/2 but we only consider values of Q away from this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Horndeski: The Horndeski action gives a general scalar-tensor gravity with second order equations of motion [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The Horndeski field equations ad- mit standard GR BH solutions under various con- ditions [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Linear perturbations around slowly rotating Kerr BHs were studied in [50] for the sub- class of Horndeski theories in which GW propagates at the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' They show that the equations are reduced to a massive scalar perturbation with an effective mass parameter µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Although the spec- trum does not correspond to perturbations in GW sector, in principle, it can be sourced by the GW sector, and we expect imprints of these frequencies in the GW signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' In this work, we only look at the QNMs in scalar sector presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='s (34)-(35) of [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Note that this spectrum reduces to the scalar perturbations of the Kerr BH in the limit µ → 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' therefore, to augment our battery of modifications, we linearly re-scale it to recover the gravitational GR Kerr BH spectrum in the limit µ → 0 and pro- mote it as yet another modified spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We re- mind that for this work, we are interested in study- ing the performance of BH spectroscopy to distin- guish a non-Kerr GR spectrum and do not aim to put bounds on any given modified theory/spectrum in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' dCS: In dynamical Chern-Simons (dCS) theory [51], a scalar field is non-minimally coupled to the higher-curvature Pontryagin invariant, resulting in a breakdown of parity symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Rotating BHs in dCS have a secondary hair in the form of a monopole scalar charge [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The QNM spectrum of dCS BHs were computed in [53] at the leading order in the spin and at second order in the non- minimal coupling constant of the scalar field αdCS (see also [54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' In the following we will use the di- mensionless coupling ζdCS = α2 dCS/M 4 f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Here, we are extrapolating the spectrum in [53] beyond the small spin approximation but this is not a critical concern for our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Also, the non-minimal cou- pling of the scalar field breaks isospectrality and therefore, similar to EdGB we consider the polar sector of the dCS spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Delta: We generate an ad-hoc phenomenological 4 spectrum by modifying the frequencies of all modes by a constant relative shift, flmn = f Kerr lmn (1 + ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We also choose to leave all damping times un- changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Delta220: We modify only the the frequency of the dominant mode f220 = f Kerr 220 (1 + ∆220) and leave all other mode frequencies and damping times un- changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The above scenarios are distinct modifications to the GR Kerr BH spectra where the no-hair hypothesis can be violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' While Kerr-Newman BHs deviate from the Kerr background due to the presence of a “primary hair” (the electric charge), in the case of EdGB and dCS BHs the background possesses a “secondary hair” (the monopole scalar charge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Further in the Horndeski BHs we consider here, the background coincides with Kerr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Moreover, note that in all the theories here, the devia- tions from GR appear also at the level of the field equa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The Delta and Delta220 are simplistic ad-hoc unrealis- tic modification schemes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' in a realistic physical scenario, we typically expect all or at least a subset of frequen- cies and damping times to be modified and it is unlikely that all modes are modified by the exact same amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We study them as they are simpler to implement and to interpret the performance of BH spectroscopy and as a benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' It also allows us to check if any of the realistic spectra can be approximated as simpler ad-hoc modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' For the EdGB, Kerr-Newman, and dCS spectra, we opt to impose consistency with the Kerr BH spectrum by linear rescaling, similar to the Horndeski case described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' If we take the limit of ζEdGB → 0 for the EdGB spectrum in [40], we do not recover the GR Kerr BH QNMs (which would be the case if the EdGB QNMs could be computed non-perturbatively) because the spec- trum is derived at O(χ2 f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Additionally, each spectrum is derived within its own set of approximations, and there- fore, they return a different approximation to the GR Kerr BH spectrum in the limit of the vanishing deviation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We provide a procedure for imposing consistency across the spectra by redefining the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We illus- trate our procedure on EdGB below – f EdGB lmn (Mf, χf, ζEdGB) = f Kerr lmn (Mf, χf) � ˆf EdGB lmn (Mf, χf, ζEdGB) ˆf EdGB lmn (Mf, χf, 0) � (9) We enforce the GR Kerr BH spectrum in the limit ζEdGB → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Here, the hat denotes the expression of EdGB spectrum presented in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Then, we defined the relative shifts as – δflmn = ˆf EdGB lmn (Mf, χf, ζEdGB) ˆf EdGB lmn (Mf, χf, 0) − 1 (10) and parametrized f EdGB lmn = f Kerr lmn (1 + δflmn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' (11) The definition (10) preserves the values of the shifts given by the traditional parametrization of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We also repeat the same procedure for the damping times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Here, we re-emphasis that this procedure is only neces- sary because the QNM spectra in these modified theories are calculated perturbatively to a limited order in the fi- nal spin of the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' In the absence of the exact spectra, we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' (11) as a fiducial definition for all the modified spectra considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 1 we plot the relative deviations δflmn and effective deviations ˜δflmn in the QNM frequencies for the modified spectra listed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We remind that it is the effective deviations ˜δflmn that are measured when performing BH spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Interestingly, we find that for some spectra even when the actual spectrum has de- viations at a percent level, the measurable effective spec- trum deviates from GR Kerr BH at a much smaller sub- percent level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' This is particularly evident for the Delta spectrum where all the true frequencies are shifted by the same amount but the measurable deviations turn out to be much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We also see this in the Kerr- Newman spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' For each spectrum and mass ratio, we quantify these differences in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We present the values of the deviation parameters α such that the devi- ation in the dominant mode frequency is at 1% level i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=', |δf220| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' V, using the values in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' I, we study the ability of BH spectroscopy to constrain ˜δflmn away from zero i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=', exclude the GR Kerr BH spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' METHODS AND IMPLEMENTATIONS We construct ringdowns that comprise of (l, m, n) ∈ {(2, 2, 0), (3, 3, 0), (2, 1, 0)} modes with the modified QNM spectra described in the previous Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We remind that the coupling parameters of the theory or the extra charges that modify the QNM spectra are cho- sen such that the dominant-mode frequency f220 differs from the GR Kerr value f Kerr 220 by 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' This is a heuris- tic choice but studying the effect of 1% deviation in f220 is a reasonable goal from the data analysis perspective as the next-generation detectors is expected to measure f220 with sub-percent accuracy [8, 12, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' For Delta and Delta220 modifications, the sign of the deviation is cho- sen to be positive i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=', fractional frequency shift of +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We fix the final mass Mf = 70M⊙ in the detector frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' To compute ρRD, we consider events with ex- trinsic parameters listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' These choices are 2 We do not consider overtones for simplicity as their measurablity and physical interpretation is still debated even when assuming that the underlying theory of gravity is GR [27, 30, 55–59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='35 EdGB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='000 flmn q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4 q = 3 q = 5 (2, 2, 0) (3, 3, 0) (2, 1, 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='35 EdGB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='005 0.' 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+page_content='0175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='0150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='0000 flmn q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4 q = 3 q = 5 (3, 3, 0) (2, 1, 0) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Rows 1-4: Relative deviations δflmn (left) and relative effective deviations ˜δflmn (right) from the GR Kerr BH spectrum, as induced by the different modified spectra discusses in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' III: EdGB, Kerr-Newman, Horndeski and dCS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Last row: Relative effective deviations ˜δflmn induces by the Delta spectrum (left) and the Delta220 spectrum (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 6 Spectrum α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 ˜ Mf (M⊙) ˜χf ˜δf330 ˜δf210 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4 , χf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='67 EdGB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='28 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='68 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='1 × 10−3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='7 × 10−3 Kerr-Newman 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='25 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='68 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='6 × 10−5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='6 × 10−5 Horndeski 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='16 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='69 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='3 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='1 × 10−2 dCS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='055 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='68 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='7 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='3 × 10−2 Delta 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='68 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='2 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='8 × 10−3 Delta220 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='68 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='5 × 10−3 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='1 × 10−3 q = 3 , χf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='54 EdGB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='31 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='54 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4 × 10−3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='8 × 10−3 Kerr-Newman 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='26 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='55 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='7 × 10−5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='5 × 10−4 Horndeski 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='15 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='57 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='3 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='1 × 10−2 dCS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='058 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='54 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='0 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='8 × 10−2 Delta 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='6 × 10−4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='1 × 10−3 Delta220 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='55 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='5 × 10−3 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='8 × 10−3 q = 5 , χf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='42 EdGB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='34 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='40 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='5 × 10−3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='2 × 10−5 Kerr-Newman 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='27 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='43 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='1 × 10−4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='2 × 10−4 Horndeski 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='14 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='46 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='3 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='1 × 10−2 dCS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='063 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='2 × 10−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='2 × 10−2 Delta 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='51 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='9 × 10−3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='1 × 10−2 Delta220 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='51 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='7 × 10−2 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='0 × 10−2 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Values α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 of the deviation parameter α inducing a 1% shift in f220, alongside the effective measurable final mass and spin { ˜ Mf, ˜χf} as well as the effective shifts ˜δflmn in the frequencies of the subdominant modes, for the different spectra con- sidered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Here α is a collective name for the additional theory parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' It represents {ζEdGB, Q, µ, ζdCS, ∆, ∆220} depending on the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 7 compatible with the GW150914 event, with the only ex- ception of the inclination angle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' the posteriors distribu- tion for the inclination angle of GW150914 prefers a face- on/face-off orientation for which the subdominant modes excitation is suppressed significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Instead, we use a more optimal inclination angle for BH spectroscopy and set ι = π/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' For each modified QNM scenario, we study mass ratio q ∈ {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4, 3, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The final spin of the remnat and the mode excitation amplitudes Almn are computed from the q using the numerical fits provided in [61] and [28], respectively, by assuming non-spinning progenitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The absolute amplitude scale A220 is set by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' (5) in [22]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' In particular, χf ≈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='67, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='54, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='52} for the three values of q chosen here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The phases φlmn do not signif- icantly affect the recovery of the QNM frequencies and damping times (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=', [30, 32]) and therefore for this study, we set it to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The ringdown waveform is modeled as h = h+ − ih×, with h+ = � l,m>0,n Almn −2Ylm,+ (ι)e−t/τlmn cos Φlmn , (12a) h× = � l,m>0,n Almn −2Ylm,× (ι)e−t/τlmn sin Φlmn , (12b) where Φlmn = 2πflmnt + φlmn, Almn and φlmn are the (real) excitation amplitudes and phases of the modes, and the plus and cross spherical harmonics are defined by4 −2Ylm,+ (ι) = −2Ylm (ι, 0) + (−1)l −2Ylm (ι, 0) , (13a) −2Ylm,+ (ι) = −2Ylm (ι, 0) − (−1)l −2Ylm (ι, 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' (13b) In writing the expression (12), we assume a non-precessing quasi-circular binary progenitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' For these systems equatorial reflection symmetries gives Al−mneiφl−mn = (−1)lAlmne−iφlmn and we simultane- ously sum over ±m using the symmetry relation ωl−mn = −ω∗ lmn 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We assume a circularly polarized ringdown as the numerical simulations favour it (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=', [25] for a de- tails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We also use spherical harmonics instead of the more natural spheroidal harmonics basis function for 3 Here we emphasize that depending on the modified theory, this assumption may not hold to differing extents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Amplitudes are governed dominantly by the plunge-merger dynamics and we typ- ically need a fully numerical simulation to infer them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' However, we do not have numerical simulations for most of the theories considered here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' thus, we do not have knowledge of the ampli- tudes of mode excitation in these theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Instead, we use the GR amplitudes as an approximate proxy for practicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 4 A global phase in the definition of the spherical harmonics can be reabsorbed in the definition of the phases φlmn of the ringdown modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 5 We use ωlmn to denote the prograde modes of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The QNM solutions also contain a set of retrograde modes, but in the numerical simulations it is seen that these modes are not excited significantly [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' See also [56, 62] on the role of retrograde modes in describing the ringdown waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' ringdown;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' this is a fairly standard approximation which is known to introduces substantial errors only in high spin limits [63, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' To predict the statistical uncertainty in the measure- ment of ˜δflmn, we employ a Fisher matrix formalism following our work in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We use the power spectral density (PSD) of the next-generation ground-based de- tector – the Einstein Telescope, assuming a triangular geometry, located in Sardinia [65] and operating at the ET-D sensitivity [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' However, note that all the quali- tative statements and ball-park numbers reported in this study should hold for any detector for a given ρRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Our study is not heavily influenced by the choice of the detec- tor as long as the modifications in the spectra are cho- sen to have 1% deviation of f220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Note that while the results in this study will be approximately similar for the next space-based detector LISA, LISA will be sen- sitive to supermassive binary BH signals and therefore, the theory-specific parameters i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=', coupling constants or extra charges, that lead to 1% deviation of f220 will be different from the choice made here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' However, we expect theory-parameters similar to the choice made here for the Cosmic Explorer, as it is sensitive to stellar mass ranges similar to the Einstein Telescope [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' For computing the Fisher covariance matrix, we pa- rameterize the waveform as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='s (6)-(7), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=', using ef- fective parameters { ˜ Mf, ˜χf, ˜δflmn, ˜δτlmn} instead of the traditionally used {flmn, τlmn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Next, for a choice of the- ory parameters that produces a modified spectra such that f220 deviates by 1% w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' GR Kerr BH, we map the true values {Mf, χf} to the measured values { ˜ Mf, ˜χf} using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='s (2) and (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Note that once we compute the values of { ˜ Mf, ˜χf}, deviation parameters for all the sub- dominant modes get fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We construct the covariance matrix for a set of 4N parameters, where N is the number of modes as – θ = { ˜ Mf, ˜χf, ˜δflmn, ˜δτlmn, log10 Almn, φlmn} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' (14) For the dominant mode, we estimate { ˜ Mf, ˜χf}, and not {˜δf220, ˜δτ220}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Next, we quantify the departure of the modified QNM spectra from GR Kerr QNM spectra using QGR as a mea- sure, where QGR is defined as the quantile at which the posterior distribution of ˜δ excludes zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Higher the value of QGR, more confidently we can exclude the null hypoth- esis that the ringdown contains QNMs corresponding to a GR Kerr BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' QGR can either be defined mode-wise over a marginalized 1-d probability distribution P(˜δflmn) or defined on a joint probability distribution of all the mea- sured modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Here, we use QGR defined on the 1-d pos- terior distributions corresponding to – a) ˜δf330 : Q330 GR, b) ˜δf210 : Q210 GR and c) on a joint posterior distribution of ˜δf330 and ˜δf210 : Q2D GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' In the Fisher matrix formalism, by construction, P(˜δflmn) is a Gaussian distribution centered at the true value of ˜δflmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' From this, Qlmn GR can be easily computed 8 ra dec ψ ι tGPS dL(Mpc) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='16 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='12 π/3 1126259462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='423 403 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Right ascension, declination, polarization an- gle, inclination angle, GPS time and luminosity distance for GW150914 that were used to compute the SNR and the Fisher covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' as Qlmn GR = erf � p/ √ 2 � , (15) where the p-value is given as p = |˜δflmn| σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' (16) Further, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' (16) can be generalized to a 2-dimensional case of the joint posterior P(˜δf330, ˜δf210).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' In the cases of 2-D posteriors – Q2D GR = 1 − exp � −1 2⃗µT · Σ−1 · ⃗µ � (17) where ⃗µ is the vector corresponding to the true values of {˜δf330, ˜δf210} and Σ is the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' RESULTS Before turning our attention to the modified spectra, we first look at the measurability of a GR Kerr BH spec- trum with the next generation ground-based detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We used ET-D PSD to illustrate the expected orders of magnitudes and trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Specifically, we consider ring- downs corresponding to q ∈ {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4, 3, 5} with Mf = 70M⊙ and the extrinsic parameters displayed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' III shows ρRD and the expected measurement uncertainties on the deviation parameters σ(˜δflmn) — note that for a GR Kerr BH spectrum, ˜δflmn = δflmn and we use tilde here for consistency of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We report the normal- ized uncertainties κ ≡ ρRDσ, where ρRD is the SNR in the ringdown and σ is the expected statistical uncertainty in the parameter computed with a Fisher information ma- trix framework 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Further, we confirm that the uncer- tainty in the measurement of the deviations in damping times σ(˜δτlmn) is large;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' therefore, we concentrate on the subdominant mode frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' To appreciate the quantitative results, we begin with il- lustrating the performance of BH spectroscopy for a) the best-case scenario, the dCS spectrum, and b) the worst- case scenario, the Kerr-Newman spectrum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' For 6 Note, the relative excitation amplitudes Almn/A220 of the sub- dominant modes increase monotonically with q for non-spinning systems [21–23, 25, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' However, ρRD decreases with q [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Therefore, while spectroscopy with high q is favourable for fixed ρRD, it need not be the case for fixed luminosity distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' q ρRD κ = ρRD σ ˜ Mf (M⊙) ˜χf ˜δf330 ˜δf210 ˜δτ220 ˜δτ330 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4 87 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='63 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='56 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='78 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='43 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='14 3 74 294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='63 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='27 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='17 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='73 5 57 436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='37 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='38 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='80 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='26 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' ρRD and uncertainties σ over mass, spin and the deviations parameters for a Kerr BH spectrum, as measured by a triangular configuration of ET detector with ET-D PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The ringdown corresponds to a BH whose detector-frame final mass is Mf = 70M⊙ and the extrinsic system parameters are enlisted in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='60 f 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='5 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='0 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='0 Mf [M ] Kerr-Newman f220 220 f330 f210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='65 f 65 70 75 80 85 90 Mf [M ] dCS f220 220 f330 f210 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Projection of BH spectroscopy on the mass-spin plane for the Kerr-Newman (top) and dCS (bottom) spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We set ρRD = 300 and q = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The bands correspond to the 90% credible intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The black and red markers indicate the measurable and true values for mass and spin, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' both cases, we use ringdowns with ρRD = 300 and q = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Specifically, we show the projections of the 90% credi- bility bands of f220, τ220, f330 and f210 on the mass-spin plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The common region of intersection of f220 and τ220 corresponds to the measurable mass and spin { ˜ Mf, ˜χf} estimate (indicated by a black marker), while the true value of {Mf, χf} (indicated by a red marker) lies outside the intersection region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' For the Kerr-Newman spectrum, all bands have a common intersection region, and there- fore, BH spectroscopy fails to detect deviations from the GR BH Kerr QNMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' On the contrary, for the dCS spec- 9 trum, there is no common intersection region and BH spectroscopy can be used to identify that this spectrum is incompatible with GR at the 90% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Next, looking at the values of { ˜ Mf, ˜χf} and of ˜δflmn in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' I, we observe that { ˜ Mf, ˜χf} does not differs sig- nificantly from {Mf, χf} and ˜δflmn ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Consequently, the results reported in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' III for the GR Kerr BH spec- tra can be expected to approximately hold for the mod- ified QNM spectra too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' This allows to derive a back-of- the-envelope ρRD estimate required to distinguish vari- ous modified spectra from the GR Kerr BH spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Further, we can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='s (15)-(16) to find the ρRD nec- essary to exclude 0 from P(˜δf330) at 90% confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='For instance, in the case of an EdGB spectrum, from (15) we observe that Qlmn GR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='9 corresponds to p ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Invert- ing (16) for ρRD = κ/σ and using the results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='s I and III, we get an approximate minimum value of ρRD as – ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='9 RD ≈ � � � � � 502 (q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4) 256 (q = 3) 278 (q = 5) (18) If we repeat this exercise for the dCS spectrum, we find ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='9 RD ≈ � � � � � 264 (q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4) 164 (q = 3) 174 (q = 5) (19) Note that it is only for the sake of demonstrating a fast and easy approximate estimation that we derive (18) us- ing the values in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' III, which assumed a GR Kerr BH spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' However, there is no fundamental obstruction to being fully consistent by applying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='s (15) and (17) using the covariance matrices from the modified spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 3, we depict marginalized 2-d posterior esti- mates of ˜δf330 and ˜δf210 for all the modified ringdowns studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Here again, we use ringdowns with ρRD = 300 to illustrate QGR as a measure to distinguish a modified QNM spectrum from the GR Kerr BH spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' When the posterior distributions are compatible with (0, 0) (in- dicated with a black dot), the QNM spectra for the f330 and f210 are compatible with the GR Kerr BH spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Note also that the shape of the contours changes with q (and that the contours do not trivially shrink with chang- ing in q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' This foreshadows the non-trivial dependence of QGR on q which will be further emphasised in fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We can identify the modifications to various level of con- fidence – Q2D GR = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='33, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='85, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='97} for Horndeski, Q2D GR = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='39, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='91, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='96} for EdGB, Q2D GR = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='85, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='99, 1} for dCS for q = {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4, 3, 5} respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We can contrast this to the case of Kerr-Newman where we do not expect to detect any deviations – we have Q2D GR = O(10−6) for q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4, O(10−3) for q = 3 and O(10−2) for q = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We will see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 4 that this is true even at very high ρRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 4, we present our main results on ρRD required to identify deviations from the GR Kerr QNM spectrum for the various modification schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The first two columns show QGR computed using a 1-d posterior distribution of f330 and f210, respectively, and the right- most column corresponds to the 2-d joint posterior distri- bution on f330 and f210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Again, we study systems corre- sponding to three mass ratios: in the first row, we study the near-equal-mass scenario of q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' in the second row, we study q = 3 and in the last row, we study q = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Let us examine the ρRD at which QGR ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='9 for all the modified QNMs considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We observe that ρRD required to distinguish the modified QNMs from GR Kerr BH spectrum depends acutely on the details of the modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Modification of the spectra, such as that predicted by Kerr-Newman and Delta, cannot be confi- dently differentiated from the GR spectrum, even when ρRD = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' This happens because these modifications can be approximated by the GR by suitably selecting the values of ˜ Mf and ˜χf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' In other words, these modi- fied spectra are highly degenerate with the GR Kerr BH QNM in the mass-spin space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' This can be quantitatively seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' For instance, ˜δflmn for these modified spectra is an order of magnitude smaller than ˜δflmn for other modifications such as dCS or EdGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' In contrast to this, modifications predicted by the EdGB and Horndeski theories can be distinguished confidently from the GR at a ρRD ∈ [250, 103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Furthermore, the dCS or Delta220- like QNM spectra could be confidently identified with an even lower ρRD ∈ [250, 400].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We highlight two non-trivial trends from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 4 that we observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' While all of them are valid measures to identify modifications in a QNM spectrum, we find that the ability to distinguish a modified theory from GR using 1-d Q330 GR, Q210 GR and 2-d joint Q2D GR are fairly different and depends on theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' While modifica- tions in the spectra predicted by dCS and Delta220 benefit by using Q330 GR, Kerr-Newmann and Delta are more distinguishable using Q210 GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Furthermore, certain types of modifications will only manifest in a detectable fashion in one of the QGR measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' For example, EdGB spectra with ρRD ≤ 103 cannot be confidently distinguished if we used Q210 GR while it can be using Q330 GR or Q2D GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The distinguishability of a given modified spectrum does not have a monotonic behaviour with q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' This occurs because q dictates both the mode excita- tion amplitudes and the final spin of the remnant BH, and the interplay between the two produces the non-monotonic trend we see here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' For instance, at a given ρRD, Q330 GR for dCS increases monotoni- cally with q, while Q330 GR for EdGB and Horndeski spectra seem to perform better for the case of q = 3 than for q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4 or 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' It is also worth noting that the performance of Q330 GR, Q210 GR and Q2D GR for differ- ent theory can differ with q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 f330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='10 f210 EdGB q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4 q = 3 q = 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='02 f330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='10 f210 Kerr-Newman q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4 q = 3 q = 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 f330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='10 f210 Horndeski q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4 q = 3 q = 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='03 f330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='10 f210 dCS q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4 q = 3 q = 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='02 f330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='10 f210 Delta q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4 q = 3 q = 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='01 f330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='10 f210 Delta220 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4 q = 3 q = 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Density plots of {˜δf330, ˜δf210} for the modified spectra considered in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' III and different mass ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Solid and dashed lines indicate the 90% and 50% probability contours respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We assume a true injected mass Mf = 70M⊙ and vanishing initial spins (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' I for the values of the corresponding effective measurable masses and spins).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We set to ρRD = 300 to clearly display the different trends of the spectra in distinguishing deviations from GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The null hypothesis under test is denoted by a black marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 11 101 102 103 RD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='9 1.' metadata={'source': 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EdGB Kerr-Newman dCS Horndeski Delta Delta220 101 102 103 RD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='5 0.' metadata={'source': 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Values of the GR quantile QGR as a function of the ρRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We consider quantiles for the marginalised single parameter posteriors P(˜δf330) (leftmost column) and P(˜δf210) (central column) and for the 2D joint posterior P(˜δf330, ˜δf210) (rightmost column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The quantiles are computed using the exact expressions (15) and (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 12 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' DISCUSSION AND CONCLUSION We performed a comprehensive study on the ability of BH spectroscopy to distinguish a modified spectra from a GR Kerr BH QNMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We studied theory-motivated mod- ifications for QNM spectra that were publicly available – specifically, EdGB, Kerr-Newman, Horndeski and dCS theory, and for two phenomenologically modified QNM spectra – Delta and Delta220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' To investigate ρRD nec- essary to distinguish these modified spectra from a GR Kerr BH spectrum, we assessed the performance of BH spectroscopy using a Fisher information matrix formal- ism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The ringdowns with the modified spectra are gen- erated such that in each case f220 deviates from the cor- responding GR Kerr QNM by 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' First, we re-iterate that we can only measure the effec- tive deviation parameters ˜δflmn and not the absolute de- viation δflmn as the mass and the spin estimation can be chosen suitably to compensate for the deviations of the QNMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Further, we show that in many theories ˜δflmn significantly differs from δflmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Therefore, to study the ability BH spectroscopy to distinguish modified spectra, the framework must be setup using the measurable effec- tive deviation parameters ˜δflmn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We found that ρRD necessary to distinguish a mod- ified spectrum from the GR Kerr one depends on the details of the modification and on the mass ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Fur- ther, we used three different measures that quantify the amount by which: a) the 1-d posterior estimate of ˜δf330 excludes 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Q330 GR, b) the 1-d posterior estimate of ˜δf210 excludes 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Q210 GR and c) the 2-d joint posterior estimation of ˜δf330−˜δf210 excludes (0,0) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Q2D GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' De- pending on the spectrum and the magnitudes of ˜δflmn, the performance of Q330 GR, Q210 GR and Q2D GR can vary signif- icantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Roughly, we find that a ρRD ≥ 150 is required to identify deviations from GR at a 90% credibility level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' This range of ρRD is attainable with the next generation detectors and BH spectroscopy will be a powerful tool to constrain GR as well as for identifying modifications to Kerr BH spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' In previous works [8, 9], we studied the measurability of QNM parameters, that is, the statistical uncertainty with which a QNM mode parameter can be estimated from a signal to assess the landscape of BH spectroscopy with a next-generation detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' However, we note here that the subdominant QNM mode with the best mea- surability may not always be the optimal mode for dis- tinguishing a modified QNM spectrum from a GR Kerr BH QNM spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The best mode to identify a depar- ture from GR Kerr spectrum depends on the interplay between measurablity and the details of modification of the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' For instance, measurablity of f330 is better than f210 in a non-spinning binary BH ringdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' How- ever, if the compact object is a Kerr-Newman BH instead of a Kerr BH, f330 measured in the ringdown would be compatible with the Kerr BH even for ringdown with ρRD ∼ 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The modifications in the spectra in this case can be observed predominantly by looking at f210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Therefore, using a setup that uses information in all mea- surable QNM mode parameters is optimal while looking for departure from the GR Kerr spectrum in a ringdown signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Greater the number of QNM mode parameters measured, greater the chances that we can distinguish a modified QNM spectra from the GR Kerr spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Furthermore, among the modified QNM spectra we have studied, we see the theories separate out into two groups : EdGB, Horndeski, dCS, Delta220: For these spec- tra, a GR Kerr ringdown can be excluded at 90% confidence level with a SNR ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='9 RD ∈ [150, 500].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' The deviations manifest more prominently in the (3, 3, 0) mode and the (2, 1, 0) mode offers a rela- tively poor constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Kerr-Newman, Delta: here the deviations manifest almost exclusively in the (2, 1, 0) mode, with Q210 GR returning the most performative no-hair test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' How- ever, the ρRD required to exclude 0 at 90% confi- dence is so high that these deviations are indistin- guishable, even at ρRD = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Finally, works such as this is essential for gauging the potential of ringdown-based tests of gravity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' but unfor- tunately, we are restricted by publicly available modified QMN spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' While it is difficult to solve the QNM spec- tra in modified theories of gravity, to adequately prepare for test of GR with next-generation detectors including designing the implementation of BH spectroscopy, we feel that the field would benefit substantially from investing effort in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Pierini for clarifications about the work in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' We thank E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Berti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Gualtieri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Maselli and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Pani for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' is supported by Eu- ropean Union’s H2020 ERC Starting Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 945155– GWmining and by Cariplo Foundation Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 2021- 0555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' would like to acknowledge the UKRI Stephen Hawking Fellowship Engineering and Physical Sciences Research Council (EPSRC), Science and Technology Fa- cilities Council (STFC), with grant reference number EP/W005727 for support during this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Teukolsky, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' 29, 1114 (1972).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Foffa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Garc´ı a-Bellido, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Grimm, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Harms, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Hin- derer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} +page_content=' Matarrese, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE0T4oBgHgl3EQfVwAY/content/2301.02267v1.pdf'} 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We show that a generic levelset of the viscosity solution +to mean curvature flow is a distributional solution in the framework +of sets of finite perimeter by Luckhaus and Sturzenhecker, which in +addition saturates the optimal energy dissipation rate. This extends the +fundamental work of Evans and Spruck (J. Geom. Anal. 1995), which +draws a similar connection between the viscosity solution and Brakke +flows. +1. Introduction +The mean curvature flow is a parabolic geometric evolution equation with +numerous applications in the sciences and engineering. It is the most basic +model describing slow relaxation driven by surface tension and can be viewed +as the diffusion equation of surfaces relating the normal velocity V and the +mean curvature H of an evolving surface via the equation V = −H. +Since the mean curvature flow develops singularities in finite time, it is +natural to consider weak solutions which are able to describe the evolution +through singular events such as pinch-off. However, after such singularities, +the flow may become non-unique [3] and it is in general difficult to compare +different weak solutions. The aim of this paper is to better understand weak +solutions and in particular draw a new connection between different weak +solution concepts. We will show that generically, they are consistent with +each other. +More precisely, we consider the viscosity solution introduced indepen- +dently by Evans and Spruck [6], and Chen, Giga and Goto [4]. This concept +is based on the levelset formulation of Osher and Sethian [16] in which all +levelsets of a function are evolved simultaneously. We show that almost ev- +ery levelset of the viscosity solution is an evolving set of finite perimeter +which moves according to the distributional formulation of mean curvature +flow due to Luckhaus and Sturzenhecker [14]. In addition, we show that the +evolving sets of finite perimeter satisfy an optimal energy dissipation rela- +tion, which is crucial for the weak-strong uniqueness of the solution concept, +Key words and phrases. Mean curvature flow, weak solution concepts, viscosity so- +lution, distributional solution, compensated compactness. MSC2020: 53E10 (Primary); +35D40; 35D30 (Secondary). +1 +arXiv:2301.01097v1 [math.AP] 3 Jan 2023 + +2 +TIM LAUX AND ANTON ULLRICH +cf. [9]. Additionally, together with the distributional identity between nor- +mal velocity and mean curvature operator, the solution can in addition be +viewed as curve of maximal slope in the spirit of De Giorgi, cf. [5, 2]. More +precisely, our solution also recovers the De Giorgi formulation in the setting +of BV solutions to mean curvature flow which was introduced by Otto and +one of the authors [12] to describe limits of the thresholding scheme. +It is a by now classical result due to Evans and Spruck [6] that the vis- +cosity solution is consistent with the classical solution in the sense that, +when starting from the same initial data, the two solutions coincide on their +common time interval of existence. More recently, such a consistency – or +weak-strong uniqueness – has also been shown between BV solutions and +the classical solution to mean curvature flow in [9]. Our result extends those +statements by stating that also past singularities, when the classical solution +ceases to exist, these two weak solution concepts are consistent with each +other. More precisely, we prove the following statement. +Theorem 1.1. Let u ∈ C(Rd × [0, T)) be a viscosity solution to mean cur- +vature flow in the sense of Definition 2.2 with well-prepared initial data ac- +cording to Definition 3.1. Then almost every levelset of u is a BV solution +to mean curvature flow in the sense of Definition 2.3. +The idea of our proof is rather simple. We use the vanishing-viscosity +approximation by Evans and Spruck [6]. The strengthened convergence nec- +essary to carry out our argument comes from the surprising estimate from [8] +in the spirit of +sup +t>0 +ˆ +Rd |H| dx < ∞. +This makes possible a compensated compactness argument to show that +the energies, which are modeled after +´ +Rd |∇u| dx, converge in the vanishing- +viscosity limit. By lower semi-continuity and the coarea-formula, this implies +that almost every levelset converges strictly, i.e., the perimeter does not +drop down in the limit. +Such an energy convergence is an essential tool +for convergence proofs for approximations of mean curvature flow, see [14, +11, 13]. Therefore, it is a natural question whether one can verify the BV +formulation also for the viscosity solution of the levelset flow. We do this +by first deriving a distributional formulation of levelset mean curvature flow +that is satisfied by the viscosity solution. Then we disintegrate the levelset +parameter in this formulation. On a technical level, this is done by using +the coarea-formula for surface-type integrals and the layer-cake formula for +volume-type integrals. +2. Definitions of the distributional and generic formulation +In this section we recall the definition of the levelset solution to mean +curvature flow and how to generalize this concept to viscosity solutions. +Afterwards, the notion of distributional solutions is stated and explained. + +GENERIC LEVEL SETS IN MCF +3 +Figure 1. Example of a time-instance of a levelset function. +Levelsets for four different values are marked in red. +To give an intuition of the levelset equation of mean curvature flow, let +u: Rd × [0, T) → R be a smooth function such that each levelset evolves +by mean curvature flow. We aim to find a partial differential equation for +u. +To this end, let x(t), t ∈ (t1, t2) ⊂ (0, T), be a trajectory following +one levelset, i.e, u(x(t), t) = const. and suppose ∇u(x(t), t) ̸= 0 for all +t ∈ (t1, t2). The outer unit normal vector to the superlevelsets is given by +ν(x, t) = − ∇u(x,t) +|∇u(x,t)| and since the mean curvature is the divergence of the +outer unit normal, H = ∇ · ν, the mean curvature flow equation can be +expressed by ˙x · ν = −∇ · ν. Thus +0 = d +dtu(x(t), t) = ∂tu + ˙x · ∇u = ∂tu + |∇u|∇ · ν +and hence, we obtain the levelset equation for u: +∂tu = −|∇u|∇ · ν = ∆u − ∇u +|∇u| · ∇2u ∇u +|∇u|. +This leads to the following definition which is equivalent to the fact that +the levelsets of u evolve by mean curvature flow, cf. [6]. +Definition 2.1 (Levelset equation). A function u: Rd × (0, T) → R with +u ∈ C1 +t ∩ C2 +x is called a solution to the levelset equation of mean curvature +flow if it solves the initial value problem +� +∂tu = ∆u − ∇u +|∇u| · ∇2u ∇u +|∇u| +in Rd × (0, T), +u = g +on Rd × {t = 0}. +(1) +This solution concept can be extended to the non-smooth case even after +the onset of singularities using viscosity solutions, cf. [6, 4]. These use the +comparison principle of the mean curvature flow to control the evolution of +the levelsets. Here, one considers continuous levelset functions. Intuitively, + +9 +8 +7 +6~ +5 ~ +4 +3 +2~ +1 . +3 +2 +0 +-1 +0 +2 +-1 +2 +-3 +-34 +TIM LAUX AND ANTON ULLRICH +the idea is to take smooth functions locally approximating the levelset func- +tion in an ordered way and impose the respective inequality on these smooth +functions. This solution concept is consistent with the levelset solutions for +smooth functions u. One way to prove this is by the characterization using +sub- and superjets, cf. [6, Section 2.3]. +Definition 2.2 (Viscosity solution). A continuous function u ∈ C(Rd × +[0, T)) is called a viscosity super-solution to (1) if, for any ϕ ∈ C∞(Rd × +(0, T)) and (x0, t0) ∈ Rd × (0, T) such that u − ϕ has a local minimum at +(x0, t0), the following inequality holds, +i) for non-critical points ∇ϕ(x0, t0) ̸= 0 : +∂tϕ ≥ ∆ϕ − ∇ϕ +|∇ϕ| · ∇2ϕ ∇ϕ +|∇ϕ| +at (x0, t0) +ii) and for critical points ∇ϕ(x0, t0) = 0 there exists ξ ∈ Rd, |ξ| ≤ 1 : +∂tϕ ≥ ∆ϕ − ξ · ∇2ϕξ +at (x0, t0). +We say that u is a viscosity sub-solution if −u is a viscosity super-solution. +Finally, u is called a viscosity solution if it is a viscosity sub- as well as a +super-solution. +Viscosity solutions are unique, i.e., it can be shown that there exists at +most one viscosity solution to mean curvature flow, cf. [6, Theorem 3.2]. +However, the levelsets may develop an interior, a phenomenon called fatten- +ing. +Another way to extend the solution concept of mean curvature flow are so- +called BV solutions due to Luckhaus and Sturzenhecker [14]. This definition +uses distributional characterizations of the mean curvature and the normal +velocity. Additionally, we impose a sharp energy dissipation which extends +the definition of Luckhaus and Sturzenhecker. +In the following the Gauß–Green measure µΩ denotes the (negative) spa- +tial distributional derivative of the characteristic function µΩ(t) = −∇χΩ(t) +for every time t ∈ (0, T) and |µΩ(t)| denotes its total variation. Moreover, +the perimeter of a set Ω(t) is defined as +P(Ω(t)) := |µΩ(t)|(Rd) = sup +� +� +� +� +� +ˆ +Ω(t) +∇ · ξ dx: ξ ∈ C1 +c (Rd; Rd), |ξ| ≤ 1 +� +� +� +� +� +. +Note that by De Giorgi’s structure theorem (cf. Theorem 6.1), integration +against the Gauß–Green measure precisely corresponds to integration over +the reduced boundary ∂∗Ω(t) with respect to the Hausdorff measure Hd−1. +Definition 2.3 (BV solution). A family of finite perimeter sets (Ω(t))t∈[0,T) +depending measurably on t ∈ [0, T) is called distributional solution to mean + +GENERIC LEVEL SETS IN MCF +5 +curvature flow if the sets admit a uniform perimeter bound +ess sup +t∈[0,T) +P(Ω(t)) < ∞ +(2) +and there is a d|µΩ(t)| dt-measurable L2 bounded function V : Rd×(0, T) → R +ˆ +Rd×(0,T) +V 2 d|µΩ(t)| dt < ∞ +(3) +such that the following holds: +i) The function V (·, t) is the normal velocity of Ω(t) in the sense that +for each test function ζ ∈ C1 +c (Rd × [0, T)) it satisfies +T +ˆ +0 +ˆ +Ω(t) +∂tζ dx dt = − +ˆ +Rd×(0,T) +ζV d|µΩ(t)| dt − +ˆ +Ω(0) +ζ(x, 0) dx. +(4) +ii) Furthermore, for any test vector field ξ ∈ C1 +c (Rd ×(0, T); Rd) it holds +ˆ +Rd×(0,T) +(∇ · ξ − ν · ∇ξν) d|µΩ(t)| dt = − +ˆ +Rd×(0,T) +V ξ · ν d|µΩ(t)| dt. +(5) +iii) For almost any time T ′ ∈ (0, T) the sharp energy–dissipation holds +P(Ω(T ′)) + +ˆ +Rd×(0,T ′) +V 2 d|µΩ(t)| dt ≤ P(Ω(0)). +(6) +Equation (4) characterizes the normal velocity for smooth sets Ω(t), which +is a direct consequence of the computation of ∂t +´ +Ω(t) ζ dx. +The operator +on the left-hand side of Equation (5) encodes the mean curvature via an +integration by parts on the surface, which can be easily seen for smooth sets. +This solution concept represents the mean curvature flow since V is the +distributional normal velocity and simultaneously the negative distributional +mean curvature. +In addition to the distributional formulation by Luckhaus and Sturzen- +hecker, we also have a sharp energy-dissipation relation given by Inequal- +ity (6). +3. Recap of results by Evans and Spruck +In this section, we recall the important results from [6], [7] and [8] that +are needed to prove Theorem 1.1. We will consider the vanishing viscosity +approximation to our levelset formulation +∂tuε = ∆uε − +∇uε +� +|∇uε|2 + ε2 · ∇2uε +∇uε +� +|∇uε|2 + ε2 . +(7) +This approximation is non-degenerate and has sufficient convergence to the +levelset solution as we will see in Theorem 3.4. In addition, it has a beautiful + +6 +TIM LAUX AND ANTON ULLRICH +geometric interpretation as the mean curvature flow of graphs in a higher +dimensional space, cf. [6]. +Throughout, we will assume that the initial datum for the viscous approx- +imation (7) is well-prepared in the following sense. +Definition 3.1. We call an initial levelset data g well-prepared if g ∈ +C3(Rd), g is constant outside a ball of radius R > 0, and g possesses an +integrable approximate mean curvature with a uniform bound in ε in the +sense that +sup +0<ε≤1 +ˆ +Rd +���∇ · +� +∇g +� +|∇g|2 + ε2 +���� dx < ∞. +(8) +Remark 1. Following [8, Lemma 2.1] one can construct well-prepared initial +data g for any given compact smooth 0-levelset. +The viscous approximation (7) has better properties in the sense that it +yields smooth solutions, cf. [7, Chapter 2]. For these, the following bounds +on the solution and its derivatives hold, cf. [6, Theorem 4.1]). +Lemma 3.2. For well-prepared initial data there exists a unique smooth +solution uε to (7). Additionally, |uε|, |∇uε|, |∂tuε| are uniformly bounded in +ε, x and t. +The central quantity of our argument will be the generalized normal ve- +locity +(9) +V := ∂tu +|∇u|χ{∇u̸=0}. +We will also need the generalized outer unit normal vector +ν := +� +− ∇u +|∇u| +if ∇u ̸= 0, +e1 +else. +To prove the convergences of the vanishing viscosity approximation, the most +crucial step is to utilize the following unexpected estimate due to Evans and +Spruck [8]. +Theorem 3.3. We have the uniform L∞ +t L1 +x-bound +sup +0≤t≤T +sup +0<ε≤1 +ˆ +Rd +|Hε(x, t)| dx < ∞, +(10) +where Hε(x, t) := ∇ · νε is the approximate mean curvature for the levelsets +of uε and νε = − +∇uε +√ +|∇uε|2+ε2 the approximate outer normal vector. +This statement can be found in [8, Theorem 2.2] and follows from the +fact that the quantity t �→ +´ +Rd |Hε(x, t)| dx is non-increasing. This uniform +L∞ +t L1 +x-bound on the approximate mean curvature has strong consequences as +can be seen in the following theorem, cf. [8, Chapter 3]. For a subsequence + +GENERIC LEVEL SETS IN MCF +7 +the approximate mean curvature Hε converges in the weak-∗ topology of +measures. Together with the relation to the approximate normal this yields +the additional convergences in the following. +One can show that u is bounded and Lipschitz in space and time, cf. [7]. +Thus, by Rademacher’s theorem the derivatives are defined almost every- +where. +Theorem 3.4 (Convergence, cf. [8]). Considering subsequences by Lemma 3.2 +we can infer by Arzelà–Ascoli that +uε → u locally uniformly. +Similarly, +∇uε +∗⇀ ∇u in L∞(Rd) +and +∂tuε +∗⇀ ∂tu in L∞(Rd). +Using Theorem 3.3 for a compensated compactness argument with νε and +the definition we conclude +|∇uε| +∗⇀ |∇u| in L∞(Rd) +and +νε +∗⇀ ν in L∞(Rd). +Finally, by an excess decay argument it follows +νε → ν in L2({∇u ̸= 0}). +The key argument for this theorem is that once one writes +|∇uε| ≈ −νε · ∇uε, +one can pass to the limit in the product on the right-hand side via compen- +sated compactness since uε → u uniformly and ∇ · νε = Hε is uniformly +bounded in ε as Radon measures. Another useful lemma is the following, +which tells us that certain integral quantities are bounded. It will be impor- +tant in taking the limit and proving estimates for u given the corresponding +estimates for uε. +This is Theorem 4.1 in [8] and follows from the bounds of uε and its +derivatives (Lemma 3.2). +Lemma 3.5. Let K ⊂ Rd × (0, T) be a compact set. Then the following +bound on the approximate mean curvature holds +sup +0<ε≤1 +T +ˆ +0 +ˆ +Rd +χK|Hε|2� +|∇uε|2 + ε2 dx dt < ∞. + +8 +TIM LAUX AND ANTON ULLRICH +This estimate is more intuitive than (10) as for smooth manifolds evolv- +ing by mean curvature flow +´ +|H|2 is the first variation of the perimeter +functional and the factor |∇u| is the coarea-factor. +Additionally, we will use a relabeling property which is stated in the fol- +lowing lemma and originates from [8, Theorem 6.3]. +Lemma 3.6 (Relabeling property). Let g be well-prepared initial data, cf. +Definition 3.1 and u the unique viscosity solution to (1) as well as Φ ∈ +C∞(R) be a smooth testfunction with positive derivative Φ′ > 0. Then Φ ◦ u +is the unique viscosity solution to the initial data Φ◦g. Moreover, the initial +data Φ ◦ g is well-prepared. +4. Generic levelsets are BV solutions +Next, we will begin with the proof of Theorem 1.1, i.e., that almost every +levelset of the viscosity solution is a distributional solution in the sense of +BV solutions. +The following first theorem can be understood as an integrated version of +the distributional equations for normal velocity and mean curvature. The +main novelty is Equation (11), which encodes the normal velocity while the +second part (Equation (12)) can be found as Theorem 5.1 in [8] with H in- +stead of V . Having proved these identities for the levelset function one can +then proceed by disintegrating the levelset parameter to get the correspond- +ing one identities for almost every levelset. This is done below in Theorem 4.4 +using the layercake-formula and the coarea-formula (Lemma 6.2). +Theorem 4.1. Let u ∈ C(Rd × [0, T)) be the unique viscosity solution with +some well-prepared initial data g. Then there exists a |∇u| dx dt-measurable +function V : Rd × (0, T) → R such that +i) For any test function ζ ∈ C1 +c (Rd × [0, T)) it holds +T +ˆ +0 +ˆ +Rd +∂tζu dx dt = − +T +ˆ +0 +ˆ +Rd +ζV |∇u| dx dt − +ˆ +Rd +g ζ(·, 0) dx. +(11) +ii) For any test vector field ξ ∈ C1 +c (Rd × (0, T); Rd) we have +T +ˆ +0 +ˆ +Rd +(∇ · ξ − ν · ∇ξν) |∇u| dx dt = − +T +ˆ +0 +ˆ +Rd +ξ · νV |∇u| dx dt. +(12) +Proof. The idea is to prove an approximate version for uε using integration by +parts and the definition of the vanishing viscosity equation. Afterwards, one +uses Lemma 3.5 and the convergences in Theorem 3.4 to get the equation +for u. Here, the convergence up to the set {∇u = 0} suffices since their +contribution will vanish. +Argument for (11). Fix a test function ζ ∈ C1 +c (Rd ×[0, T)). We first claim +that ∂tu = V |∇u| almost everywhere on Rd × (0, T). Then we may conclude + +GENERIC LEVEL SETS IN MCF +9 +via +T +ˆ +0 +ˆ +Rd +∂tζu dx dt = − +T +ˆ +0 +ˆ +Rd +ζ∂tu dx dt − +ˆ +Rd +gζ(·, 0) dx += − +T +ˆ +0 +ˆ +Rd +ζV |∇u| dx dt − +ˆ +Rd +g ζ(·, 0) dx, +which is precisely (11). +Now we argue that indeed ∂tu = V |∇u| almost everywhere. First, ob- +serve that by definition of the normal velocity V , cf. (9), the identity holds +on the set {∇u ̸= 0}. Thus, we only need to argue that ∂tu = 0 almost +everywhere on {∇u = 0}. This in turn follows from the next computation, +cf. [8, Lemma 4.2]. Note that we can express the evolution equation (7) as +∂tuε = −Hε +� +|∇uε|2 + ε2. Then for every compact subset A of the critical +set {∇u = 0}: +������ +ˆ +A +∂tu dx dt +������ +≤ lim +ε↘0 +ˆ +A +|∂tuε| dx dt += lim +ε↘0 +ˆ +A +|Hε| +� +|∇uε|2 + ε2 dx dt +≤ lim inf +ε↘0 +� +� +ˆ +A +H2 +ε +� +|∇uε|2 + ε2 dx dt +� +� +1 +2 +× +� +� +ˆ +A +� +|∇uε|2 + ε2 dx dt +� +� +1 +2 +≤ C +ˆ +A +|∇u| dx dt = 0, +where in the last step we have used Lemma 3.5 to bound the first term and +Theorem 3.4 to pass to the limit in the second term. Now by inner regularity +this proves ∂tu = 0 almost everywhere on {∇u = 0}. +Argument for (12). As stated above, the following argument can be found +in [8, Theorem 5.1], but for the convenience of the reader we highlight the +main idea. Testing the equation (7) with ξ · νε for a given test vector field +ξ ∈ C1 +c (Rd×(0, T); Rd) and integrating by parts yields the following ε-variant +of (12): +T +ˆ +0 +ˆ +Rd +(∇ · ξ − νε · ∇ξνε) +� +|∇uε|2 + ε2 dx dt = − +T +ˆ +0 +ˆ +Rd +ξ · νε∂tuε dx dt. + +10 +TIM LAUX AND ANTON ULLRICH +Now Theorem 3.4 and Lemma 3.5 allow us to pass to the limit ε ↘ 0 on +both sides of this identity. +□ +We additionally need the optimal energy dissipation. This will be a con- +sequence of the following theorem for the levelset function u, which follows +from [8, Theorem 5.2] where again V is called H. +Theorem 4.2. For the viscosity solution u and any two time instances 0 ≤ +t1 < t2 ≤ T, we have a version of the optimal energy dissipation: +ˆ +Rd +|∇u| dx +������ +t2 ++ +t2 +ˆ +t1 +ˆ +Rd +V 2|∇u| dx dt ≤ +ˆ +Rd +|∇u| dx +������ +t1 +. +(13) +From the previous two theorems, one can extract the levelset case for +almost every levelset by use of the coarea-formula (Lemma 6.2) and the +layercake-formula. To this end, it is necessary to define a notion for the +levelsets and superlevelsets. +Definition 4.3 (Ωs, Σs). For a viscosity solution u to (1) and any s ∈ R +and t ∈ [0, ∞), one defines the superlevelsets Ωs(t) := {x : u(x, t) > s} and +the levelsets Σs(t) := {x : u(x, t) = s}. +The following theorem states the disintegrated version of Theorem 4.1. +Theorem 4.4. Let u be a viscosity solution to the levelset equation (1) and +let Ωs, Σs be given as in Definition 4.3. Then for almost every value s ∈ R +we have +i) The normal velocity of Σs is given by V in the sense that for all test +functions ζ ∈ C1 +c (Rd × (0, T)): +T +ˆ +0 +ˆ +Ωs(t) +∂tζ dx dt = − +T +ˆ +0 +ˆ +Σs(t) +ζV dHd−1(x) dt − +ˆ +Ωs(0) +ζ(·, 0) dx, +(14) +ii) The equation V = −H is satisfied for Σs in the sense that for all test +vector fields ξ ∈ C1 +c (Rd × (0, T); Rd): +T +ˆ +0 +ˆ +Σs(t) +(∇ · ξ − ν · ∇ξν) dHd−1(x) dt = − +T +ˆ +0 +ˆ +Σs(t) +ξ · νV dHd−1(x) dt. +(15) +Equation (15) is similar to the one found in [8] now on a levelset basis +while (14) is new. +Proof. The idea is to use Theorem 4.1 combined with the relabeling property. +By the coarea-formula and layercake-formula one can split the integrals into +the (super-)levelsets and then use the relabeling to separate these and get +the equality for almost every levelset separately. + +GENERIC LEVEL SETS IN MCF +11 +Argument for (14): Fix Φ ∈ C∞(R) with Φ′ ≥ 0 and observe that Φ ◦ u +is the unique viscosity solution to the well-prepared initial data Φ ◦ g by +Lemma 3.6. Now we can use equation (11) with VΦ = +∂tu +|∇u| = V . By the +coarea-formula the right-hand side is equal to +ˆ +R +Φ′(s) +T +ˆ +0 +ˆ +Σs(t) +ζV dHd−1(x) dt ds − +ˆ +Rd +gζ(·, 0) dx. +For the left-hand side and the initial data term one can use the layercake- +formula with a transformation, where K ∈ R, K < umin and umin is the +infimum of u in space and time (see Lemma 3.2): +Φ ◦ u = +Φ◦u +ˆ +Φ(K) +1 dτ + Φ(K) = +∞ +ˆ +K +χ{u>s}Φ′(s) ds + Φ(K). +This yields for the left-hand side of (11) +T +ˆ +0 +ˆ +Rd +∂tζΦ ◦ u dx dt = +T +ˆ +0 +ˆ +Rd +∂tζ +∞ +ˆ +K +χ{u>s}Φ′(s) ds dx dt + +T +ˆ +0 +ˆ +Rd +∂tζΦ(K) dx dt += +∞ +ˆ +K +Φ′(s) +T +ˆ +0 +ˆ +Ωs(t) +∂tζ dx dt ds + 0. +Now since Φ was arbitrary under the above assumptions one can use Φ(s) = +s´ +−∞ +ϕ(s′) ds′ for ϕ ∈ C∞ +c (R; R≥0) a bump-function. Then by the fundamental +theorem of the calculus of variations, one has for almost every levelset value s +(for s < umin both sides equal 0) and for all test functions ζ ∈ C1 +c (Rd×(0, T)) +T +ˆ +0 +ˆ +Ωs(t) +∂tζ dx dt = − +T +ˆ +0 +ˆ +Σs(t) +ζV dHd−1(x) dt − +ˆ +Ωs(0) +ζ(·, 0) dx. +Argument for (15): Fix again Φ ∈ C∞(R) with Φ′ > 0 and well-prepared +initial data g. By Lemma 3.6, Φ ◦ g is also well-prepared. +Then, by (12) using νΦ = − ∇(Φ◦u) +|∇(Φ◦u)| = ν, |∇(Φ ◦ u)| = Φ′ ◦ u|∇u|, we +obtain: +T +ˆ +0 +ˆ +Rd +Φ′ ◦ u (∇ · ξ − ν · ∇ξν) |∇u| dx dt += − +T +ˆ +0 +ˆ +Rd +Φ′ ◦ u ξ · νV |∇u| dx dt + +12 +TIM LAUX AND ANTON ULLRICH +and by the coarea-formula, this can be written as +ˆ +R +Φ′(s) +T +ˆ +0 +ˆ +Σs(t) +(∇ · ξ − ν · ∇ξν) dHd−1(x) dt ds += − +ˆ +R +Φ′(s) +T +ˆ +0 +ˆ +Σs(t) +ξ · νV dHd−1(x) dt ds. +As before, for almost every s this yields the desired equality +T +ˆ +0 +ˆ +Σs(t) +(∇ · ξ − ν · ∇ξν) dHd−1(x) dt = − +T +ˆ +0 +ˆ +Σs(t) +ξ · νV dHd−1(x) dt, +which concludes the proof. +□ +Similarly, one can prove the levelset version of Theorem 4.2. This version +will become the sharp energy dissipation for the BV solution. +Theorem 4.5. For almost every levelset value s ∈ R and time instances +0 ≤ t1 < t2 ≤ T the sharp energy dissipation holds: +Hd−1(Σs(t2)) + +t2 +ˆ +t1 +ˆ +Σs(t) +V 2 dHd−1(x) dt ≤ Hd−1(Σs(t1)). +(16) +Proof. This proof is similar to the previous theorem. We use the coarea- +formula in Equation (13) to show that for almost every levelset the difference +of the sides of the inequality has a value of less or equal to zero: +By the relabeling property for Φ ∈ C∞(R) with Φ′ ≥ 0 applied to Equa- +tion (13) one gets: +0 ≥ +ˆ +Rd +Φ′ ◦ u|∇u| dx +������ +t2 ++ +t2 +ˆ +t1 +ˆ +Rd +Φ′ ◦ uV 2|∇u| dx dt − +ˆ +Rd +Φ′ ◦ u|∇u| dx +������ +t1 += +ˆ +R +Φ′(s) +� +�Hd−1(Σs(t2)) + +t2 +ˆ +t1 +ˆ +Σs +V 2 dHd−1(x) dt − Hd−1(Σs(t1)) +� +� ds. +Since Φ′ ≥ 0 and by the fundamental theorem of the calculus of variations, +this yields the desired inequality. +□ +To prove now that almost every levelset is a distributional solution, some +well-posedness conditions have to be satisfied. The following lemma shows +that our sets are sets of finite perimeter and that u is differentiable a.e. on +these, which is stated in the following lemma that can be found in [8] as +Lemma 6.1. + +GENERIC LEVEL SETS IN MCF +13 +Lemma 4.6. For every time 0 ≤ t ≤ T and almost every levelset value +s ∈ R the viscosity solution u is differentiable Hd−1-almost everywhere on +Σs with non-vanishing gradient |∇u| ̸= 0. +Proof. For fixed 0 ≤ t ≤ T the function u is Lipschitz and thus by the +coarea-formula for ζ : Rd → R integrable one has +ˆ +Rd +ζ|∇u| dx = +ˆ +R +ˆ +Σs +ζ dHd−1 ds. +For ζ = χA, A := {u not diff. or ∇u = 0} it yields Hd−1(A ∩ Σs) = 0. +□ +Proof of Theorem 1.1. Let u ∈ C(Rd × (0, T)) be a viscosity solution to the +levelset equation (1) of mean curvature flow with well-prepared initial data +g ∈ C3(Rd). Then, by Inequality (16) in Theorem 4.5 the L2-bound (3) of +V is satisfied (when plugging in t1 = 0 and t2 = T). This is due to the fact +that for almost all levelsets the reduced boundary of the superlevelsets is the +levelset ∂∗Ωs(t) = Σs(t). Similarly, the perimeters are uniformly bounded +(2) since they are monotonically decreasing. +Moreover, the distributional characterizations (4) and (5) of the normal +velocity V and the mean curvature H were proven in Theorem 4.4 in Equa- +tions (14) and (15), respectively. +Lastly, the sharp energy dissipation (6) can be inferred by plugging t1 = +0, t2 = T ′ into Inequality (16) in Theorem 4.5. +□ +5. Extensions and open problems +5.1. Extension to homogeneous Neumann boundary conditions. In +this paper, we worked in the setting of Rd with initial data which are constant +outside a ball. One could also consider the case of homogeneous Neumann +data, i.e., the surface Σ satisfies V = −H and is orthogonal to a given fixed +domain boundary Σ ⊥ ∂D. +Similarly to before, one can define levelset solutions as well as viscosity +solutions and an analogous vanishing viscosity approximation, cf. [1]. The +levelset equation becomes +� +� +� +� +� +∂tu = ∆u − ∇u +|∇u| · ∇2u ∇u +|∇u| +in D × (0, T), +∇u · ν∂D = 0 +on ∂D × (0, T), +u = g +on D × {t = 0}. +Here, the initial data g also has to satisfy the boundary condition ∇g ·ν∂D = +0. Now, the vanishing viscosity approximation can be constructed in the +same way satisfying the condition ∇uε · ν∂D = 0. The major difference in +this context lies in the formulation of the distributional solution. To encode +the boundary conditions, we use again integration by parts, see also [10]. +Indeed, the homogeneous Neumann condition ν∂D · ∇u = 0 can be encoded + +14 +TIM LAUX AND ANTON ULLRICH +by asking that for any test vector field ξ ∈ C1 +c (D×(0, T); Rd) with ξ·ν∂D = 0 +it holds +T +ˆ +0 +ˆ +Rd +(∇ · ξ − νu · ∇ξνu) |∇u| dx dt = − +T +ˆ +0 +ˆ +Rd +ξ · νuV |∇u| dx dt. +This weak formulation follows from from the following calculation (in +which we suppress the t-dependence) for the vanishing viscosity approxi- +mation +− +ˆ +D +ξ · νε∂tuε dx = − +ˆ +D +ξ · ∇uε∇ · νε dx += +ˆ +D +νε · ∇ξ∇uε dx + +ˆ +D +ξ · ∇2uενε dx +− +ˆ +∂D +ξ · ∇uενε · ν∂D dHd−1(x) += +ˆ +D +νε · ∇ξ∇uε dx − +ˆ +D +ξ · ∇ +� +|∇uε|2 + ε2 dx += +ˆ +D +νε · ∇ξ∇uε dx + +ˆ +D +∇ · ξ +� +|∇uε|2 + ε2 dx +− +ˆ +∂D +ξ · ν∂D +� +|∇uε|2 + ε2 dHd−1(x) += +ˆ +D +(∇ · ξ − νε · ∇ξνε) +� +|∇uε|2 + ε2 dx +− +ˆ +∂D +ξ · ν∂D +� +|∇uε|2 + ε2 dHd−1(x). +For test vector fields ξ with ξ · ν∂D, the additional boundary term vanishes +exactly. Then, one can pass to the limit ε ↘ 0. With these changes, we can +modify the proof to show Theorem 1.1 for homogeneous Neumann conditions. +Due to [1, Lemma 2.3] the existence of well-prepared initial data to a 0- +levelset which meets ∂D orthogonally is ensured. Moreover, in this paper, the +integrability of the approximate mean curvature as well as the convergences +and the L2-integrability with coarea-factor is shown, c.f. [1, Theorem 3.2, +Theorem 3.3 and Theorem 3.4]. Theorems 4.2 and 4.1 follow with the same +reasoning with the sole difference that the convergences now use that we are +on the bounded domain D. + +GENERIC LEVEL SETS IN MCF +15 +5.2. Further directions and open questions. We showed that almost +every levelset of the viscosity solution is a BV solution. It remains an in- +teresting open question whether the inverse assertion holds. Let g be well- +prepared initial data and u ∈ C(Rd × (0, T)) such that every levelset of u is +a distributional solution to mean curvature flow. Is u then a viscosity solu- +tion to mean curvature flow? By the weak-strong uniqueness result in [9], +this is true as long as a classical solution exists. However, after the onset of +singularities this is still an open question. +A further interesting generalization would be to incorporate non-homoge- +neous Neumann condition, i.e., ∇u · νD = cos α. Geometrically speaking, +this fixes the angle between the free interface and the boundary ∂D. The +angle condition is the result of a boundary energy, which enforces the an- +gle condition via Young’s law. For similar results regarding the phase-field +approximation, we refer to [10]. +It seems feasible to generalize our present proof to the anisotropic case, +just as Tonegawa [17] generalized the proof of Evans and Spruck [8] to the +anisotropic setting. +6. Appendix +The following theorem is taken from [15, Part 2] and gives a characteriza- +tion of the distributional derivative of characteristic functions. +Theorem 6.1 (De Giorgi’s structure theorem). Let A is a set of finite +perimeter. Then the distributional derivative of its characteristic function +can be expressed using the generalized outer unit normal vector and the re- +duced boundary via ∇χA = νAHd−1⌊∂∗A. +The coarea-formula can also be found in [15, Chapter 13]. +Lemma 6.2 (Coarea-formula). Let u : Rd → R be Lipschitz, Ω ⊆ Rd open +and f ∈ L1(Rd) integrable. Then +ˆ +Ω +|∇u(x)|f(x) dx = +ˆ +R +ˆ +{x∈Ω:u(x)=s} +f(x) dHd−1(x) ds. +Acknowledgments +The present paper is an extension of the second author’s master’s the- +sis at the University of Bonn. This project has received funding from the +Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) un- +der Germany’s Excellence Strategy – EXC-2047/1 – 390685813. +References +[1] S. Aimi. Level set mean curvature flow with Neumann boundary conditions, 2021. +arXiv:2103.16386. +[2] L. Ambrosio, N. Gigli, and G. Savaré. Gradient flows in metric spaces and in the space +of probability measures. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, +Basel, 2005. + +16 +TIM LAUX AND ANTON ULLRICH +[3] S. Angenent, T. Ilmanen, and D. Chopp. A computed example of nonuniqueness of +mean curvature flow in R3. Comm. Partial Differ. Equations, 20(11-12):1937–1958, +1995. +[4] Y.-G. Chen, Y. Giga, and S. Goto. Uniqueness and existence of viscosity solutions of +generalized mean curvature flow equations. J. Differ. Geom., 33(3):749–786, 1991. +[5] E. De Giorgi, A. Marino, and M. Tosques. Problems of evolution in metric spaces and +maximal decreasing curve. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), +68(3):180–187, 1980. +[6] L. C. Evans and J. Spruck. Motion of level sets by mean curvature. I. J. Differ. +Geom., 33(3):635–681, 1991. +[7] L. C. Evans and J. Spruck. Motion of level sets by mean curvature. III. J. Geom. +Anal., 2(2):121–150, 1992. +[8] L. C. Evans and J. Spruck. Motion of level sets by mean curvature. IV. J. Geom. +Anal., 5(1):79–116, 1995. +[9] J. Fischer, S. Hensel, T. Laux, and T. M. Simon. The local structure of the energy +landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability +of evolutions. arXiv preprint, 2020. arXiv:2003.05478v2. +[10] S. Hensel and T. Laux. BV solutions for mean curvature flow with constant contact +angle: Allen–Cahn approximation and weak-strong uniqueness. Indiana Univ. Math. +J. (online first), 2021. +[11] T. Laux and F. Otto. Convergence of the thresholding scheme for multi-phase mean- +curvature flow. Calc. Var. Partial Differ. Equ., 55(5):74, 2016. Id/No 129. +[12] T. Laux and F. Otto. The thresholding scheme for mean curvature flow and de Giorgi’s +ideas for minimizing movements. In The role of metrics in the theory of partial dif- +ferential equations. Proceedings of the 11th Mathematical Society of Japan, Seasonal +Institute (MSJ-SI), Nagoya University, Japan, July 2–13 2018, pages 63–93. Tokyo: +Mathematical Society of Japan, 2021. +[13] T. Laux and T. M. Simon. Convergence of the Allen–Cahn equation to multiphase +mean curvature flow. Comm. Pure Appl. Math., 71(8):1597–1647, 2018. +[14] S. Luckhaus and T. Sturzenhecker. Implicit time discretization for the mean curvature +flow equation. Calc. Var. Partial Differ. Equ., 3(2):253–271, 1995. +[15] F. Maggi. Sets of finite perimeter and geometric variational problems. An introduction +to geometric measure theory., volume 135. Cambridge: Cambridge University Press, +2012. +[16] S. Osher and J. A. Sethian. Fronts propagating with curvature-dependent speed: +Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys., 79(1):12–49, +1988. +[17] Y. Tonegawa. Some remarks on the level set flow by anisotropic curvature. Calc. Var. +Partial Differ. Equ., 10(2):101–118, 2000. +T. L. Hausdorff Center for Mathematics and Institute for Applied Math- +ematics, University of Bonn, Villa Maria, Endenicher Allee 62, 53115 Bonn, +Germany +Email address: +tim.laux@hcm.uni-bonn.de +A. U. Max Planck Institute for Mathematics in the Sciences, Inselstraße +22, 04103 Leipzig, Germany +Email address: +anton.ullrich@mis.mpg.de + diff --git a/XdAzT4oBgHgl3EQfKftJ/content/tmp_files/load_file.txt b/XdAzT4oBgHgl3EQfKftJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..49eb405579ba8cda9b2c54c0203349315f3daa26 --- /dev/null +++ b/XdAzT4oBgHgl3EQfKftJ/content/tmp_files/load_file.txt @@ -0,0 +1,588 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf,len=587 +page_content='GENERIC LEVEL SETS IN MEAN CURVATURE FLOW ARE BV SOLUTIONS TIM LAUX AND ANTON ULLRICH Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' We show that a generic levelset of the viscosity solution to mean curvature flow is a distributional solution in the framework of sets of finite perimeter by Luckhaus and Sturzenhecker, which in addition saturates the optimal energy dissipation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' This extends the fundamental work of Evans and Spruck (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' 1995), which draws a similar connection between the viscosity solution and Brakke flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Introduction The mean curvature flow is a parabolic geometric evolution equation with numerous applications in the sciences and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' It is the most basic model describing slow relaxation driven by surface tension and can be viewed as the diffusion equation of surfaces relating the normal velocity V and the mean curvature H of an evolving surface via the equation V = −H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Since the mean curvature flow develops singularities in finite time, it is natural to consider weak solutions which are able to describe the evolution through singular events such as pinch-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' However, after such singularities, the flow may become non-unique [3] and it is in general difficult to compare different weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' The aim of this paper is to better understand weak solutions and in particular draw a new connection between different weak solution concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' We will show that generically, they are consistent with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' More precisely, we consider the viscosity solution introduced indepen- dently by Evans and Spruck [6], and Chen, Giga and Goto [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' This concept is based on the levelset formulation of Osher and Sethian [16] in which all levelsets of a function are evolved simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' We show that almost ev- ery levelset of the viscosity solution is an evolving set of finite perimeter which moves according to the distributional formulation of mean curvature flow due to Luckhaus and Sturzenhecker [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' In addition, we show that the evolving sets of finite perimeter satisfy an optimal energy dissipation rela- tion, which is crucial for the weak-strong uniqueness of the solution concept, Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Mean curvature flow, weak solution concepts, viscosity so- lution, distributional solution, compensated compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' MSC2020: 53E10 (Primary);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' 35D40;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' 35D30 (Secondary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='01097v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='AP] 3 Jan 2023 2 TIM LAUX AND ANTON ULLRICH cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Additionally, together with the distributional identity between nor- mal velocity and mean curvature operator, the solution can in addition be viewed as curve of maximal slope in the spirit of De Giorgi, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' [5, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' More precisely, our solution also recovers the De Giorgi formulation in the setting of BV solutions to mean curvature flow which was introduced by Otto and one of the authors [12] to describe limits of the thresholding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' It is a by now classical result due to Evans and Spruck [6] that the vis- cosity solution is consistent with the classical solution in the sense that, when starting from the same initial data, the two solutions coincide on their common time interval of existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' More recently, such a consistency – or weak-strong uniqueness – has also been shown between BV solutions and the classical solution to mean curvature flow in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Our result extends those statements by stating that also past singularities, when the classical solution ceases to exist, these two weak solution concepts are consistent with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' More precisely, we prove the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Let u ∈ C(Rd × [0, T)) be a viscosity solution to mean cur- vature flow in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='2 with well-prepared initial data ac- cording to Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Then almost every levelset of u is a BV solution to mean curvature flow in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' The idea of our proof is rather simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' We use the vanishing-viscosity approximation by Evans and Spruck [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' The strengthened convergence nec- essary to carry out our argument comes from the surprising estimate from [8] in the spirit of sup t>0 ˆ Rd |H| dx < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' This makes possible a compensated compactness argument to show that the energies, which are modeled after ´ Rd |∇u| dx, converge in the vanishing- viscosity limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' By lower semi-continuity and the coarea-formula, this implies that almost every levelset converges strictly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=', the perimeter does not drop down in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Such an energy convergence is an essential tool for convergence proofs for approximations of mean curvature flow, see [14, 11, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Therefore, it is a natural question whether one can verify the BV formulation also for the viscosity solution of the levelset flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' We do this by first deriving a distributional formulation of levelset mean curvature flow that is satisfied by the viscosity solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Then we disintegrate the levelset parameter in this formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' On a technical level, this is done by using the coarea-formula for surface-type integrals and the layer-cake formula for volume-type integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Definitions of the distributional and generic formulation In this section we recall the definition of the levelset solution to mean curvature flow and how to generalize this concept to viscosity solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Afterwards, the notion of distributional solutions is stated and explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' GENERIC LEVEL SETS IN MCF 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Example of a time-instance of a levelset function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Levelsets for four different values are marked in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' To give an intuition of the levelset equation of mean curvature flow, let u: Rd × [0, T) → R be a smooth function such that each levelset evolves by mean curvature flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' We aim to find a partial differential equation for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' To this end, let x(t), t ∈ (t1, t2) ⊂ (0, T), be a trajectory following one levelset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='e, u(x(t), t) = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' and suppose ∇u(x(t), t) ̸= 0 for all t ∈ (t1, t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' The outer unit normal vector to the superlevelsets is given by ν(x, t) = − ∇u(x,t) |∇u(x,t)| and since the mean curvature is the divergence of the outer unit normal, H = ∇ · ν, the mean curvature flow equation can be expressed by ˙x · ν = −∇ · ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Thus 0 = d dtu(x(t), t) = ∂tu + ˙x · ∇u = ∂tu + |∇u|∇ · ν and hence, we obtain the levelset equation for u: ∂tu = −|∇u|∇ · ν = ∆u − ∇u |∇u| · ∇2u ∇u |∇u|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' This leads to the following definition which is equivalent to the fact that the levelsets of u evolve by mean curvature flow, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1 (Levelset equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' A function u: Rd × (0, T) → R with u ∈ C1 t ∩ C2 x is called a solution to the levelset equation of mean curvature flow if it solves the initial value problem � ∂tu = ∆u − ∇u |∇u| · ∇2u ∇u |∇u| in Rd × (0, T), u = g on Rd × {t = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' (1) This solution concept can be extended to the non-smooth case even after the onset of singularities using viscosity solutions, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' [6, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' These use the comparison principle of the mean curvature flow to control the evolution of the levelsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Here, one considers continuous levelset functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Intuitively, 9 8 7 6~ 5 ~ 4 3 2~ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' 3 2 0 1 0 2 1 2 3 34 TIM LAUX AND ANTON ULLRICH the idea is to take smooth functions locally approximating the levelset func- tion in an ordered way and impose the respective inequality on these smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' This solution concept is consistent with the levelset solutions for smooth functions u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' One way to prove this is by the characterization using sub- and superjets, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' [6, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='2 (Viscosity solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' A continuous function u ∈ C(Rd × [0, T)) is called a viscosity super-solution to (1) if, for any ϕ ∈ C∞(Rd × (0, T)) and (x0, t0) ∈ Rd × (0, T) such that u − ϕ has a local minimum at (x0, t0), the following inequality holds, i) for non-critical points ∇ϕ(x0, t0) ̸= 0 : ∂tϕ ≥ ∆ϕ − ∇ϕ |∇ϕ| · ∇2ϕ ∇ϕ |∇ϕ| at (x0, t0) ii) and for critical points ∇ϕ(x0, t0) = 0 there exists ξ ∈ Rd, |ξ| ≤ 1 : ∂tϕ ≥ ∆ϕ − ξ · ∇2ϕξ at (x0, t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' We say that u is a viscosity sub-solution if −u is a viscosity super-solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Finally, u is called a viscosity solution if it is a viscosity sub- as well as a super-solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Viscosity solutions are unique, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=', it can be shown that there exists at most one viscosity solution to mean curvature flow, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' [6, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' However, the levelsets may develop an interior, a phenomenon called fatten- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Another way to extend the solution concept of mean curvature flow are so- called BV solutions due to Luckhaus and Sturzenhecker [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' This definition uses distributional characterizations of the mean curvature and the normal velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Additionally, we impose a sharp energy dissipation which extends the definition of Luckhaus and Sturzenhecker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' In the following the Gauß–Green measure µΩ denotes the (negative) spa- tial distributional derivative of the characteristic function µΩ(t) = −∇χΩ(t) for every time t ∈ (0, T) and |µΩ(t)| denotes its total variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Moreover, the perimeter of a set Ω(t) is defined as P(Ω(t)) := |µΩ(t)|(Rd) = sup � � � � � ˆ Ω(t) ∇ · ξ dx: ξ ∈ C1 c (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Rd), |ξ| ≤ 1 � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Note that by De Giorgi’s structure theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1), integration against the Gauß–Green measure precisely corresponds to integration over the reduced boundary ∂∗Ω(t) with respect to the Hausdorff measure Hd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='3 (BV solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' A family of finite perimeter sets (Ω(t))t∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='T) depending measurably on t ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' T) is called distributional solution to mean GENERIC LEVEL SETS IN MCF 5 curvature flow if the sets admit a uniform perimeter bound ess sup t∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='T) P(Ω(t)) < ∞ (2) and there is a d|µΩ(t)| dt-measurable L2 bounded function V : Rd×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' T) → R ˆ Rd×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='T) V 2 d|µΩ(t)| dt < ∞ (3) such that the following holds: i) The function V (·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' t) is the normal velocity of Ω(t) in the sense that for each test function ζ ∈ C1 c (Rd × [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' T)) it satisfies T ˆ 0 ˆ Ω(t) ∂tζ dx dt = − ˆ Rd×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='T) ζV d|µΩ(t)| dt − ˆ Ω(0) ζ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' 0) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' (4) ii) Furthermore, for any test vector field ξ ∈ C1 c (Rd ×(0, T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Rd) it holds ˆ Rd×(0,T) (∇ · ξ − ν · ∇ξν) d|µΩ(t)| dt = − ˆ Rd×(0,T) V ξ · ν d|µΩ(t)| dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' (5) iii) For almost any time T ′ ∈ (0, T) the sharp energy–dissipation holds P(Ω(T ′)) + ˆ Rd×(0,T ′) V 2 d|µΩ(t)| dt ≤ P(Ω(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' (6) Equation (4) characterizes the normal velocity for smooth sets Ω(t), which is a direct consequence of the computation of ∂t ´ Ω(t) ζ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' The operator on the left-hand side of Equation (5) encodes the mean curvature via an integration by parts on the surface, which can be easily seen for smooth sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' This solution concept represents the mean curvature flow since V is the distributional normal velocity and simultaneously the negative distributional mean curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' In addition to the distributional formulation by Luckhaus and Sturzen- hecker, we also have a sharp energy-dissipation relation given by Inequal- ity (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Recap of results by Evans and Spruck In this section, we recall the important results from [6], [7] and [8] that are needed to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' We will consider the vanishing viscosity approximation to our levelset formulation ∂tuε = ∆uε − ∇uε � |∇uε|2 + ε2 · ∇2uε ∇uε � |∇uε|2 + ε2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' (7) This approximation is non-degenerate and has sufficient convergence to the levelset solution as we will see in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' In addition, it has a beautiful 6 TIM LAUX AND ANTON ULLRICH geometric interpretation as the mean curvature flow of graphs in a higher dimensional space, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Throughout, we will assume that the initial datum for the viscous approx- imation (7) is well-prepared in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' We call an initial levelset data g well-prepared if g ∈ C3(Rd), g is constant outside a ball of radius R > 0, and g possesses an integrable approximate mean curvature with a uniform bound in ε in the sense that sup 0<ε≤1 ˆ Rd ���∇ · � ∇g � |∇g|2 + ε2 ���� dx < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' (8) Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Following [8, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1] one can construct well-prepared initial data g for any given compact smooth 0-levelset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' The viscous approximation (7) has better properties in the sense that it yields smooth solutions, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' [7, Chapter 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' For these, the following bounds on the solution and its derivatives hold, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' [6, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' For well-prepared initial data there exists a unique smooth solution uε to (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Additionally, |uε|, |∇uε|, |∂tuε| are uniformly bounded in ε, x and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' The central quantity of our argument will be the generalized normal ve- locity (9) V := ∂tu |∇u|χ{∇u̸=0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' We will also need the generalized outer unit normal vector ν := � − ∇u |∇u| if ∇u ̸= 0, e1 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' To prove the convergences of the vanishing viscosity approximation, the most crucial step is to utilize the following unexpected estimate due to Evans and Spruck [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' We have the uniform L∞ t L1 x-bound sup 0≤t≤T sup 0<ε≤1 ˆ Rd |Hε(x, t)| dx < ∞, (10) where Hε(x, t) := ∇ · νε is the approximate mean curvature for the levelsets of uε and νε = − ∇uε √ |∇uε|2+ε2 the approximate outer normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' This statement can be found in [8, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='2] and follows from the fact that the quantity t �→ ´ Rd |Hε(x, t)| dx is non-increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' This uniform L∞ t L1 x-bound on the approximate mean curvature has strong consequences as can be seen in the following theorem, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' [8, Chapter 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' For a subsequence GENERIC LEVEL SETS IN MCF 7 the approximate mean curvature Hε converges in the weak-∗ topology of measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Together with the relation to the approximate normal this yields the additional convergences in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' One can show that u is bounded and Lipschitz in space and time, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Thus, by Rademacher’s theorem the derivatives are defined almost every- where.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='4 (Convergence, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Considering subsequences by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='2 we can infer by Arzelà–Ascoli that uε → u locally uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Similarly, ∇uε ∗⇀ ∇u in L∞(Rd) and ∂tuε ∗⇀ ∂tu in L∞(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='3 for a compensated compactness argument with νε and the definition we conclude |∇uε| ∗⇀ |∇u| in L∞(Rd) and νε ∗⇀ ν in L∞(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Finally, by an excess decay argument it follows νε → ν in L2({∇u ̸= 0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' The key argument for this theorem is that once one writes |∇uε| ≈ −νε · ∇uε, one can pass to the limit in the product on the right-hand side via compen- sated compactness since uε → u uniformly and ∇ · νε = Hε is uniformly bounded in ε as Radon measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Another useful lemma is the following, which tells us that certain integral quantities are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' It will be impor- tant in taking the limit and proving estimates for u given the corresponding estimates for uε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' This is Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1 in [8] and follows from the bounds of uε and its derivatives (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Let K ⊂ Rd × (0, T) be a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Then the following bound on the approximate mean curvature holds sup 0<ε≤1 T ˆ 0 ˆ Rd χK|Hε|2� |∇uε|2 + ε2 dx dt < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' 8 TIM LAUX AND ANTON ULLRICH This estimate is more intuitive than (10) as for smooth manifolds evolv- ing by mean curvature flow ´ |H|2 is the first variation of the perimeter functional and the factor |∇u| is the coarea-factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Additionally, we will use a relabeling property which is stated in the fol- lowing lemma and originates from [8, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='6 (Relabeling property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Let g be well-prepared initial data, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1 and u the unique viscosity solution to (1) as well as Φ ∈ C∞(R) be a smooth testfunction with positive derivative Φ′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Then Φ ◦ u is the unique viscosity solution to the initial data Φ◦g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Moreover, the initial data Φ ◦ g is well-prepared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Generic levelsets are BV solutions Next, we will begin with the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=', that almost every levelset of the viscosity solution is a distributional solution in the sense of BV solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' The following first theorem can be understood as an integrated version of the distributional equations for normal velocity and mean curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' The main novelty is Equation (11), which encodes the normal velocity while the second part (Equation (12)) can be found as Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1 in [8] with H in- stead of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Having proved these identities for the levelset function one can then proceed by disintegrating the levelset parameter to get the correspond- ing one identities for almost every levelset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' This is done below in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='4 using the layercake-formula and the coarea-formula (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Let u ∈ C(Rd × [0, T)) be the unique viscosity solution with some well-prepared initial data g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Then there exists a |∇u| dx dt-measurable function V : Rd × (0, T) → R such that i) For any test function ζ ∈ C1 c (Rd × [0, T)) it holds T ˆ 0 ˆ Rd ∂tζu dx dt = − T ˆ 0 ˆ Rd ζV |∇u| dx dt − ˆ Rd g ζ(·, 0) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' (11) ii) For any test vector field ξ ∈ C1 c (Rd × (0, T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Rd) we have T ˆ 0 ˆ Rd (∇ · ξ − ν · ∇ξν) |∇u| dx dt = − T ˆ 0 ˆ Rd ξ · νV |∇u| dx dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' (12) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' The idea is to prove an approximate version for uε using integration by parts and the definition of the vanishing viscosity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Afterwards, one uses Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='5 and the convergences in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='4 to get the equation for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Here, the convergence up to the set {∇u = 0} suffices since their contribution will vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Argument for (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Fix a test function ζ ∈ C1 c (Rd ×[0, T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' We first claim that ∂tu = V |∇u| almost everywhere on Rd × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Then we may conclude GENERIC LEVEL SETS IN MCF 9 via T ˆ 0 ˆ Rd ∂tζu dx dt = − T ˆ 0 ˆ Rd ζ∂tu dx dt − ˆ Rd gζ(·, 0) dx = − T ˆ 0 ˆ Rd ζV |∇u| dx dt − ˆ Rd g ζ(·, 0) dx, which is precisely (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Now we argue that indeed ∂tu = V |∇u| almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' First, ob- serve that by definition of the normal velocity V , cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' (9), the identity holds on the set {∇u ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Thus, we only need to argue that ∂tu = 0 almost everywhere on {∇u = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' This in turn follows from the next computation, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' [8, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Note that we can express the evolution equation (7) as ∂tuε = −Hε � |∇uε|2 + ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Then for every compact subset A of the critical set {∇u = 0}: ������ ˆ A ∂tu dx dt ������ ≤ lim ε↘0 ˆ A |∂tuε| dx dt = lim ε↘0 ˆ A |Hε| � |∇uε|2 + ε2 dx dt ≤ lim inf ε↘0 � � ˆ A H2 ε � |∇uε|2 + ε2 dx dt � � 1 2 × � � ˆ A � |∇uε|2 + ε2 dx dt � � 1 2 ≤ C ˆ A |∇u| dx dt = 0, where in the last step we have used Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='5 to bound the first term and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='4 to pass to the limit in the second term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Now by inner regularity this proves ∂tu = 0 almost everywhere on {∇u = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Argument for (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' As stated above, the following argument can be found in [8, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1], but for the convenience of the reader we highlight the main idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Testing the equation (7) with ξ · νε for a given test vector field ξ ∈ C1 c (Rd×(0, T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Rd) and integrating by parts yields the following ε-variant of (12): T ˆ 0 ˆ Rd (∇ · ξ − νε · ∇ξνε) � |∇uε|2 + ε2 dx dt = − T ˆ 0 ˆ Rd ξ · νε∂tuε dx dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' 10 TIM LAUX AND ANTON ULLRICH Now Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='4 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='5 allow us to pass to the limit ε ↘ 0 on both sides of this identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' □ We additionally need the optimal energy dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' This will be a con- sequence of the following theorem for the levelset function u, which follows from [8, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='2] where again V is called H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' For the viscosity solution u and any two time instances 0 ≤ t1 < t2 ≤ T, we have a version of the optimal energy dissipation: ˆ Rd |∇u| dx ������ t2 + t2 ˆ t1 ˆ Rd V 2|∇u| dx dt ≤ ˆ Rd |∇u| dx ������ t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' (13) From the previous two theorems, one can extract the levelset case for almost every levelset by use of the coarea-formula (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='2) and the layercake-formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' To this end, it is necessary to define a notion for the levelsets and superlevelsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='3 (Ωs, Σs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' For a viscosity solution u to (1) and any s ∈ R and t ∈ [0, ∞), one defines the superlevelsets Ωs(t) := {x : u(x, t) > s} and the levelsets Σs(t) := {x : u(x, t) = s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' The following theorem states the disintegrated version of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Let u be a viscosity solution to the levelset equation (1) and let Ωs, Σs be given as in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Then for almost every value s ∈ R we have i) The normal velocity of Σs is given by V in the sense that for all test functions ζ ∈ C1 c (Rd × (0, T)): T ˆ 0 ˆ Ωs(t) ∂tζ dx dt = − T ˆ 0 ˆ Σs(t) ζV dHd−1(x) dt − ˆ Ωs(0) ζ(·, 0) dx, (14) ii) The equation V = −H is satisfied for Σs in the sense that for all test vector fields ξ ∈ C1 c (Rd × (0, T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Rd): T ˆ 0 ˆ Σs(t) (∇ · ξ − ν · ∇ξν) dHd−1(x) dt = − T ˆ 0 ˆ Σs(t) ξ · νV dHd−1(x) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' (15) Equation (15) is similar to the one found in [8] now on a levelset basis while (14) is new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' The idea is to use Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1 combined with the relabeling property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' By the coarea-formula and layercake-formula one can split the integrals into the (super-)levelsets and then use the relabeling to separate these and get the equality for almost every levelset separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' GENERIC LEVEL SETS IN MCF 11 Argument for (14): Fix Φ ∈ C∞(R) with Φ′ ≥ 0 and observe that Φ ◦ u is the unique viscosity solution to the well-prepared initial data Φ ◦ g by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Now we can use equation (11) with VΦ = ∂tu |∇u| = V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' By the coarea-formula the right-hand side is equal to ˆ R Φ′(s) T ˆ 0 ˆ Σs(t) ζV dHd−1(x) dt ds − ˆ Rd gζ(·, 0) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' For the left-hand side and the initial data term one can use the layercake- formula with a transformation, where K ∈ R, K < umin and umin is the infimum of u in space and time (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='2): Φ ◦ u = Φ◦u ˆ Φ(K) 1 dτ + Φ(K) = ∞ ˆ K χ{u>s}Φ′(s) ds + Φ(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' This yields for the left-hand side of (11) T ˆ 0 ˆ Rd ∂tζΦ ◦ u dx dt = T ˆ 0 ˆ Rd ∂tζ ∞ ˆ K χ{u>s}Φ′(s) ds dx dt + T ˆ 0 ˆ Rd ∂tζΦ(K) dx dt = ∞ ˆ K Φ′(s) T ˆ 0 ˆ Ωs(t) ∂tζ dx dt ds + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Now since Φ was arbitrary under the above assumptions one can use Φ(s) = s´ −∞ ϕ(s′) ds′ for ϕ ∈ C∞ c (R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' R≥0) a bump-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Then by the fundamental theorem of the calculus of variations, one has for almost every levelset value s (for s < umin both sides equal 0) and for all test functions ζ ∈ C1 c (Rd×(0, T)) T ˆ 0 ˆ Ωs(t) ∂tζ dx dt = − T ˆ 0 ˆ Σs(t) ζV dHd−1(x) dt − ˆ Ωs(0) ζ(·, 0) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Argument for (15): Fix again Φ ∈ C∞(R) with Φ′ > 0 and well-prepared initial data g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='6, Φ ◦ g is also well-prepared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Then, by (12) using νΦ = − ∇(Φ◦u) |∇(Φ◦u)| = ν, |∇(Φ ◦ u)| = Φ′ ◦ u|∇u|, we obtain: T ˆ 0 ˆ Rd Φ′ ◦ u (∇ · ξ − ν · ∇ξν) |∇u| dx dt = − T ˆ 0 ˆ Rd Φ′ ◦ u ξ · νV |∇u| dx dt 12 TIM LAUX AND ANTON ULLRICH and by the coarea-formula, this can be written as ˆ R Φ′(s) T ˆ 0 ˆ Σs(t) (∇ · ξ − ν · ∇ξν) dHd−1(x) dt ds = − ˆ R Φ′(s) T ˆ 0 ˆ Σs(t) ξ · νV dHd−1(x) dt ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' As before, for almost every s this yields the desired equality T ˆ 0 ˆ Σs(t) (∇ · ξ − ν · ∇ξν) dHd−1(x) dt = − T ˆ 0 ˆ Σs(t) ξ · νV dHd−1(x) dt, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' □ Similarly, one can prove the levelset version of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' This version will become the sharp energy dissipation for the BV solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' For almost every levelset value s ∈ R and time instances 0 ≤ t1 < t2 ≤ T the sharp energy dissipation holds: Hd−1(Σs(t2)) + t2 ˆ t1 ˆ Σs(t) V 2 dHd−1(x) dt ≤ Hd−1(Σs(t1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' (16) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' This proof is similar to the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' We use the coarea- formula in Equation (13) to show that for almost every levelset the difference of the sides of the inequality has a value of less or equal to zero: By the relabeling property for Φ ∈ C∞(R) with Φ′ ≥ 0 applied to Equa- tion (13) one gets: 0 ≥ ˆ Rd Φ′ ◦ u|∇u| dx ������ t2 + t2 ˆ t1 ˆ Rd Φ′ ◦ uV 2|∇u| dx dt − ˆ Rd Φ′ ◦ u|∇u| dx ������ t1 = ˆ R Φ′(s) � �Hd−1(Σs(t2)) + t2 ˆ t1 ˆ Σs V 2 dHd−1(x) dt − Hd−1(Σs(t1)) � � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Since Φ′ ≥ 0 and by the fundamental theorem of the calculus of variations, this yields the desired inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' □ To prove now that almost every levelset is a distributional solution, some well-posedness conditions have to be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' The following lemma shows that our sets are sets of finite perimeter and that u is differentiable a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' on these, which is stated in the following lemma that can be found in [8] as Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' GENERIC LEVEL SETS IN MCF 13 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' For every time 0 ≤ t ≤ T and almost every levelset value s ∈ R the viscosity solution u is differentiable Hd−1-almost everywhere on Σs with non-vanishing gradient |∇u| ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' For fixed 0 ≤ t ≤ T the function u is Lipschitz and thus by the coarea-formula for ζ : Rd → R integrable one has ˆ Rd ζ|∇u| dx = ˆ R ˆ Σs ζ dHd−1 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' For ζ = χA, A := {u not diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' or ∇u = 0} it yields Hd−1(A ∩ Σs) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Let u ∈ C(Rd × (0, T)) be a viscosity solution to the levelset equation (1) of mean curvature flow with well-prepared initial data g ∈ C3(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Then, by Inequality (16) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='5 the L2-bound (3) of V is satisfied (when plugging in t1 = 0 and t2 = T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' This is due to the fact that for almost all levelsets the reduced boundary of the superlevelsets is the levelset ∂∗Ωs(t) = Σs(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Similarly, the perimeters are uniformly bounded (2) since they are monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Moreover, the distributional characterizations (4) and (5) of the normal velocity V and the mean curvature H were proven in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='4 in Equa- tions (14) and (15), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Lastly, the sharp energy dissipation (6) can be inferred by plugging t1 = 0, t2 = T ′ into Inequality (16) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Extensions and open problems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Extension to homogeneous Neumann boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' In this paper, we worked in the setting of Rd with initial data which are constant outside a ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' One could also consider the case of homogeneous Neumann data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=', the surface Σ satisfies V = −H and is orthogonal to a given fixed domain boundary Σ ⊥ ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Similarly to before, one can define levelset solutions as well as viscosity solutions and an analogous vanishing viscosity approximation, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' The levelset equation becomes � � � � � ∂tu = ∆u − ∇u |∇u| · ∇2u ∇u |∇u| in D × (0, T), ∇u · ν∂D = 0 on ∂D × (0, T), u = g on D × {t = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Here, the initial data g also has to satisfy the boundary condition ∇g ·ν∂D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Now, the vanishing viscosity approximation can be constructed in the same way satisfying the condition ∇uε · ν∂D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' The major difference in this context lies in the formulation of the distributional solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' To encode the boundary conditions, we use again integration by parts, see also [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Indeed, the homogeneous Neumann condition ν∂D · ∇u = 0 can be encoded 14 TIM LAUX AND ANTON ULLRICH by asking that for any test vector field ξ ∈ C1 c (D×(0, T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Rd) with ξ·ν∂D = 0 it holds T ˆ 0 ˆ Rd (∇ · ξ − νu · ∇ξνu) |∇u| dx dt = − T ˆ 0 ˆ Rd ξ · νuV |∇u| dx dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='This weak formulation follows from from the following calculation (in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='which we suppress the t-dependence) for the vanishing viscosity approxi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='mation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ξ · νε∂tuε dx = − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ξ · ∇uε∇ · νε dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='νε · ∇ξ∇uε dx + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ξ · ∇2uενε dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='∂D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ξ · ∇uενε · ν∂D dHd−1(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='νε · ∇ξ∇uε dx − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ξ · ∇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='|∇uε|2 + ε2 dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='νε · ∇ξ∇uε dx + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='∇ · ξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='|∇uε|2 + ε2 dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='∂D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ξ · ν∂D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='|∇uε|2 + ε2 dHd−1(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='(∇ · ξ − νε · ∇ξνε) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='|∇uε|2 + ε2 dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='∂D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ξ · ν∂D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='|∇uε|2 + ε2 dHd−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' For test vector fields ξ with ξ · ν∂D, the additional boundary term vanishes exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Then, one can pass to the limit ε ↘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' With these changes, we can modify the proof to show Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1 for homogeneous Neumann conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Due to [1, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='3] the existence of well-prepared initial data to a 0- levelset which meets ∂D orthogonally is ensured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Moreover, in this paper, the integrability of the approximate mean curvature as well as the convergences and the L2-integrability with coarea-factor is shown, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' [1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='3 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1 follow with the same reasoning with the sole difference that the convergences now use that we are on the bounded domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' GENERIC LEVEL SETS IN MCF 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Further directions and open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' We showed that almost every levelset of the viscosity solution is a BV solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' It remains an in- teresting open question whether the inverse assertion holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Let g be well- prepared initial data and u ∈ C(Rd × (0, T)) such that every levelset of u is a distributional solution to mean curvature flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Is u then a viscosity solu- tion to mean curvature flow?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' By the weak-strong uniqueness result in [9], this is true as long as a classical solution exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' However, after the onset of singularities this is still an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' A further interesting generalization would be to incorporate non-homoge- neous Neumann condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=', ∇u · νD = cos α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Geometrically speaking, this fixes the angle between the free interface and the boundary ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' The angle condition is the result of a boundary energy, which enforces the an- gle condition via Young’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' For similar results regarding the phase-field approximation, we refer to [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' It seems feasible to generalize our present proof to the anisotropic case, just as Tonegawa [17] generalized the proof of Evans and Spruck [8] to the anisotropic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Appendix The following theorem is taken from [15, Part 2] and gives a characteriza- tion of the distributional derivative of characteristic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='1 (De Giorgi’s structure theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Let A is a set of finite perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Then the distributional derivative of its characteristic function can be expressed using the generalized outer unit normal vector and the re- duced boundary via ∇χA = νAHd−1⌊∂∗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' The coarea-formula can also be found in [15, Chapter 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='2 (Coarea-formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Let u : Rd → R be Lipschitz, Ω ⊆ Rd open and f ∈ L1(Rd) integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Then ˆ Ω |∇u(x)|f(x) dx = ˆ R ˆ {x∈Ω:u(x)=s} f(x) dHd−1(x) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Acknowledgments The present paper is an extension of the second author’s master’s the- sis at the University of Bonn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' This project has received funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) un- der Germany’s Excellence Strategy – EXC-2047/1 – 390685813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Aimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Level set mean curvature flow with Neumann boundary conditions, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='16386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Ambrosio, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Gigli, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Savaré.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Gradient flows in metric spaces and in the space of probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Lectures in Mathematics ETH Zürich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Birkhäuser Verlag, Basel, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' 16 TIM LAUX AND ANTON ULLRICH [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Angenent, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Ilmanen, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Chopp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' A computed example of nonuniqueness of mean curvature flow in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Partial Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Equations, 20(11-12):1937–1958, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' [4] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Giga, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Goto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=', 33(3):749–786, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' [5] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' De Giorgi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Marino, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Tosques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Problems of evolution in metric spaces and maximal decreasing curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Atti Accad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Naz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Lincei Rend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Fis.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' [17] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Tonegawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Some remarks on the level set flow by anisotropic curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Partial Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=', 10(2):101–118, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Hausdorff Center for Mathematics and Institute for Applied Math- ematics, University of Bonn, Villa Maria, Endenicher Allee 62, 53115 Bonn, Germany Email address: tim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='laux@hcm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='uni-bonn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='de A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content=' Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany Email address: anton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='ullrich@mis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQfKftJ/content/2301.01097v1.pdf'} +page_content='mpg.' metadata={'source': 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Wu1, P. Shi2, F. Subba1, H. Sun2, M. Wischmeier3, R. Zanino1, the ASDEX Upgrade Team +1.NEMO Group, Dipartimento Energia, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 +Torino, Italy +2.United Kingdom Atomic Energy Authority, Culham Centre for Fusion Energy, Culham Science Centre, +Abingdon, Oxon OX14 3DB, United Kingdom. +3.Max-Planck-Institut für Plasmaphysik, Boltzmannstraße 2, D-85748 Garching, Germany +Abstract: +The High Field High Density Region (HFSFD) has been experimentally observed in both divertor and +limiter tokamak devices. In order to numerically reproduced the HFSHD region in the limiter tokamak +device J-TEXT, we first performed SOLPS-ITER simulations on the J-TEXT limiter tokamak with the +activation of drifts, which is associated with the HFSHD region. The validated physical models, which is +from ASDEX Upgrade (AUG) divertor modelling including full drifts and currents, were applied. Through +a gas puffing rate scan, both attached and detached regimes were numerically obtained in the AUG divertor +and J-TEXT limiter. The key plasma parameters in the J-TEXT limiter are evaluated with and without drifts +that have a qualitative performance like AUG except the roll-overment of the total ion flux at the targets. +The drift effects on the target profiles are investigated in which the maximum electron temperature at the +outer targets is 15eV and 5eV respectively. When the outer targets are attached in the J-TEXT limiter and +AUG divertor, the drifts result in the partial detachment of the inner targets. In the detached regimes, the +drifts decrease the electron temperature on AUG divertor targets. However, for the J-TEXT limiter, the +electron temperature only decreases at the far SOL region. The effect of drifts on the neutral density is also +presented. +1. Introduction +Edge plasma code packages, e.g. SOLPS-ITE [1], UEDGE[2] and SOLEDGE2D[3], are widely used to +study boundary plasma behavior in current tokamak devices, e.g. ASDEX Upgrade (AUG)[4] and Alcator +C-mod[5], and to the design of future tokamak devices, e.g. DTT [6], SPARC[7] and EU-DEMO[8]. In +SOLPS-ITER, two main modules are included: the multi-fluid plasma solver B2[9] for charged species +transport toroidal symmetry, and the Monte Carlo code EIRENE[10], which describes kinetic neutral +transport. +The High Field Side High Density region, in which the volume electron density ne is at least 10 times higher +than the upstream averaged and target values, has been observed in the Scrape-Off layer(SOL) of the +divertor Tokamak device ASDEX Upgrade (AUG)[11], and has been numerically reproduced by SOLPS- +ITER with the activation of drifts [4][12]. Recently, a similar high-density front has been observed in the +SOL of the limiter tokamak device J-TEXT, in which the high-density front is seven times higher than the +density in the low field side region [14]. +In order to investigate the formation of high-density front in the J-TEXT, we first performed a SOLPS- +ITER numerical simulation on the J-TEXT limiter configuration with the activation of full drifts and +currents[15][16]. The effect of drifts in the J-TEXT limiter is evaluated through a preliminary comparison +with AUG single-null divertor configuration simulation results, in which the physical models [4] have been +validated against experimental data. The rest of the paper is organized as follows. Section 2 describes the + +modelling setup. Section 3 compares the effect of drifts in the limiter J-TEXT and divertor AUG, not only +in the key parameters, e.g., upstream density and temperature etc., but also the targets profiles in the +attached and detached regimes. The conclusions are the section 4. +2. Modelling setup +The 96×36 quadrilateral mesh for plasma, both for AUG divertor and J-TEXT limiter, and triangular mesh +for neutral particle transport are shown in Figure1. It should be mentioned that there is no private flux region +(PRF) in J-TEXT limiter, and it only contains the core and SOL computational regions. In this study, the +point in where the LCFS is tangentially contacted with the Limiter targets is name as O-point that is similar +to the X-point in divertor configuration. +For the AUG divertor, the gas puffing position is at the outer mid-plane and the pumping and settings are +inherited from previous study [4]. For the J-TEXT limiter, the gas puffing location is at the bottom of wall +according to experiments[14]. Considering the Carbon material of the first wall in the J-TEXT, a pumping +albedo with 0.9 is used at all wall elements to mimic the Carbon absorption. Only pure deuterium is +considered. At the core boundary, the ion fluxes are equal to the neutral particle flux cross the core boundary +and no other core fueling. Other boundary conditions which are not same with our previous modelling[4] +are summarized in Table 1. This is because we found with such values, current J-TEXT Limiter simulation +results are not far away from experimental measurements, which are helpful for our future validation. The +default set of EIRENE[17] reactions in SOLPS-ITER is employed which include Kotov-2008 model +[18]and +Deuterium +neutral-neutral +collision +[19]. + +Figure1 Computational meshes for (a) AUG divertor and (b) J-TEXT limiter. + + + + + + +AUGDivertor +J-TEXT Limiter +0.4 +1 +0.3 +0.2 +0.5 +0.1 +BT O +BT O +(m) +(m) +0 +Bpol O +0 +N +Bpol O +-0.1 +-0.5 +-0.2 +-0.3 +-1 +-0.4 +(a) +(b) +1.5 +2 +2.5 +0.7 +0.8 +0.9 +1 +1.1 +1.2 +1.3 +1.4 +R (m) +R (m) +Table 1 Boundary conditions for the AUG divertor and J-TEXT limiter simulation. + +AUG divertor +J-TEXT limiter +Input Power +0.6MW +0.316MW +Leakage factor for ion density at the north boundary +-0.01 +-0.001 +Leakage factor for ion temperature at the north +boundary +-0.01 +-0.001 +Leakage factor for ion thermal velocity at the north +boundary +-1.0e-4 +-1.0e-4 + +In this study, the typical AUG L-mode transport coefficients that 𝐷⊥ = 0.5 𝑚2/𝑠, 𝜒⊥,𝑖𝑜𝑛 = 𝜒⊥,𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛 = +1.6 𝑚2/𝑠 [20][21] were selected for both AUG divertor and J-TEXT limiter simulations, instead of +adjusting transport coefficients to match experimental upstream profiles. The validated physical models +[4][22], including the E×B and diamagnetic drifts and all currents (parallel electric current, anomalous +current, diamagnetic current, inertial current, ion-neutral current, current due to perpendicular and parallel +viscosity, current due to viscosity tensor), are used for both AUG divertor and J-TEXT limiter. +3. Results and discussions +3.1 +Gas puffing scan +We performed a gas puffing rate scan [23] with and without drifts for both AUG divertor and J-TEXT +limiter. The gas puffing rate Γpuff,D2 is from 1.0×1021 D/s to 6.0 ×1021 D/s and from 3.0×1020 D/s to 1.2 ×1021 +D/s for the AUG divertor and J-TEXT limiter respectively. The performance of key plasma parameters, +including outer mid-plane (OMP) electron density ne,omp and OMP temperature Te,OMP, maximum electron +temperature at the inner and outer targets Te, IT and Te,OT, the total ion flux at the inner and outer targets +Γion,IT and Γion,OT , are presented in Figure 2. In Figure 2 (a) and (b) it can be found that as gas puffing rate +increase, the ne,omp increase and Te,OMP decrease with and without drifts both in the AUG divertor and J- +TEXT limiter. When the gas puffing rate is low, the drifts effect is not strong. However, when the gas +puffing value is high, the drifts effect on ne,omp is obvious that result in the ne,OMP is ~15% higher than the +non-drift cases. Through gas puffing rate scan with the activation of drifts, the max Te,OT is from 16eV to +3eV and 25eV to 4eV in the AUG divertor and J-TEXT limiter respectively in Figure 2 (c) and (d). From +attached regimes to detached regimes [24] are numerically achieved in the AUG divertor and J-TEXT +limiter. The differences between with and without drifts are within 20%. From Figure 2 (e) and (f), for the +Γion,IT and Γion,OT, there is no roll-overment in the J-TEXT limier which is only observed in the AUG divertor. +This is because the recycled neutrals from J-TEXT limiter targets are directly cross the Last Closed Flux +Surface (LCFS) and into the core region, then ionize as ions that transport along the closed magnetic field +lines [21]. No recombination front [21][25] is formed in the J-TEXT limiter SOL. Even there is no roll- +overment of total ion flux in J-TEXT limiter, with a high gas puffing rate, the max Te,OT can be below 5eV +both with and without drifts. Thus, we believe that at such a situation the J-TEXT limiter is in the detach +regime [24]. Compared to the non-drifts cases, in the detached regime in both AUG divertor and J-TEXT +limiter, the drifts result in a lower total ion flux at both the inner and outer targets. + + + +Figure 2 The AUG divertor and J-TEXT limiter key plasma parameters behavior with and without drifts +under D2 gas puffing rate scan, including (a) Electron density at the outer mid-plane ne,OMP, (b) Electron +temperature at the outer mid-plane Te,OMP, (c) Maximum electron temperature at the inner target Te,IT , (d) +Maximum electron temperature at the at the outer target Te,OT, (e) Total ion flux at the inner target Γion,IT and +(f) Total ion flux at the inner target Γion,OT. + +3.2 +Target profiles +In order to evaluate the drifts effect on the targets profiles, two types of cases are considered as attached +cases and detached cases, in which the maximum Te,OT (with drifts) are 15eV and 5eV for both AUG +divertor and J-TEXT limiter. For the attached cases, the electron pressure balance [26][27] and electron +temperature at the targets for the AUG divertor and J-TEXT limiter are shown in Figure 3 and 4. The drifts +have no impact on the upstream electron pressure but result in the decrease of electron pressure and + +OMP n +OMP T +X1019 +50 +e +J-TEXT Full Drifts +J-TEXT Full Drifts +4 +..... .-TEXT No Drifts +..... ..TEXT No Drifts. +AUGFullDrifts +40 +(eV) +AUG Full Drifts +...AUG No Drifts. +..... AUG No Drifts. +2 +T +20 +(a) +(b) +0 +10 +0 +10 +20 +30 +40 +50 +60 +0 +10 +20 +30 +40 +50 +60 +(102° D/s) +(102° D/s) +Max T at Inner Target +Max T, at Outer Target +30 +30 +25 +25 +20 +20 +15 +15 +Max +Max T +10 +10 +5 +5 +(c) +(d) +0 +0 +0 +10 +20 +30 +40 +50 +60 +0 +10 +20 +30 +40 +50 +60 +(102° D/s) +Total T: +at Inner Target +Totalr +at Outer Target +X1022 +ion +X1041 +ion +2 +2.5 +1.5 +ion,IT +ion,OT +1.5 +Total I. +1 +Total I. +0.5 +0.5 +e +0 +0 +0 +10 +20 +30 +40 +50 +60 +0 +10 +20 +30 +40 +50 +60 +(1020 D/s) +(1020 D/s) +Puff,D +Puff,D,temperature at the inner target both for the AUG divertor and J-TEXT limiter, which means when the outer +targets are attached, the inner targets are partially detached and this is consistent with previous study[4][12]. + +Figure 3 AUG divertor target profiles for the attached cases. (a) is electron pressure balance at the inner +target, (b) is electron pressure balance at the outer target, (c) is the electron temperature at the inner target +and (d) is the electron temperature at the outer target. + +Figure 4 J-TEXT Limiter target profiles for the attached cases. (a) is electron pressure balance at the inner +target, (b) is electron pressure balance at the outer target, (c) is the electron temperature at the inner target +and (d) is the electron temperature at the outer target. + + +InnerTargetProfiles +OuterTargetProfiles +5X1020 +5X1020 + Full drifts n,T, at OMP + Full drifts n, T, at OMP + Full drifts 2n Te +4 +-. No drifts n.T at OMP +4 +..No difts n. at OMP +eV) +...No drifts 2n,.T. +eV) +O... No drifts 2n . . +? +3 +(m +e +1 +1 +a +(b) +0 +0 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +Distance along target (m) +Distance along target (m) +14 +15 +12 +10 +10 +(eV) +8 +(eV) +6 +5 +4 +Full drifts T +Full drifts T +2 +..No drifts T +..No drifts T +(c) +(d) +0 +0 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +Distancealongtarget(m) +Distance along target (m)X1020 +InnerTargetProfiles +X1020 +OuterTargetProfiles +Full drifts n,T,at OMP +FulldriftsnTatOMP +5 +Full drifts 2n,Te +6 +Full drifts 2n, T +.No drifts nTatOMP +....No drifts n.T. at OMP +-..No drifs n,T. +eV) +5 +.. No drifts 2n,T +4 +4 +3 +B +n +2 +1 +1 +(a) +(b) +0 +0 +0 +0.05 +0.1 +0.15 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +Distance alongtarget (m) +Distance along target (m) +14 +16 +Full driftsT +FulldriftsT +12 +NodriftsT +NodriftsT +14 +8 +10 +6 +(c) +(d) +4. +8 +0 +0.05 +0.1 +0.15 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +Distance alongtarget (m) +Distance along target (m)For the detached cases, the simulation results about target profiles are shown in Figure 5 and 6. Compared +with non-drift cases, the drifts result in a lower electron pressure at the upstream and targets both in the +AUG divertor and J-TEXT limiter. However, the drifts lead to a higher electron temperature near O-point +in the J-TEXT limiter, which is the opposite of the AUG divertor. We believe this is because in the detached +regime, the recycled neutrals are ionized in the core region that an ionization front is formed in the core +region near the O-point. However, with the activation of drifts, the strong poloidal drifts [28] moves the +plasma to the upstream. This is also consistent with the fact that drifts increase upstream density in Figure +2. Besides, the length of the cell in the J-JTEXT limiter SOL near O-point is larger than the others that +might leads a numeric uncertainty. + + +Figure 5 AUG divertor target profiles for the detached cases. (a) is electron pressure balance at the inner +target, (b) is electron pressure balance at the outer target, (c) is the electron temperature at the inner target +and (d) is the electron temperature at the outer target. + +Inner TargetProfiles +X1020 +Outer Target Profiles +6 +Full drifts n,T, at OMP + Full drifts n,T。 at OMP + Full drfts 2n Te + Full drifts 2n, T。 +5 +5 +. No drifts ..T at OMP +... No drifts n,.T at OMP +... No drifts n . T. +4 +eV) +4 +-.. No drifts 2n..T. +? +? +3 +3 +me2 +me2 +1 +1 +(a) +(b) +0 +0 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +Distance along target (m) +Distance along target (m) +2 +6 +5 +1.5 +4 +(eV) +2 +0.5 +Full drifts T +Full drifts T +1 +...No drifts T +. No drifts T +(c) +(d) +0 +0 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +Distance along target (m) +Distance along target (m) +Figure 6 J-TEXT Limiter target profiles for the detached cases. (a) is electron pressure balance at the inner +target, (b) is electron pressure balance at the outer target, (c) is the electron temperature at the inner target +and (d) is the electron temperature at the outer target. +3.3 +Neutral density distribution +Figure 7 shows the neutral particle density (D+2D2) distributions in the AUG divertor and J-TEXT limiter +for the attached and detached cases with the activation of drifts. The neutral density in SOL region near +targets in AUG divertor is ~two order magnitude higher than the one in J-TEXT limiter. This is consistent +with the fact that the roll-overment of total ion flux is only observed in AUG divertor due to the strong +recombination front. The complicated sub-divertor structures in AUG [4][13], which are not exist in the J- +TEXT limiter, may also contributes to the high neutral density. Figure 8 shows the ratio of neutral density +between with and without drifts. For the attached cases, the drifts result in a higher neutral density in the +high field side, which is consistent that the drifts lead to the inner target partially detached. For the detached +cases, in AUG divertor, the drifts result in a higher neutral density in the divertor entrance. However, for +the J-TEXT limiter, the drifts lead to a lower neutral density near the O-point in the core region and SOL +region. In the future, we will investigate in detail about how the ExB drifts and diamagnetic drifts affect the +particle flux which is related to the lower neutral density in the J-TEXT limiter. + +X1020 +Inner Target Profiles +X1020 +Outer Target Profiles +Full drifts n,T at OMP +Full drifts n,T。 at OMP +6 + Full drifts 2n, Te +6 + Full drifts 2n, T。 +.... No drifts n.T.at OMP +.... No drifts n.T. at OMP +'eV) +5 +... No drifts n . T. +eV) +5 +4 +? +4 +e3 +e +2 +n +2 +1 +1 +(a) +(b) +0 +0 +0 +0.05 +0.1 +0.15 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +Distance along target (m) +Distance along target (m) +4.5 +5.5 +4 +5 +3.5 +4.5 +(eV) +(eV) +3 +Te +4 +2.5 +3.5 +Full drifts T +Full driftsT +2 +3 +...No drifts T +.... No drifts T +(c) +(d +1.5 +2.5 +0 +0.05 +0.1 +0.15 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +Distance along target (m) +Distance along target (m) +Figure 7 Neutral density (D+2D2) distributions with the activation of drifts. (a) is AUG divertor in +attached case, (b) is J-TEXT limiter in attached case, (c) is AUG divertor in detached case and (d) is +(c) J-TEXT limiter in detached case. + +Figure 8 The ratio of neutral density (D+2D2) between with and without drift. (a) is AUG divertor in +attached case, (b) is J-TEXT limiter in attached case, (c) is AUG divertor in detached case and (d) is +J-TEXT limiter in detached case. + + +Attached:n +Density (m3) +Detached:n +neu,drift +neu,drift +Density (m) +-0.4 +-0.4 +1020 +1020 +-0.6 +-0.6 +1018 +1018 +N +-1 +I- +-1.2 +10/6 +-1.2 +1016 +(a) +(b) +-1.4 +-1.4 +1 +1.5 +2 +2.5 +1 +1.5 +2 +2.5 +R (m) +R (m) +Attached:n +Density (m3) +Detached: n +neu,drift +neu,drift +Density (m3) +0.4 +102 +0.4 +102 +0.3 +1020 +0.3 +1020 +0.2 +0.2 +1019 +0.1 +0.1 +1019 +(u) +(w) +0 +1018 +0 +1018 +N +N +-0.1 +-0.1 +1017 +1017 +-0.2 +0.2 +0.3 +1016 +0.3 +1016 +-0.4 +(c) +-0.4F +(d) +1015 +1015 +0.8 +1.2 +1.4 +0.8 +1 +1.2 +1.4 +R (m) +R (m)Attached:n +/n +neu,w/odrifts +Ratio +Detached:n +neu,withdrift +/n +neu +neu,w/odrifts +Ratio +-0.4 +-0.4 +-0.6 +-0.6 +1.5 +1.5 +N +I- +-1.2 +-1.2 +(a) +(b) +-1.4 +0.5 +-1.4 +0.5 +1 +1.5 +2 +2.5 +1 +1.5 +2 +2.5 +R (m) +R (m) +Attached:n +neu,with drift +/n +neu,w/o drifts Ratio +Detached:n +neu,with drift +/n +"neu,w/odriftsRatio +0.4 +0.4 +0.3 +0.3 +0.2 +0.2 +1.5 +1.5 +0.1 +0.1 +(u) +(w) +0 +0 +N +N +-0.1 +-0.1 +-0.2 +-0.2 +0.3 +0.3 +-0.4 +(c) +-0.4F +(d) +0.5 +0.5 +0.8 +1.2 +1.4 +0.8 +1.2 +1.4 +R (m) +R (m)4. Conclusions +In this work, SOLPS-ITER numerical simulations on the J-TEXT limiter configuration are performed with +and without the activation of drifts. The simulation results are compared with ASDEX upgrade results to +assess the effect of drifts. Through a gas puffing rate scan, the key plasma parameters, except the total ion +flux, in the J-TEXT limiter have a qualitative performance compared with the AUG divertor simulation +results. The recycled neutrals from limiter targets are directly into the core region that there is no +recombination front in the limiter SOL. Attached and detached cases are selected, in which the maximum +Te,OT are 15eV and 5eV respectively, to evaluate the effect of drifts on the target profiles. In the attached +cases, the drifts result in the partial detachment of inner target both in the AUG divertor and J-TEXT limiter. +For the detached cases, the drifts decrease electron temperature in the AUG divertor targets but increase +the electron temperature near the O-point at the J-TEXT limiter targets. This is because the poloidal drifts +move the ions, which is from the ionization of recycled neutrons, to the upstream. The effect of drifts on +neutral density distribution is also discussed that the drifts lead to a higher neutral density in the attached +regime. +In the future, the J-TEXT limiter simulation results will be validated against experimental data through +adjust the transport coefficients. The Carbon impurity and different limiter positions will also be considered. + +Acknowledgements +The authors would like to thank Dr.X. Bonnin for his useful discussions about the neutral particle transport +in limiter. This work has been carried out within the framework of the EUROfusion Consortium, funded +by the European Union via the Euratom Research and Training Programme (Grant Agreement No +101052200 — EUROfusion). Views and opinions expressed are however those of the author(s) only and +do not necessarily reflect those of the European Union or the European Commission. Neither the European +Union nor the European Commission can be held responsible for them. This work has been part-funded by +the EPSRC Energy Programme (Grant No. EP/W006839/1). + +References +[1] S. Wiesen, et al., The new SOLPS-ITER code package, Journal of nuclear materials 463 (2015): +480-484. +[2] T.D. Rognlien et al., A fully implicit, time dependent 2D fluid code for modeling tokamak edge +plasmas Journal of nuclear materials. 196(1992) 347–351 +[3] H. Bufferand et al., Implementation of drift velocities and currents in SOLEDGE2D-EIRENE, Nucl. +Mater. 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Kaveeva et al., SOLPS-ITER modelling of ITER edge plasma with drifts and currents, Nucl. +Fusion 60 (2020) 046019. + diff --git a/_9E3T4oBgHgl3EQfTQlk/content/tmp_files/load_file.txt b/_9E3T4oBgHgl3EQfTQlk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bf13918231a44903a9908fed27f24164bce06cec --- /dev/null +++ b/_9E3T4oBgHgl3EQfTQlk/content/tmp_files/load_file.txt @@ -0,0 +1,523 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf,len=522 +page_content='SOLPS-ITER numerical evaluation about the effect of drifts in a divertor configuration of ASDEX-Upgrade and a limiter configuration of J-TEXT H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Wu1, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Shi2, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Subba1, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Sun2, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Wischmeier3, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Zanino1, the ASDEX Upgrade Team 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='NEMO Group, Dipartimento Energia, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='United Kingdom Atomic Energy Authority, Culham Centre for Fusion Energy, Culham Science Centre, Abingdon, Oxon OX14 3DB, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='Max-Planck-Institut für Plasmaphysik, Boltzmannstraße 2, D-85748 Garching, Germany Abstract: The High Field High Density Region (HFSFD) has been experimentally observed in both divertor and limiter tokamak devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' In order to numerically reproduced the HFSHD region in the limiter tokamak device J-TEXT, we first performed SOLPS-ITER simulations on the J-TEXT limiter tokamak with the activation of drifts, which is associated with the HFSHD region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The validated physical models, which is from ASDEX Upgrade (AUG) divertor modelling including full drifts and currents, were applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Through a gas puffing rate scan, both attached and detached regimes were numerically obtained in the AUG divertor and J-TEXT limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The key plasma parameters in the J-TEXT limiter are evaluated with and without drifts that have a qualitative performance like AUG except the roll-overment of the total ion flux at the targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The drift effects on the target profiles are investigated in which the maximum electron temperature at the outer targets is 15eV and 5eV respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' When the outer targets are attached in the J-TEXT limiter and AUG divertor, the drifts result in the partial detachment of the inner targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' In the detached regimes, the drifts decrease the electron temperature on AUG divertor targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' However, for the J-TEXT limiter, the electron temperature only decreases at the far SOL region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The effect of drifts on the neutral density is also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Introduction Edge plasma code packages, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' SOLPS-ITE [1], UEDGE[2] and SOLEDGE2D[3], are widely used to study boundary plasma behavior in current tokamak devices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' ASDEX Upgrade (AUG)[4] and Alcator C-mod[5], and to the design of future tokamak devices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' DTT [6], SPARC[7] and EU-DEMO[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' In SOLPS-ITER, two main modules are included: the multi-fluid plasma solver B2[9] for charged species transport toroidal symmetry, and the Monte Carlo code EIRENE[10], which describes kinetic neutral transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The High Field Side High Density region, in which the volume electron density ne is at least 10 times higher than the upstream averaged and target values, has been observed in the Scrape-Off layer(SOL) of the divertor Tokamak device ASDEX Upgrade (AUG)[11], and has been numerically reproduced by SOLPS- ITER with the activation of drifts [4][12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Recently, a similar high-density front has been observed in the SOL of the limiter tokamak device J-TEXT, in which the high-density front is seven times higher than the density in the low field side region [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' In order to investigate the formation of high-density front in the J-TEXT, we first performed a SOLPS- ITER numerical simulation on the J-TEXT limiter configuration with the activation of full drifts and currents[15][16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The effect of drifts in the J-TEXT limiter is evaluated through a preliminary comparison with AUG single-null divertor configuration simulation results, in which the physical models [4] have been validated against experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Section 2 describes the modelling setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Section 3 compares the effect of drifts in the limiter J-TEXT and divertor AUG, not only in the key parameters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=', upstream density and temperature etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=', but also the targets profiles in the attached and detached regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The conclusions are the section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Modelling setup The 96×36 quadrilateral mesh for plasma, both for AUG divertor and J-TEXT limiter, and triangular mesh for neutral particle transport are shown in Figure1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' It should be mentioned that there is no private flux region (PRF) in J-TEXT limiter, and it only contains the core and SOL computational regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' In this study, the point in where the LCFS is tangentially contacted with the Limiter targets is name as O-point that is similar to the X-point in divertor configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' For the AUG divertor, the gas puffing position is at the outer mid-plane and the pumping and settings are inherited from previous study [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' For the J-TEXT limiter, the gas puffing location is at the bottom of wall according to experiments[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Considering the Carbon material of the first wall in the J-TEXT, a pumping albedo with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='9 is used at all wall elements to mimic the Carbon absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Only pure deuterium is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' At the core boundary, the ion fluxes are equal to the neutral particle flux cross the core boundary and no other core fueling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Other boundary conditions which are not same with our previous modelling[4] are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' This is because we found with such values, current J-TEXT Limiter simulation results are not far away from experimental measurements, which are helpful for our future validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The default set of EIRENE[17] reactions in SOLPS-ITER is employed which include Kotov-2008 model [18]and Deuterium neutral-neutral collision [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Figure1 Computational meshes for (a) AUG divertor and (b) J-TEXT limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' AUGDivertor J-TEXT Limiter 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 BT O BT O (m) (m) 0 Bpol O 0 N Bpol O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 (a) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 R (m) R (m) Table 1 Boundary conditions for the AUG divertor and J-TEXT limiter simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' AUG divertor J-TEXT limiter Input Power 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='6MW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='316MW Leakage factor for ion density at the north boundary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='001 Leakage factor for ion temperature at the north boundary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='001 Leakage factor for ion thermal velocity at the north boundary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='0e-4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='0e-4 In this study, the typical AUG L-mode transport coefficients that 𝐷⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 𝑚2/𝑠, 𝜒⊥,𝑖𝑜𝑛 = 𝜒⊥,𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='6 𝑚2/𝑠 [20][21] were selected for both AUG divertor and J-TEXT limiter simulations, instead of adjusting transport coefficients to match experimental upstream profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The validated physical models [4][22], including the E×B and diamagnetic drifts and all currents (parallel electric current, anomalous current, diamagnetic current, inertial current, ion-neutral current, current due to perpendicular and parallel viscosity, current due to viscosity tensor), are used for both AUG divertor and J-TEXT limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Results and discussions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 Gas puffing scan We performed a gas puffing rate scan [23] with and without drifts for both AUG divertor and J-TEXT limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The gas puffing rate Γpuff,D2 is from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='0×1021 D/s to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='0 ×1021 D/s and from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='0×1020 D/s to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 ×1021 D/s for the AUG divertor and J-TEXT limiter respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The performance of key plasma parameters, including outer mid-plane (OMP) electron density ne,omp and OMP temperature Te,OMP, maximum electron temperature at the inner and outer targets Te, IT and Te,OT, the total ion flux at the inner and outer targets Γion,IT and Γion,OT , are presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' In Figure 2 (a) and (b) it can be found that as gas puffing rate increase, the ne,omp increase and Te,OMP decrease with and without drifts both in the AUG divertor and J- TEXT limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' When the gas puffing rate is low, the drifts effect is not strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' However, when the gas puffing value is high, the drifts effect on ne,omp is obvious that result in the ne,OMP is ~15% higher than the non-drift cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Through gas puffing rate scan with the activation of drifts, the max Te,OT is from 16eV to 3eV and 25eV to 4eV in the AUG divertor and J-TEXT limiter respectively in Figure 2 (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' From attached regimes to detached regimes [24] are numerically achieved in the AUG divertor and J-TEXT limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The differences between with and without drifts are within 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' From Figure 2 (e) and (f), for the Γion,IT and Γion,OT, there is no roll-overment in the J-TEXT limier which is only observed in the AUG divertor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' This is because the recycled neutrals from J-TEXT limiter targets are directly cross the Last Closed Flux Surface (LCFS) and into the core region, then ionize as ions that transport along the closed magnetic field lines [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' No recombination front [21][25] is formed in the J-TEXT limiter SOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Even there is no roll- overment of total ion flux in J-TEXT limiter, with a high gas puffing rate, the max Te,OT can be below 5eV both with and without drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Thus, we believe that at such a situation the J-TEXT limiter is in the detach regime [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Compared to the non-drifts cases, in the detached regime in both AUG divertor and J-TEXT limiter, the drifts result in a lower total ion flux at both the inner and outer targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Figure 2 The AUG divertor and J-TEXT limiter key plasma parameters behavior with and without drifts under D2 gas puffing rate scan, including (a) Electron density at the outer mid-plane ne,OMP, (b) Electron temperature at the outer mid-plane Te,OMP, (c) Maximum electron temperature at the inner target Te,IT , (d) Maximum electron temperature at the at the outer target Te,OT, (e) Total ion flux at the inner target Γion,IT and (f) Total ion flux at the inner target Γion,OT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 Target profiles In order to evaluate the drifts effect on the targets profiles, two types of cases are considered as attached cases and detached cases, in which the maximum Te,OT (with drifts) are 15eV and 5eV for both AUG divertor and J-TEXT limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' For the attached cases, the electron pressure balance [26][27] and electron temperature at the targets for the AUG divertor and J-TEXT limiter are shown in Figure 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The drifts have no impact on the upstream electron pressure but result in the decrease of electron pressure and OMP n OMP T X1019 50 e J-TEXT Full Drifts J-TEXT Full Drifts 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='-TEXT No Drifts .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='.TEXT No Drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' AUGFullDrifts 40 (eV) AUG Full Drifts .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='AUG No Drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' AUG No Drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' 2 T 20 (a) (b) 0 10 0 10 20 30 40 50 60 0 10 20 30 40 50 60 (102° D/s) (102° D/s) Max T at Inner Target Max T, at Outer Target 30 30 25 25 20 20 15 15 Max Max T 10 10 5 5 (c) (d) 0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 (102° D/s) Total T: at Inner Target Totalr at Outer Target X1022 ion X1041 ion 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 ion,IT ion,OT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 Total I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' 1 Total I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 e 0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 (1020 D/s) (1020 D/s) Puff,D Puff,D,temperature at the inner target both for the AUG divertor and J-TEXT limiter, which means when the outer targets are attached, the inner targets are partially detached and this is consistent with previous study[4][12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Figure 3 AUG divertor target profiles for the attached cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' (a) is electron pressure balance at the inner target, (b) is electron pressure balance at the outer target, (c) is the electron temperature at the inner target and (d) is the electron temperature at the outer target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Figure 4 J-TEXT Limiter target profiles for the attached cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' (a) is electron pressure balance at the inner target, (b) is electron pressure balance at the outer target, (c) is the electron temperature at the inner target and (d) is the electron temperature at the outer target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' InnerTargetProfiles OuterTargetProfiles 5X1020 5X1020 Full drifts n,T, at OMP Full drifts n, T, at OMP Full drifts 2n Te 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' No drifts n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='T at OMP 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='.No difts n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' at OMP eV) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='No drifts 2n,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' eV) O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' No drifts 2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' 3 (m e 1 1 a (b) 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 Distance along target (m) Distance along target (m) 14 15 12 10 10 (eV) 8 (eV) 6 5 4 Full drifts T Full drifts T 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='.No drifts T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='.No drifts T (c) (d) 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 Distancealongtarget(m) Distance along target (m)X1020 InnerTargetProfiles X1020 OuterTargetProfiles Full drifts n,T,at OMP FulldriftsnTatOMP 5 Full drifts 2n,Te 6 Full drifts 2n, T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='No drifts nTatOMP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='.No drifts n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' at OMP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='.No drifs n,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' eV) 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='. No drifts 2n,T 4 4 3 B n 2 1 1 (a) (b) 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='14 Distance alongtarget (m) Distance along target (m) 14 16 Full driftsT FulldriftsT 12 NodriftsT NodriftsT 14 8 10 6 (c) (d) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' 8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='14 Distance alongtarget (m) Distance along target (m)For the detached cases, the simulation results about target profiles are shown in Figure 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Compared with non-drift cases, the drifts result in a lower electron pressure at the upstream and targets both in the AUG divertor and J-TEXT limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' However, the drifts lead to a higher electron temperature near O-point in the J-TEXT limiter, which is the opposite of the AUG divertor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' We believe this is because in the detached regime, the recycled neutrals are ionized in the core region that an ionization front is formed in the core region near the O-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' However, with the activation of drifts, the strong poloidal drifts [28] moves the plasma to the upstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' This is also consistent with the fact that drifts increase upstream density in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Besides, the length of the cell in the J-JTEXT limiter SOL near O-point is larger than the others that might leads a numeric uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Figure 5 AUG divertor target profiles for the detached cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' (a) is electron pressure balance at the inner target, (b) is electron pressure balance at the outer target, (c) is the electron temperature at the inner target and (d) is the electron temperature at the outer target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Inner TargetProfiles X1020 Outer Target Profiles 6 Full drifts n,T, at OMP Full drifts n,T。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' at OMP Full drfts 2n Te Full drifts 2n, T。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' 5 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' No drifts .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='.T at OMP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' No drifts n,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='T at OMP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' No drifts n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' 4 eV) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='. No drifts 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='.T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' 3 3 me2 me2 1 1 (a) (b) 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 Distance along target (m) Distance along target (m) 2 6 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 4 (eV) 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 Full drifts T Full drifts T 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='No drifts T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' No drifts T (c) (d) 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 Distance along target (m) Distance along target (m) Figure 6 J-TEXT Limiter target profiles for the detached cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' (a) is electron pressure balance at the inner target, (b) is electron pressure balance at the outer target, (c) is the electron temperature at the inner target and (d) is the electron temperature at the outer target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 Neutral density distribution Figure 7 shows the neutral particle density (D+2D2) distributions in the AUG divertor and J-TEXT limiter for the attached and detached cases with the activation of drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The neutral density in SOL region near targets in AUG divertor is ~two order magnitude higher than the one in J-TEXT limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' This is consistent with the fact that the roll-overment of total ion flux is only observed in AUG divertor due to the strong recombination front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The complicated sub-divertor structures in AUG [4][13], which are not exist in the J- TEXT limiter, may also contributes to the high neutral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Figure 8 shows the ratio of neutral density between with and without drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' For the attached cases, the drifts result in a higher neutral density in the high field side, which is consistent that the drifts lead to the inner target partially detached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' For the detached cases, in AUG divertor, the drifts result in a higher neutral density in the divertor entrance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' However, for the J-TEXT limiter, the drifts lead to a lower neutral density near the O-point in the core region and SOL region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' In the future, we will investigate in detail about how the ExB drifts and diamagnetic drifts affect the particle flux which is related to the lower neutral density in the J-TEXT limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' X1020 Inner Target Profiles X1020 Outer Target Profiles Full drifts n,T at OMP Full drifts n,T。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' at OMP 6 Full drifts 2n, Te 6 Full drifts 2n, T。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='. No drifts n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='at OMP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='. No drifts n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=" at OMP 'eV) 5 ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' No drifts n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' eV) 5 4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' 4 e3 e 2 n 2 1 1 (a) (b) 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='14 Distance along target (m) Distance along target (m) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 4 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 (eV) (eV) 3 Te 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 Full drifts T Full driftsT 2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='No drifts T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='. No drifts T (c) (d 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='14 Distance along target (m) Distance along target (m) Figure 7 Neutral density (D+2D2) distributions with the activation of drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' (a) is AUG divertor in attached case, (b) is J-TEXT limiter in attached case, (c) is AUG divertor in detached case and (d) is (c) J-TEXT limiter in detached case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Figure 8 The ratio of neutral density (D+2D2) between with and without drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' (a) is AUG divertor in attached case, (b) is J-TEXT limiter in attached case, (c) is AUG divertor in detached case and (d) is J-TEXT limiter in detached case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Attached:n Density (m3) Detached:n neu,drift neu,drift Density (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 1020 1020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='6 1018 1018 N 1 I- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 10/6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 1016 (a) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 R (m) R (m) Attached:n Density (m3) Detached: n neu,drift neu,drift Density (m3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 1020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 1020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 1019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 1019 (u) (w) 0 1018 0 1018 N N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 1017 1017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 1016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 1016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4F (d) 1015 1015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 R (m) R (m)Attached:n /n neu,w/odrifts Ratio Detached:n neu,withdrift /n neu neu,w/odrifts Ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 N I- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 (a) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 R (m) R (m) Attached:n neu,with drift /n neu,w/o drifts Ratio Detached:n neu,with drift /n "neu,w/odriftsRatio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 (u) (w) 0 0 N N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4F (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='4 R (m) R (m)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Conclusions In this work, SOLPS-ITER numerical simulations on the J-TEXT limiter configuration are performed with and without the activation of drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The simulation results are compared with ASDEX upgrade results to assess the effect of drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Through a gas puffing rate scan, the key plasma parameters, except the total ion flux, in the J-TEXT limiter have a qualitative performance compared with the AUG divertor simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The recycled neutrals from limiter targets are directly into the core region that there is no recombination front in the limiter SOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Attached and detached cases are selected, in which the maximum Te,OT are 15eV and 5eV respectively, to evaluate the effect of drifts on the target profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' In the attached cases, the drifts result in the partial detachment of inner target both in the AUG divertor and J-TEXT limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' For the detached cases, the drifts decrease electron temperature in the AUG divertor targets but increase the electron temperature near the O-point at the J-TEXT limiter targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' This is because the poloidal drifts move the ions, which is from the ionization of recycled neutrons, to the upstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The effect of drifts on neutral density distribution is also discussed that the drifts lead to a higher neutral density in the attached regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' In the future, the J-TEXT limiter simulation results will be validated against experimental data through adjust the transport coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' The Carbon impurity and different limiter positions will also be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Acknowledgements The authors would like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Bonnin for his useful discussions about the neutral particle transport in limiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 — EUROfusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' Neither the European Union nor the European Commission can be held responsible for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' This work has been part-funded by the EPSRC Energy Programme (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} +page_content=' EP/W006839/1).' metadata={'source': 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046019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E3T4oBgHgl3EQfTQlk/content/2301.04440v1.pdf'} diff --git a/_tFIT4oBgHgl3EQf-Cuq/vector_store/index.faiss b/_tFIT4oBgHgl3EQf-Cuq/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..1130526c665a3eb39b75361f90483670820d96ce --- /dev/null +++ b/_tFIT4oBgHgl3EQf-Cuq/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:79ad1d9acfacbe000ff685007d404f04e50986cdcec645a917848829d42e244d +size 3604525 diff --git a/aNAzT4oBgHgl3EQfZPyw/content/2301.01349v1.pdf b/aNAzT4oBgHgl3EQfZPyw/content/2301.01349v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..8f15f40a5be19edae85e00d84ecdcbdcb9a528ed --- /dev/null +++ b/aNAzT4oBgHgl3EQfZPyw/content/2301.01349v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9daddd5ec08d9edcf9a6e67014c28bb69e4593a63e2c89b20712c9e0411b6d7c +size 1017614 diff 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mode 100644 index 0000000000000000000000000000000000000000..72523435292c4aab24254a979407591210dbd032 --- /dev/null +++ b/aNFOT4oBgHgl3EQf_jRR/content/tmp_files/2301.12978v1.pdf.txt @@ -0,0 +1,5147 @@ +THE RANK OF SPARSE SYMMETRIC MATRICES OVER ARBITRARY +FIELDS +Remco van der Hofstad, Noela Müller, Haodong Zhu +January 31, 2023 +ABSTRACT +Let F be an arbitrary field and (Gn,d/n)n be a sequence of sparse weighted Erd˝os-Rényi random +graphs on n vertices with edge probability d/n, where weights from F \ {0} are assigned to the +edges according to a fixed matrix Jn. We show that the normalised rank of the adjacency matrix +of (Gn,d/n)n converges in probability to a constant, and derive the limiting expression. Our result +shows that for the general class of sparse symmetric matrices under consideration, the asymptotics +of the normalised rank are independent of the edge weights and even the field, in the sense that the +limiting constant for the general case coincides with the one previously established for adjacency +matrices of sparse (non-weighted) Erd˝os-Rényi matrices over R from [8]. Our proof, which is purely +combinatorial in its nature, is based on an intricate extension of the novel perturbation approach from +[10] to the symmetric setting. +Keywords Rank · Random matrix · Erd˝os-Rényi graph +1 +Introduction +1.1 +Background and motivation +The study of matrices with random entries, going back to the 1950’s [29], is an important and lively field of +modern probability and combinatorics with close ties to a multitude of other scientific disciplines such as theoretical +physics, mathematical statistics, computer science, neuroscience or machine learning. Up to this day, the theory of +random matrices has developed into a mature field and advanced to a very precise understanding of classical models +such as Gaussian Ensembles, Bernoulli matrices or Wishart matrices. +Moreover, in the last decade, there has been a burst of progress in the theoretical understanding of random matrices +which appear naturally in the study of random graphs, such as their adjacency matrices. Especially the adjacency matrix +of the classical Erd˝os-Rényi random graph model and its spectral properties have attracted a great deal of attention +[7, 21, 22]. The Erd˝os-Rényi graph Gn,pn = ({1, . . . , n}, En), which is arguably the simplest random graph model, is +a graph on n vertices, where each edge is present independently with probability pn. Its adjacency matrix An,pn is a +symmetric n × n-matrix with entries An,pn(i, j) = 1 {{i, j} ∈ En}1. In particular, it is a symmetric Bernoulli matrix, +which, depending on the limiting behaviour of the edge probability pn, displays different asymptotic behaviour: Results +by Costello, Tao and Vu [15] and later by Basak and Rudelson [5] have shown that there is a sharp transition in the +invertibility of the adjacency matrix around ln n/n + k(n)/n, for a function k(n) that tends slowly to infinity: When +pn > ln n/n + k(n)/n, with high probability (w.h.p.) the adjacency matrix is nonsingular, while it is singular w.h.p. +for pn < ln n/n − k(n)/n. +Following this threshold result, a natural question is to determine the rank of the adjacency matrix An,pn when pn +is small enough such that the matrix is singular w.h.p. In the regime where pn ∈ [c ln n/n, 1/2] for c > 1/2, Costello +and Vu [17] show that w.h.p., the rank of An,pn is exactly equal to n minus the number of isolated vertices in the +underlying Erd˝os-Rényi random graph. They extend their result to c > 0 and arbitrary deterministic non-zero entries +(instead of 1) in [16]. This result shows that w.h.p., the rank only depends on the structure of the graph, regardless of +the precise value of the nonzero entries of the adjacency matrix. Finally, when pn = d/n for fixed d > 0, Bordenave, +Lelarge and Salez [8] derive an asymptotic rank formula for An,d/n (see (1.3) below). +1For an event B, 1{B} denotes the indicator function of B. When appropriate, we also use 1B. +arXiv:2301.12978v1 [math.CO] 30 Jan 2023 + +The rank of sparse symmetric matrices over arbitrary fields +While all these results naturally consider the rank of the adjacency matrix An,pn over R (or equivalently, Q), we +will be interested in the rank of An,pn over arbitrary fields F in the sparse regime where pn = d/n (interpreting a +1-entry as the multiplicative identity of the field, and a 0-entry as its additive identity). Moreover, inspired by [16], we +consider the more general class of matrices where the non-zero entries of An,pn are arbitrary deterministic non-zero +elements of F. Our main Theorem 1.2 shows that even under this vast generalisation, the asymptotic rank formula of +Bordenave, Lelarge and Salez still remains valid. This result suggests that the rank indeed only depends on the positions +of the non-zero entries of the adjacency values, which is reflected in our proof strategy. +Indeed, thanks to observations of Bauer and Golinelli [6], there is a by now well-known and purely combinatorial +upper bound on the asymptotic rank of An,d/n, which is based on the Karp-Sipser algorithm for finding large matchings +[26]: Start with Gn,p. At each step of the algorithm, recursively, a vertex of degree one along with its unique neighbor +is removed. The process stops once only isolated vertices and vertices of degree at least two, the so-called Karp-Sipser +core, are left. It is straightforward to check that this “leaf-removal” leaves the nullity of the graph invariant (for a proof, +see [6]). Since the nullity of the reduced graph is apparently lower bounded by its number of isolated vertices, this +number of isolated vertices provides an upper bound on the rank of the original graph that is completely oblivious to +the field or the precise values of the non-zero entries. Karp and Sipser [26] also derive a formula for the asymptotic +number of isolated vertices in the reduced graph. Moreover, for d ≤ e, all but a vanishing proportion of vertices become +isolated after running the Karp-Sipser algorithm on Gn,d/n. Thus, for d ≤ e, the question is already completely settled. +However, when d > e, w.h.p., the Karp-Sipser core is not negligible, which complicates matters significantly. +Since there is already the rank formula of [8] in the sparse case, a natural take on the problem of the missing lower +bound would be to turn to the proof methods of Bordenave, Lelarge and Salez and adapt them to our setting. However, +their analysis makes heavy use of spectral properties of real symmetric matrices, so to the best of our knowledge, there +is no possibility to follow their approach. +On the other hand, inspired by insights from statistical physics, Coja-Oghlan, Ergür, Gao, Hetterich and Rolvien +[10] found a new combinatorial approach to derive an asymptotic rank formula for a broad class of asymmetric sparse +random matrices, generalising earlier results by Cooper, Frieze and Pegden for F2 [14]. Correspondingly, the results +of [10] are valid over any field, regardless of the distribution of the non-zero entries. However, their approach cannot +straightforwardly be applied to symmetric random matrices, since these retain much less independence among the +positions of their non-zero entries. Indeed, the authors note that “an intriguing question for future research is to extend +the techniques from the present paper to symmetric random matrices.” +In this paper, we build on several of the core concepts of [10] to develop a corresponding combinatorial approach +towards rank formulas for sparse symmetric matrices. As in [10], instead of investigating the rank of An,d/n directly, +we work with a perturbed version of An,d/n. Moreover, as in [10], we use a telescoping argument to lower bound +the expected rank and relate the rank difference of matrices whose sizes differ by one to so-called “frozen” variables. +However, the symmetry of our matrices poses serious obstructions to any attempt to literally follow in the footsteps +of [10], and we therefore introduce quite a number of changes and adaptations. These changes allow us to give +a precise characterization of the rank increase when we add a row and a column, and therefore to show that the +asymptotic behavior of the rank of a broad class of random matrices, whose non-zero entries are prescribed by the +adjacency structure of a sparse Erd˝os-Rényi random graph, over any field F, is indeed the same as the rank of the simple +0/1-adjacency matrix of Gn,d/n over the field R. +This paper is organised as follows: In Section 1.2, we introduce our precise model and main result. A proof +overview, together with the most important intermediate steps, can be found in Section 2. Section 3 collects results +on our matrix perturbation. In Section 4, we investigate various properties of the different variable (or vertex) types +introduced earlier, and their relation to the rank. We then derive the fixed point equations for the asymptotic proportions +of some of the different types in Section 5. Section 6 uses these fixed point equations to derive the desired lower bound +on the asymptotic rank. In Appendix A, we provide important properties of the various functions related to the rank +formula. Appendix B explains how to derive an upper bound on the normalised rank from results on the Karp-Sipser +leaf-removal algorithm. Finally, Appendix C contains a proposition which is used to compare different conditional +expectations. +Remark 1.1 (Notation for random variables). Throughout the article, we use bold letters to indicate random variables +and regular letters to indicate deterministic quantities. +■ +1.2 +Main results +Let F be an arbitrary field and F∗ := F \ {0} its multiplicative group. For a general matrix A ∈ Fm×n, rkF(A) +specifically denotes the rank of A over F, i.e. the dimension of the linear subspace of Fn spanned by the columns of A. +Moreover, we use Symn(F∗) for the set of all symmetric n × n matrices with entries in F∗. +In the present article, we study adjacency matrices of sparse Erd˝os-Rényi random graphs with arbitrary non-zero +edge weights over F. To define the precise model, let (Jn)n≥1 be any deterministic sequence of “template” matrices +such that for all n ≥ 1, Jn ∈ Symn(F∗), and (q(i, j))i,j≥1 be an array of i.i.d. uniform random variables in [0, 1]. For +2 + +The rank of sparse symmetric matrices over arbitrary fields +p ∈ [0, 1], we then define the matrix An,p by setting +An,p(i, j) = +� +� +� +1{q(i, j) < p}Jn(i, j), +i < j; +1{q(j, i) < p}Jn(j, i), +i > j; +0, +i = j. +(1.1) +An,p can be alternatively regarded as the adjacency matrix of a weighted Erd˝os-Rényi random graph on the vertex set +[n], where each potential edge {i, j} is present independently with probability p. If it is present, it is assigned edge +weight Jn(i, j) = Jn(j, i). The construction (1.1) also incorporates a natural coupling of the positions of the nonzero +entries of the matrices An,p for all choices of n and p. +In the important special case where Jn(i, j) ≡ 1 for all i, j ∈ {1, 2, . . . , n}, An,p coincides with the adjacency +matrix of an unweighted Erd˝os-Rényi graph with n vertices and edge probability p. An asymptotic rank formula for +this model over F = R in the regime where p = d/n was given by Bordenave, Lelarge and Salez in [8]: For any d > 0, +let φd : [0, 1] → R, φd(α) := exp(d(α − 1)) be the probability generating function of a Poisson random variable with +parameter d and Rd : [0, 1] → R be defined by setting +Rd(α) = 2 − φd (1 − φd(α)) − (1 + d(1 − α))φd(α). +(1.2) +Bordenave, Lelarge and Salez [8] then show that for any d > 0, in the coupling given above, +lim +n→∞ +1 +n rkR +� +An,d/n +� += min +α∈[0,1] Rd(α) +a.s. +(1.3) +The article [8] also provides asymptotic rank formulas for the adjacency matrices of any sequence of random graphs +that converges locally to a rooted Galton-Watson tree whose degree distribution has a finite second moment. +For general fields F, of course, rkF(An,d/n) need not be identical to rkR(An,d/n) (even in the case where +Jn(i, j) ≡ 1). For example, if F = Fp is the finite field with p elements, then generally only the upper bound +rkFp(An,d/n) ≤ rkR(An,d/n) holds true. Moreover, the proof of the rank formula (1.3) is based on the rank-nullity +theorem and the fact that nulF(An,d/n) is identical to the dimension of the eigenspace of A corresponding to 0. Since +for real symmetric matrices, the geometric and algebraic multiplicities of all eigenvalues coincide, the dimension of the +eigenspace of A corresponding to 0 can be studied through an associated spectral measure in this case. On the other +hand, for symmetric matrices over Fp, there is no reason to assume the matrix to be diagonalisable. +Pursuing a purely combinatorial approach that does not rely on the analysis of a spectral measure, our main result +generalises the asymptotic rank formula of [8] to arbitrary fields F and general non-zero entries: +Theorem 1.2. For any d > 0 and any field F, rkF +� +An,d/n +� +/n converges in probability to minα∈[0,1] Rd(α) uniformly +in (Jn)n≥1 in the sense that for any ε > 0, +lim +n→∞ +sup +Jn∈Symn(F∗) +P +����� +1 +n rkF +� +An,d/n +� +− min +α∈[0,1] Rd(α) +���� ≥ ε +� += 0. +(1.4) +Remark 1.3 (Almost sure convergence). In the case where Jn(i, j) ≡ 1 and one is interested in convergence of the +sequence (An,d/n)n≥1 of adjacency matrices of a sparse Erd˝os-Rényi random graph, the convergence in probability +can easily be lifted to almost sure convergence by a standard martingale argument as given in [8, Appendix 1]. +■ +In line with previous results on the rank of sparse random asymmetric matrices [10], Theorem 1.2 illustrates that +(within the specified framework) the rank formula (1.4) solely depends on d, but not on the field F or the choice of the +sequence (Jn)n≥1. +2 +Proof overview +On the following pages, we present an overview of the proof of Theorem 1.2. After fixing some notation, we +first reduce the uniform convergence in probability in (1.4) to an upper bound in probability and a lower bound in +expectation in Section 2.2. While the upper bound is based on the leaf-removal algorithm and the results of [3, 26], the +lower bound constitutes the main contribution of our article. To lower bound the expected rank of An,d/n, we transform +it to a “symmetrised” matrix and grow the modified matrix from εn to n step by step. An essential ingredient in the +quantification of the described one-step rank change are the powerful techniques developed in [10], which allow us +to focus on the positions of the nonzero entries in the target matrix rather than their precise values. Finally, the rank +formula follows by interpreting the sum of the lower bounds as the Riemann sum of an integral, which is analytically +tractable. +2.1 +Notation +This section can be used as a reference for recurring notation that is used throughout the article. +3 + +The rank of sparse symmetric matrices over arbitrary fields +Sets. +We write [ℓ] = {1, 2, . . . , ℓ} and denote the cardinality of a set B by |B|. For two sets B1 and B2, we denote +their symmetric difference as B1∆B2 and use ⊎i∈IBi to indicate the union over pairwise disjoint sets (Bi)i∈I. If B is +a set and ℓ ≤ |B|, we write +�B +ℓ +� +for the collection of ℓ-subsets of B. +Real numbers and fields. +For a, b ∈ R, we write a∨b = max {a, b} and a∧b = min {a, b}. F is reserved to denote +a generic field, and F∗ = F \ {0} its multiplicative group. +Vectors and matrices. +For A ∈ Fm×n, we denote its transpose by AT . For a vector b = (b1, b2, . . . , bn) ∈ F1×n, +we let supp(b) = supp(bT ) = {i ∈ [n]: bi ̸= 0}. We denote by en(i) the ith standard unit vector in F1×n. +For s = (s1, s2, . . . , sℓ) ∈ R1×ℓ, define ∥s∥∞ = supi∈[ℓ] |si| and ∥s∥k = (�ℓ +i=1 |si|k)1/k. +For A ∈ Fm×n, we denote +(i) the ith row of A by A(i, ) and the jth column of A by A(, j). +(ii) the +matrix +obtained +by +removing +rows +ℓ1, ℓ2, . . . , ℓs +and +columns +ℓ′ +1, ℓ′ +2, . . . , ℓ′ +t +from +A +by +A ⟨ℓ1, ℓ2, . . . , ℓs; ℓ′ +1, ℓ′ +2, . . . , ℓ′ +t⟩. +By a slight abuse of indexing, the ith row in the diminished matrix +A ⟨ℓ1, ℓ2, . . . , ℓs; ℓ′ +1, ℓ′ +2, . . . , ℓ′ +t⟩ refers to the row vector A(i, ) ⟨; ℓ′ +1, ℓ′ +2, . . . , ℓ′ +t⟩, i.e., the ith row of A (mi- +nus the entries corresponding to columns ℓ′ +1, ℓ′ +2, . . . , ℓ′ +t). We use an analogous convention for columns. +Functions. +For a function f : Ω → R, we denote by f + its positive and by f − its negative part, i.e. f +(x) = 0∨f(x) +and f −(x) = 0 ∨ (−f(x)) for x ∈ Ω. +Random variables. +For a finite set B, we write Unif(B) to denote a discrete uniform random variable on B, Bin (n, p) +to denote a binomial random variable with n trials and success probability p and Po (d) to denote a Poisson variable +with parameter d. +For two random variables X, Y taking values in (Ω, G), we denote the total variation distance between X and Y as +dTV(X, Y ) = sup +B∈G +|P (X ∈ B) − P (Y ∈ B)| . +Notions of convergence. +Throughout the article, the order in which limits are taken matters significantly. For families +of real numbers (an,P,N,JN )n,P,N∈Z+,JN∈SymN(F∗), we write +(i) an,P,N,JN = on(1) +⇐⇒ +For all P ≥ 1 : +limn→∞ supN≥n,JN∈SymN(F∗) |an,P,N,JN | = 0; +(ii) an,P,N,JN = on,P (1) ⇐⇒ +lim supP →∞ lim supn→∞ supN≥n,JN∈SymN(F∗) |an,P,N,JN | = 0. +Given a family of real numbers (cn,P,N,JN,t)n,P,N∈Z+,JN∈SymN(F∗),t∈[0,d], we say that +(i) cn,P,N,JN,t = on(1) uniformly in t ∈ [0, d] +⇐⇒ +supt∈[0,d] cn,P,N,JN,t = on(1); +(ii) cn,P,N,JN,t = on,P (1) uniformly in t ∈ [0, d] +⇐⇒ +supt∈[0,d] cn,P,N,JN,t = on,P (1). +For a family of uniformly bounded random variables (bn,P,N,JN,t)n,P,N∈Z+,JN∈SymN(F∗),t∈[0,d], we write +(i) bn,P,N,JN,t = ¯oP(1) +⇐⇒ +E |bn,P,N,JN,t| = on,P (1) uniformly in t ∈ [0, d]; +(ii) bn,P,N,JN,t ≥ ¯oP(1) +⇐⇒ +(bn,P,N,JN,t)− = ¯oP(1). +(iii) bn,P,N,JN,t ≤ ¯oP(1) +⇐⇒ +(bn,P,N,JN,t)+ = ¯oP(1). +For a family of events (Bn,P,N,JN,t)n,P,N∈Z+,JN∈SymN(F∗),t∈[0,d], we say that Bn,P,N,JN,t occurs w.h.p. +if +P (Bn,P,N,JN,t) = 1 + on,P (1) uniformly in t ∈ [0, d]. +We extend the above notions of convergence to families of numbers and events that only depend on subsets of +the parameters. For example, for a family of real numbers (cn,P )n,P ∈Z+, by treating it as constant on the unspecified +parameters, we write cn,P = on,P (1) whenever lim supP →∞ limn→∞ cn,P = 0. +2.2 +Deduction of Theorem 1.2 from suitable upper and lower bounds +Our main result, Theorem 1.2, is a statement about convergence in probability of the normalised rank sequence +rkF +� +An,d/n +� +/n that holds uniformly in (Jn)n≥1. In this section, we show how Theorem 1.2 readily follows from the +following upper bound in probability and the subsequent lower bound in expectation: +Theorem 2.1 (Upper bound in probability). Let d > 0 and F be any field. Then for any ε > 0, +lim +n→∞ P +� +sup +Jn∈Symn(F∗) +rkF +� +An,d/n +� +n +≤ min +α∈[0,1] Rd(α) + ε +� += 1. +(2.1) +4 + +The rank of sparse symmetric matrices over arbitrary fields +Theorem 2.2 (Lower bound in expectation). For any d > 0 and any field F, +lim inf +n→∞ +inf +Jn∈Symn(F∗) E +� +rkF +� +An,d/n +� +n +� +≥ min +α∈[0,1] Rd(α). +(2.2) +While Theorem 2.1 straightforwardly follows from the fact that the nullity of an adjacency matrix remains invariant +under “leaf-removal” (see [6]) and the results of [26]2, the derivation of Theorem 2.2 is the main contribution of our +work. The central steps towards (2.2) are laid out in the remainder of Section 2. With Theorems 2.1 and 2.2 in hand, we +are in the position to prove Theorem 1.2: +Proof of Theorem 1.2 subject to Theorems 2.1 and 2.2. Let +sn = sn(Jn) = rkF +� +An,d/n +� +n +− min +α∈[0,1] Rd(α). +Then |sn| ≤ 1 + +��minα∈[0,1] Rd(α) +��. By Theorem 2.1, for any ε > 0, +> 0 lim sup +n→∞ +sup +Jn∈Symn(F∗) +E +� +s+ +n +� +≤ lim sup +n→∞ E +� +sup +Jn∈Symn(F∗) +s+ +n +� +≤ ε + +� +1 + +���� min +α∈[0,1] Rd(α) +���� +� +lim sup +n→∞ P +� +sup +Jn∈Symn(F∗) +s+ +n ≥ ε +� += ε. +Since ε can be chosen arbitrarily small, we conclude that lim supn→∞ supJn∈Symn(F∗) E [s+ +n ] = 0. On the other hand, +by Theorem 2.2, lim infn→∞ infJn∈Symn(F∗) E [sn] ≥ 0. Since sn = s+ +n − s− +n , +lim sup +n→∞ +sup +Jn∈Symn(F∗) +E +� +s− +n +� +≤ lim sup +n→∞ +sup +Jn∈Symn(F∗) +E +� +s+ +n +� +− lim inf +n→∞ +inf +Jn∈Symn(F∗) E [sn] ≤ 0. +As a consequence, lim supn→∞ supJn∈Symn(F∗) E [|sn|] = 0. The uniform convergence in probability now follows +from Markov’s inequality. +We conclude that it remains to prove Theorem 2.2 and outline the main steps in the following subsections. +2.3 +The lower bound: Building the matrix +Instead of proving Theorem 2.2 for the sequence (An,d/n)n≥1 directly, we work with a “symmetrised” version +that possesses a suitable form of joint row and column exchangeability. To define the auxiliary matrices, fix a number +N ∈ N≥1 and let τ be a uniform permutation of [N]. For n ∈ [N], define the matrix T (N) +n,p ∈ Fn×n by setting +T (N) +n,p (i, j) = +� +� +� +1{q(τ(i), τ(j)) < p}JN(τ(i), τ(j)), +i < j; +1{q(τ(j), τ(i)) < p}JN(τ(j), τ(i)), +i > j; +0, +i = j. +(2.3) +For any N ∈ N≥1, this construction yields N matrices T (N) +1,p , T (N) +2,p , . . . , T (N) +N,p of growing dimension. Specifically, +we have T (N) +N,p (i, j) = AN,p(τ(i), τ(j)) and rkF +� +T (N) +N,p +� += rkF (AN,p), so that Theorem 2.2 would follow from the +lower bound +lim inf +n→∞ +inf +Jn∈Symn(F∗) E +� 1 +n rkF +� +T (n) +n,d/n +�� +≥ min +α∈[0,1] Rd(α). +However, for technical reasons that will become apparent later, we actually show the stronger statement +lim inf +n→∞ +inf +N≥n +inf +JN∈SymN(F∗) E +� 1 +n rkF +� +T (N) +n,d/n +�� +≥ min +α∈[0,1] Rd(α). +Correspondingly, in the following, we focus on the derivation of a lower bound on E[rkF(T (N) +n,d/n)]/n for N ≥ n. +Nonetheless, for a lighter notation, we omit the superscript N in the matrices below. The basic idea in this derivation is +2See Appendix B. +5 + +The rank of sparse symmetric matrices over arbitrary fields +rather simple: Fix a small number ε ∈ (0, 1) and trace the rank change when the matrix Tεn,d/n is grown to Tn,d/n +step by step. Then, by a telescoping sum, +1 +nE +� +rkF +� +Tn,d/n +�� +≥ 1 +n +n−1 +� +m=εn +� +E +� +rkF +� +Tm+1,d/n +�� +− E +� +rkF +� +Tm,d/n +��� +. +(2.4) +The last expression thus reduces the problem of lower bounding E[rkF +� +Tn,d/n +� +]/n to lower bounding +�n−1 +m=εn +� +E +� +rkF +� +Tm+1,d/n +�� +− E +� +rkF +� +Tm,d/n +��� +/n. +Since this bound is based on a comparison of the two matrices Tm+1,d/n and Tm,d/n whose sizes differ by one, +our approach might superficially resemble the Aizenman-Sims-Starr scheme from mathematical physics, which had +previously found its application in the study of the rank of random matrices in [10]. The Aizenman-Sims-Starr scheme, +whose basic idea is to compare a system of n variables to a system of n + 1 variables and to study the influence on the +(n + 1)st variable, has originally been developed to tackle the Sherrington-Kirkpatrick spin glass model [2]. However, +our approach cannot straightforwardly be interpreted as a cavity computation for the original matrix sequence, since +we do not (directly or indirectly) compare two matrices of the form An,d/n and An−1,d/(n−1). Instead, we compare +matrices Tm,d/n and Tm+1,d/n whose sizes differ by one, but who are of a purely auxiliary nature and do not represent +copies of the original matrix model. +2.4 +Taming linear relations +While a comparison of the rather similar matrices Tm+1,d/n and Tm,d/n might look innocuous at first glance, +obtaining good control over the ensuing rank change is not a simple task, since it requires detailed knowledge of the +intricate linear dependencies of the matrix Tm,d/n. Following and extending core ideas of [10], this section collects the +main tools that are necessary to deal with these relations and to accurately describe the change in rank from Tm,d/n to +Tm+1,d/n. +The following definition from [10] contains a collection of terminology that will turn out useful in the coming +considerations on linear dependencies. +Definition 2.3 (Linear relations: [10, Definition 2.1]). Let A ∈ Fm×n. +(i) A set ∅ ̸= I ⊆ [n] is a relation of A if there exists a row vector y ∈ F1×m such that ∅ ̸= supp(yA) ⊆ I. If +furthermore supp(yA) = I, then we call y a representation of I in A. +(ii) If I = {i} is a relation of A, then we call i a frozen variable in A. Let F(A) be the set of all frozen variables. +(iii) A relation I ⊆ [n] is a proper relation of A if I\F(A) is a relation of A. +(iv) For δ > 0, ℓ ≥ 2, we say that A is (δ, ℓ)-free if there are no more than δnℓ proper relations I ⊆ [n] of size +|I| = ℓ. +♦ +Remark 2.4 (Frozen variables). +(i) The terminology frozen variable refers to the role that the corresponding +coordinate plays in the kernel of A: Frozen variables are exactly those coordinates that are invariably 0 in all +vectors of kerF(A) (see [10, Fact 2.2]). +(ii) In Lemma 4.1 (also in [18, Lemma 4.7]), we will see yet another convenient characterization of frozen variables +in terms of column removal as follows: +i ∈ F(A) ⇐⇒ rkF (A) − rkF (A ⟨; i⟩) = 1. +(2.5) +■ +Let A ∈ Fm×n be any matrix and b ∈ F1×n be a non-zero row vector, and suppose that we want to attach b to A +and characterise the ensuing rank change. This is a simpler operation than what we actually need (attaching both a row +and a column), but still instructive. In terms of frozen variables and proper relations, we can say the following about the +rank increase of attaching b to A: If all variables of supp (b) are frozen, then surely b lies in the linear span of the rows +of A, since it can be linearly combined using the representations of its non-zero coordinates. On the other hand, if b is +contained in the linear span of the rows of A, then because of the existence of a linear combination, either all variables +of supp (b) are frozen or they form a proper relation of A. As a consequence, we have the following key implications: +supp (b) ⊆ F(A) +=⇒ +b is in the span of the rows of A +=⇒ +supp (b) ⊆ F(A) or supp (b) is a proper relation of A. +(2.6) +These implications are useful for our purposes since the concept of a relation only takes into account the locations +of non-zero entries, but not their entries. However, unfortunately, (2.6) does not come in form of an equivalence, since +supp (b) being a proper relation of A does not imply that b lies in the span of the rows of A. +6 + +The rank of sparse symmetric matrices over arbitrary fields +To remedy this issue, based on ideas from [10], we use a matrix perturbation that greatly reduces the overall +number of short proper linear relations in the resulting matrix, such that morally, an equivalence of the form “supp (b) ⊆ +F(A) ⇐⇒ b is in the span of the rows of A” holds. While the perturbation from [10] is based on the attachment of unit +rows, we will augment this definition by the attachment of unit columns to account for the symmetry of our matrices. +The basic idea is that the attachment of unit rows at the bottom of a given matrix A can eliminate short proper relations +in the augmented matrix, while the attachment of unit columns to the left of A can eliminate short proper relations in its +transpose. +The details of the perturbation are considerably more subtle. We split its definition into two main parts, since it +involves two stages of randomness. In the first definition, we present the basic row and column attachment matrices. +Their non-zero entries may be confined to fixed initial segments of the column set [n] and row set [m], respectively: +Definition 2.5 (Perturbation matrices). +(i) Let θr, n1, n2 ∈ N with n1 ≤ n2. The row-perturbation matrix +Θr[θr, n1|n2] ∈ {0, 1}θr×n2 with parameters θr, n1, n2 is defined by setting exactly one entry in each of its +θr rows equal to 1, where the choice of this entry is uniform among the first n1 out of its n2 columns. More +precisely, the unique 1-entry of row k ∈ [θr] is in column jk, where j1, . . . , jθr ∈ [n1] are i.i.d. uniformly +distributed random variables. +(ii) Let θc, m1, m2 ∈ N with m1 ≤ m2. The column-perturbation matrix Θc[m1|m2, θc] ∈ {0, 1}m2×θc with +parameters θc, m1, m2 is defined by setting exactly one entry in each of its θc columns equal to 1, where the +choice of this entry is uniform among the first m1 out of its m2 rows. More precisely, the unique 1-entry of +column k ∈ [θc] is in row ik, where i1, . . . , iθc ∈ [m1] are i.i.d. uniformly distributed random variables. +♦ +Figure 1: Schematic representation of the row-perturbation matrix Θr[θr, n1|n2]. +For A ∈ Fm×n and a row perturbation matrix Θr[θr, n1|n] with non-zero column-coordinates j1, . . . , jθr ∈ [n1] +of its θr rows, consider the perturbed matrix +A′ := +� +A +Θr[θr, n1|n] +� +. +Then in A′, the non-zero columns j1, . . . , jθr ∈ [n1] of Θr[θr, n1|n] are part of the set of frozen variables: Since js is +the index of the only non-zero entry in the (m + s)th row, the Boolean row vector em+θr(m + s) is a representation of +js in A′. In this sense, one can view the attachment of Θr[θr, n1|n] at the bottom of a matrix as explicitly freezing the +variables corresponding to non-zero columns. +On the other hand, appending Θc[m1|m, θc] to the right of A has quite a contrary and more subtle effect upon +the set of frozen variables: In a sense, additional columns have the same impact as row removals and therefore can +“unfreeze” coordinates (see Lemma 4.3 for a proof). The necessity of column perturbation matrices constitutes the main +difference to the previously employed perturbation from [10]. +Before we introduce a second level of randomness to the perturbation, in the next lemma, we construct a coupling +of the row-perturbation matrices Θr[θr, n1|n2] for all possible sizes θr × n2 and subsets of freezable coordinates +[n1] ⊆ [n2]. The benefit of this coupling is twofold. First, perturbation matrices of increasing size, but with fixed subset +of freezable coordinates, will be nested. Second, the probability that matrices of fixed dimension, but with different +subsets of freezable coordinates, disagree, can be bounded explicitly. This coupling ensures that with high probability, +we can apply the same perturbation to both Tm,d/n and Tm+1,d/n, and still get the desired properties: +Lemma 2.6 (Coupling of perturbation matrices). There is a coupling of the family {Θr[θr, n1|n2] : θr, n2 ≥ 1, n1 ∈ +[n2]} with the following properties: +(i) For any θr, n2 ≥ 1 and n1 ∈ [n2], Θr[θr, n1|n2 + 1] ⟨; n2 + 1⟩ = Θr[θr, n1|n2]. +(ii) For any θr, n2 ≥ 1 and n1 ∈ [n2], Θr[θr + 1, n1|n2] ⟨θr + 1;⟩ = Θr[θr, n1|n2]. +(iii) For any θr, n2 ≥ 1 and n0 ≤ n1 ≤ n2, P (Θr[θr, n0|n2] = Θr[θr, n1|n2]) = (n0/n1)θr. +Similarly, there is a coupling with analogous properties for the family {Θc[m1|m2, θc] : θc, m2 ≥ 1, m1 ∈ [m2]}. +7 + +n1 +Or[0r, ni|n2] = +n2The rank of sparse symmetric matrices over arbitrary fields +Remark 2.7. From now on, we assume that the perturbation families {Θr[θr, n1|n2] : θr, n2 ≥ 1, n1 ∈ [n2]} and +{Θc[m1|m2, θc] : θc, m2 ≥ 1, m1 ∈ [m2]} are coupled as in Lemma 2.6 and independent of each other. +■ +Lemma 2.6 is proved in Section 3.1. Based on the ensembles {Θr[θr, n1|n2] : θr, n2 ≥ 1, n1 ∈ [n2]} and +{Θc[m1|m2, θc] : θc, m2 ≥ 1, m1 ∈ [m2]} from Lemma 2.6, we finally introduce the central perturbation of this +article: +Definition 2.8 (Canonical perturbation). For A ∈ Fm×n and θ = (θr, θc) ∈ N2, we write +A[θ] = +� +A +Θc[m|m, θc] +Θr[θr, n|n] +0θr×θc +� +. +For the canonical choice θ = (θr, θc) ∼ Unif([P]2), where P ∈ N is fixed and θ is independent of the couplings +{Θr[θr, n1|n2] : θr, n2 ≥ 1, n1 ∈ [n2]} and {Θc[m1|m2, θc] : θc, m2 ≥ 1, m1 ∈ [m2]}, we simply write A[θ]. +♦ +Remark 2.9. In the rest of this paper, θ always denotes a random vector chosen uniformly at random from [P]2. It is +important to keep in mind that the random vector θ is always understood to depend on the parameter P, even though +this is omitted from the notation (in line with the notation in [10]). +■ +As advertised earlier, perturbation typically greatly reduces the number of short proper relations. The next +proposition shows that, for any fixed L ∈ N≥2, A[θ] and A[θ]T are w.h.p. (δ, ℓ)-free for all 2 ≤ ℓ ≤ L (observe that +even if A is symmetric, the perturbed matrix A[θ] generally is not). This makes the matrices A[θ] and A[θ]T much +more convenient to study in comparison to A: +Proposition 2.10 (Perturbation eliminates most short proper relations). Fix δ > 0, L ∈ N≥2 and s ∈ Z. Then +sup +A∈F(n+s)×n P +� +A[θ] or A[θ]T is not (δ, ℓ)-free for some 2 ≤ ℓ ≤ L +� += on,P (1). +(2.7) +The proof of Proposition 2.10 is given in Section 3.2. Proposition 2.10 is the symmetric version of [10, Proposition +2.3]. It is remarkable in the sense that it shows that the simple perturbation of attaching a bounded number of unit rows +and columns eliminates a large proportion of short proper relations both column- and row-wise. +Rather than lower bounding E +� +rkF +� +Tn,d/n +�� +/n as indicated in (2.4), in the next subsections, we will outline how +to lower bound the expected normalised rank of the perturbed matrix Tn,d/n[θ]. This also gives a lower bound for +E +� +rkF +� +Tn,d/n +�� +/n, since if we add a row or a column to a matrix, its rank stays unchanged or increases by 1, and +therefore, +rk(A) ≤ rk(A[θ]) ≤ rk(A) + θr + θc. +(2.8) +Thus, as long as θr, θc are bounded random variables, all results on the asymptotic rank of the perturbed matrices +transfer to the unperturbed ones. +2.5 +Rank increase for the perturbed matrix and obstructions due to symmetry +At this point, our strategy rests on lower bounding the differences +E +� +rkF +� +Tm+1,d/n[θ] +�� +− E +� +rkF +� +Tm,d/n[θ] +�� +for m ≥ εn. Since m grows linearly in n, a reparametrisation yields the more convenient expression +E +� +rkF +� +Tn+1,t/n[θ] +�� +− E +� +rkF +� +Tn,t/n[θ] +�� +, +(2.9) +where now t ∈ [εd, d]. While all the perturbed matrices use the same vector θ that fixes the dimensions of the perturba- +tion, the positions of the non-zero entries in the perturbation part may change from matrix to matrix. Conveniently, this +does not happen frequently, since thanks to the coupling from Lemma 2.6, with high probability, +Tn+1,t/n[θ] ⟨n + 1; n + 1⟩ = Tn,t/n[θ]. +(2.10) +On the event (2.10), observation (2.5) on column removal and frozen variables implies that +rkF +� +Tn+1,t/n[θ] +� +− rkF +� +Tn,t/n[θ] +� +(2.11) +=1{n + 1 ∈ F(Tn+1,t/n[θ]T )} + 1{n + 1 ∈ F(Tn+1,t/n[θ] ⟨n + 1;⟩)}, +where we first remove the (n + 1)st row and then the (n + 1)st column to go from Tn+1,t/n[θ] to Tn,t/n[θ]. In the +analysis of (2.11), both the benefits of working with the matrix Tn,t/n and then its perturbation Tn,t/n[θ] become +apparent. We next explain how these ideas can be used effectively in the evaluation of the r.h.s. of (2.11). +8 + +The rank of sparse symmetric matrices over arbitrary fields +For p ∈ (0, 1), let αn,p and αT +n,p be the proportions of frozen variables i ∈ [n] in Tn,p[θ] and Tn,p[θ]T , +respectively. By the distributional invariance of the matrix Tn+1,t/n[θ] under joint row- and column-relabelling3, +conditionally on αT +n+1,t/n, the probability that n + 1 is frozen in Tn+1,t/n[θ]T is simply given by +P(n + 1 ∈ F(Tn+1,t/n[θ]T ) | αT +n+1,t/n) = αT +n+1,t/n. +This provides a simple expression for the first indicator in r.h.s. of (2.11). We next consider the second indicator +that n + 1 is frozen in Tn+1,t/n[θ] ⟨n + 1;⟩. Again by observation (2.5), this event is the same as the event the (n + 1)st +column of Tn+1,t/n[θ] ⟨n + 1;⟩ lies in the span of the columns of Tn,t/n[θ]. Considering the transposed matrix, this +translates to the event that the (n + 1)st row of Tn+1,t/n[θ] ⟨n + 1;⟩T lies in the span of the rows of Tn,t/n[θ]T . By +this chain of equivalences, we have turned the original event into one that we can handle very well thanks to (2.6) +and the perturbation: Since the perturbation effectively excludes the possibility that the non-zero components of the +(n + 1)st row of Tn+1,t/n[θ] ⟨n + 1;⟩T form a proper relation, the event in question roughly corresponds to the event +that all the non-zero components of the (n + 1)st row of Tn+1,t/n[θ] ⟨n + 1;⟩T are frozen in Tn,t/n[θ]T . +For a lighter notation, we abbreviate b := Tn+1,t/n[θ](n+1, ), so that b is the (n+1)st row of Tn+1,t/n[θ]. Since +the positions of the non-zero entries of b are chosen uniformly at random and independently of Tn,t/n[θ], conditionally +on αn,t/n and |supp (b) |, the probability that all the non-zero components of the (n+1)st row of Tn+1,t/n[θ] ⟨n + 1;⟩T +are frozen in Tn,t/n[θ]T should be close to 1 − α|supp(b)| +n,t/n +. On the other hand, |supp (b) | asymptotically follows a +Po (t) distribution, so that after taking expectation with respect to |supp (b) |, we arrive at the approximation +P(n + 1 ∈ F(Tn+1,t/n[θ] ⟨n + 1;⟩)|αT +n,t/n) ≈ 1 − φt(αT +n,t/n). +In the above, recall that φt is the probability generating function of a Po (t) variable. Thus, on a heuristic level, +E +� +rkF +� +Tn+1,t/n[θ] +� +|αT +n+1,t/n +� +− E +� +rkF +� +Tn,t/n[θ] +� +|αT +n,t/n +� +≈ αT +n+1,t/n + 1 − φt +� +αT +n,t/n +� +. +(2.12) +This expression has two flaws: First of all, rather than depending on one random variable, it depends on both αT +n,t/n +and αT +n+1,t/n. Secondly, even though we trace the rank change in Tn,t/n[θ], the left hand side of (2.12) comes in +terms of the proportions in the transposed matrices. Fortunately, in Section 4, we will show that in expectation, the +difference αn+1,t/n − αn,t/n is small, which allows us to reduce the r.h.s. of (2.12) to one parameter. On the other +hand, αT +n+1,t/n and αn+1,t/n are identically distributed, so the second problem is solved as well. With ht : [0, 1] → R, +ht (α) := α + 1 − φt (α) , +(2.13) +we have thus heuristically derived the following result: +Proposition 2.11 (The rank increase). For any d > 0, +E +� +rkF +� +Tn+1,t/n[θ] +� +− rkF +� +Tn,t/n[θ] +�� += E +� +ht +� +αn,t/n +�� ++ on,P (1), uniformly in t ∈ [0, d]. +(2.14) +We give a full proof of (2.14) in Proposition 2.11 in Section 6. Proposition 2.11 lays the basis for the targeted +lower bound on E[rkF(Tn,d/n)]/n. In view of the rank formula (1.3), it might be tempting to just take the minimum +over all α ∈ [0, 1] on the r.h.s. of (2.14). Unfortunately, this is not sufficient to arrive at (1.3), and we need means to +restrict the potential values of αn,t/n. +It thus “only” remains to get our hands on αn,t/n. With (2.14) in mind, it is natural to suspect that αn,t/n +converges and to try to calculate its limit. However, the situation is not that simple, and based on results for a similar +class of asymmetric sparse matrices [9], it is not reasonable to expect αn,t/n to stabilise. Instead, our strategy will be to +derive an asymptotic fixed point equation for αn,t/n. The ensuing characterisation will finally allow us to make the +connection to the rank formula (1.3). +To motivate the desired equation for αn,t/n, we again take a look at the evaluation of the second indicator in the +derivation of (2.12) above: +P(n + 1 ∈ F(Tn+1,t/n[θ] ⟨n + 1;⟩)|αn,t/n, αT +n,t/n) ≈ 1 − φt(αT +n,t/n). +Since the matrix Tn+1,t/n[θ] ⟨n + 1;⟩ is rather similar to Tn,t/n[θ], one might make the bold assumption that +P(n + 1 ∈ F(Tn+1,t/n[θ] ⟨n + 1;⟩)|αn,t/n, αT +n,t/n) ≈ P(n + 1 ∈ F(Tn+1,t/n[θ])|αn,t/n, αT +n,t/n). +3For the precise arguments, see Section 4.3.1. +9 + +The rank of sparse symmetric matrices over arbitrary fields +On the other hand, +P(n + 1 ∈ F(Tn+1,t/n[θ])|αn+1,t/n, αT +n+1,t/n) ≈ αn+1,t/n. +Based on the previous assumption, we can again argue that αn+1,t/n ≈ αn,t/n and use a handy proposition on the +comparison of conditional expectations4 to conclude that +1 − φt(αT +n,t/n) ≈ αn,t/n. +Along the same lines, we can conclude that 1 − φt(αn,t/n) ≈ αT +n,t/n. Combining the two approximations, we +heuristically deduce that αn,t/n should approximately satisfy the equation +αn,t/n ≈ 1 − φt(1 − φt(αn,t/n)). +(2.15) +While (2.15) is surely based on a plausible line of arguments, crucially, the very first step in its derivation might have +been too bold. Indeed, this approximation was in essence based on the assumption that w.h.p., for any fixed i ∈ [n], +i ̸∈ F(Tn,t/n[θ] ⟨i;⟩)∆F(Tn,t/n[θ]). +(2.16) +Does (2.16) hold w.h.p.? We believe so5. Sadly, we cannot prove it, and therefore (2.15) is just a conjecture at this +point. Nevertheless, the heuristic approximation illustrates the pivotal role of events of the form (2.16) for symmetric +matrices, which motivates a more fine-grained description of frozen variables as introduced in the following section. +This description will finally allow us to find another, more indirect route towards (2.15), while still, the belief in (2.16) +lies at the heart of the argument. +2.6 +Frozen variables revisited +As discussed in Section 2.5, we cannot prove that w.h.p., removal of row i from Tn,t/n[θ] does not unfreeze i. To +keep track of those “problematic” variables where removal of row i unfreezes variable i, we now give a name to them: +Definition 2.12 (Frailly, firmly and completely frozen variables). For any matrix A ∈ Fm×n and i ∈ [m ∧ n], we say +that +(i) i is frailly frozen in A if i ∈ F(A)\F (A ⟨i;⟩)6; +(ii) i is firmly frozen in A if i ∈ F (A ⟨i;⟩); +(iii) i is completely frozen in A if i is firmly frozen in both A and AT . +♦ +In addition, in Section 4.1 we show that variables which are frailly frozen in A are also frailly frozen in the +transpose AT . So indeed, we can partition the set of coordinates into five disjoint sets as follows: +Definition 2.13. (Typecasting of variables) For any matrix A ∈ Fm×n, we partition the set [m ∧ n] into +(i) the set X(A) of frailly frozen variables; +(ii) the set Y(A) of completely frozen variables; +(iii) the set Z(A) of variables that are neither frozen in A or AT ; +(iv) the set U(A) of variables that are not frozen in A and firmly frozen in AT ; +(v) the set V(A) of variables that are firmly frozen in A and not frozen in AT . +For each i ∈ [m ∧ n], we refer to the category it belongs to with respect to the above partition as its type. +♦ +This distinction between different types of frozen variables is a chief ingredient in our calculation of the lower +bound, and the main difference with respect to the preceding works [9, 10]. Notably, it allows us to extend core ideas of +these articles to symmetric matrices. For example, with the terminology of Definition 2.13, we can now express the +rank increase of interest alternatively as +rkF +� +Tn+1,t/n[θ] +� +− rkF +� +Tn,t/n[θ] +� += 1{n + 1 ∈ X(Tn+1,t/n[θ])} + 2 · 1{n + 1 ∈ Y(Tn+1,t/n[θ])} ++ 1{n + 1 ∈ U(Tn+1,t/n[θ])} + 1{n + 1 ∈ V(Tn+1,t/n[θ])} +(for a proof of this identity, see Lemma 4.7). +Returning to the discussion at the end of Section 2.5, the typecasting allows us to the derive fixed point equations +not for αn,t/n, but for some of the proportions of the finer types. Thereby, we gain a better understanding of the +proportion of frailly frozen variables and of αn,t/n. And, what is more, these fixed point equations provide enough +information to derive the desired lower bound on ht(αn,t/n) as given below in Proposition 2.14, and therefore to bypass +(2.16), which is precisely what we need. The derivation of the fixed point equations is the content of Section 5. +4See Proposition C.1 in the appendix. +5Our belief is underpinned by the fact that removal of row i has the same effect as attachment of a unit column (see Lemma 4.3), +which is akin to a pinning operation. +6This is equivalent to what we need, see Corollary 4.4. +10 + +The rank of sparse symmetric matrices over arbitrary fields +2.7 +The heuristic fixed point equation and its connection to Rd(α) +Let us return to the heuristic fixed point equation (2.15), which suggests that only zeroes of the function Gd : +[0, 1] �→ R, +Gd(α) := α + φd (1 − φd (α)) − 1, +(2.17) +constitute viable candidates for αn,t/n. And indeed, for any d ≥ 0, Gd has at least one zero: If α0(d) ∈ [0, 1] is such +that α0(d) = 1 − φd (α0(d))7, then +Gd(α0(d)) = α0(d) + φd (1 − φd (α0(d))) − 1 = −φd(α) + φd(α) = 0. +(2.18) +Unfortunately, for some d ≥ 0, Gd has more zeroes: Let α⋆(d) and α⋆(d) denote the smallest and largest zeroes of +Gd(α) in [0, 1], respectively. The existence of α⋆(d) and α⋆(d) is guaranteed by (2.18). A detailed analysis of the +function Gd, its zeroes and relation to the function 1 − Rd is carried out in [9], where the asymmetric counterpart of +An,p with all non-zero entries being identical to 1 was studied. In [9], the authors show that Gd has at most the three +zeroes α⋆(d) ≤ α0(d) ≤ α⋆(d). +From the analysis of the finer types as described in Section 2.6, it will become apparent that in the limit, only the +two zeroes α⋆(d) and α⋆(d) correspond to possible values of αn,d/n. For the asymmetric case, where no perturbation +is necessary, the connection between Gd and the proportion of frozen variables has been studied in [9]. While we +cannot derive a picture as detailed as in [9], we can show that ht(αn,t/n) is no less than ht evaluated at one of the +zeroes, which provides a sufficient substitute for the exact asymptotic characterisation of αn,t/n: +Proposition 2.14 (Lower bound on the rank increase). For any d > 0, +ht +� +αn,t/n +� +≥ ht (α⋆(t)) + ¯oP(1). +(2.19) +Recall that the principal aim of (2.15) was to establish a connection to the rank formula (1.3), which comes in +terms of an optimization problem over [0, 1]. The function Rd attains its minimum on [0, 1] either for α ∈ {0, 1} or for +α ∈ (0, 1) such that +R′ +d(α) = d2φd (α) (α + φd (1 − φd (α)) − 1) = 0. +(2.20) +The little calculation of (2.20) shows that R′ +d(α) = 0 if and only if Gd(α) = 0. Indeed, α⋆(d) and α⋆(d) are the two +only minimizers of Rd on [0, 1]: +Rd (α⋆(d)) = Rd (α⋆(d)) = min +α∈[0,1] Rd(α). +(2.21) +Thus doubtlessly, the lower bound (2.19) establishes a connection to the minimizers of Rd. Given Propositions 2.11 +and 2.14, it is now a matter of analysis to prove (2.2), which we complete next. +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +d +α⋆(d) +α0(d) +α⋆(d) +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +d +minα∈[0,1] Rd(α) +Figure 2: Left: Plot of α⋆(d), α0(d) and α⋆(d), which are distinct for d > e. Right: Plot of the function d �→ +minα∈[0,1] Rd(α). +7The existence and uniqueness of α0(d) are straightforward to check, see Lemma A.1 in Appendix A. +11 + +The rank of sparse symmetric matrices over arbitrary fields +2.8 +Lower bound on the expected rank: Proof of Theorem 2.2 subject to Propositions 2.11 and 2.14 +An application of Propositions 2.11 and 2.14 now gives the following lower bound for 1 +nE +� +rkF +� +Tn,d/n[θ] +�� +: +1 +nE +� +rkF +� +Tn,d/n[θ] +�� +≥ 1 +n +n−1 +� +m=εn +� +E +� +rkF +� +Tm+1,(dm/n)/m[θ] +�� +− E +� +rkF +� +Tm,(dm/n)/m[θ] +��� +(2.22) += 1 +n +n−1 +� +m=εn +E +� +hdm/n +� +αm,d/n +�� ++ on,P (1) ≥ 1 +n +n−1 +� +m=εn +hdm/n (α⋆(dm/n)) + on,P (1). +The sum 1 +n +�n−1 +m=εn hmd/n (α⋆(md/n)) can be treated as a Riemann sum, i.e., +1 +nE +� +rkF +� +Tn,d/n[θ] +�� +≥ +� 1 +ε +hds (α∗ (ds)) ds + on,P (1) = 1 +d +� d +εd +ht (α⋆(t)) dt + on,P (1). +Taking the appropriate limits on both sides gives +lim inf +P →∞ lim inf +n→∞ +inf +N≥n, +JN∈SymN(F∗) +1 +nE +� +rkF +� +Tn,d/n[θ] +�� +≥ 1 +d +� d +εd +ht (α⋆(t)) dt +and since we can choose ε arbitrarily small, we conclude that +lim inf +P →∞ lim inf +n→∞ +inf +N≥n, +JN∈SymN(F∗) +1 +nE +� +rkF +� +Tn,d/n[θ] +�� +≥ 1 +d +� d +0 +ht (α⋆(t)) dt. +Then indeed, as we prove in Section 6.3, the derived integral expression coincides with the desired rank formula: +Lemma 2.15 (Integral evaluation). For any d ≥ 0, +� d +0 +ht (α⋆(t)) dt = d · Rd (α⋆(d)) . +Now, the combination of (2.21) and Lemma 2.15 yields that +1 +d +� d +0 +ht (α⋆(t)) dt = Rd (α⋆(d)) = min +α∈[0,1] Rd(α) +and therefore by (2.8) that +lim inf +n→∞ +inf +N≥n, +JN∈SymN(F∗) +1 +nE +� +rkF +� +Tn,d/n +�� +≥ lim inf +P →∞ lim inf +n→∞ +inf +N≥n, +JN∈SymN(F∗) +1 +nE +� +rkF +� +Tn,d/n[θ] +�� +≥ min +α∈[0,1] Rd(α). +(2.23) +By definition of Tn,p = T (N) +n,p , we have rkF +� +T (N) +N,p +� += rkF (AN,p) and consequently +inf +Jn∈Symn(F∗) +1 +nE +� +rkF +� +An,d/n +�� += +inf +N=n, +JN∈SymN(F∗) +1 +nE +� +rkF +� +T (N) +n,d/n +�� +≥ +inf +N≥n, +JN∈SymN(F∗) +1 +nE +� +rkF +� +T (N) +n,d/n +�� +. (2.24) +Theorem 2.2 now follows from the combination of eqs. (2.23) and (2.24). +2.9 +Discussion +The understanding of the ensemble of adjacency matrices of Erd˝os-Rényi random graphs has seen major advances +during the last two decades, in particular with respect to its real rank and spectral properties. Prominently, ln(n)/n is +a threshold for the singularity of these matrices [5, 15]. More generally, in the regime where pn ∈ [c ln(n)/n, 1/2] +for c > 0 and for more general real matrix entries as considered in the current article, Costello and Vu [16] show +that with high probability, the nullity of An,pn is exactly equal to the number of isolated vertices in the underlying +Erd˝os-Rényi random graph. In the same spirit, DeMichele, Moreira and Glasgow [18] show that for pn = ω(1/n), +with high probability, the nullity of An,pn coincides with the number of isolated vertices in the graph that arises from +Gn,d/n after an application of the Karp-Sipser algorithm described in Section 1.1. In an associated random matrix +process where edges are revealed one after the other, Addario-Berry and Eslava [1] derive a hitting time theorem in the +12 + +The rank of sparse symmetric matrices over arbitrary fields +sense that with high probability, the matrix becomes singular at the exact moment when there are no zero rows and +columns left. +In the challenging sparse regime where pn = d/n for fixed d > 0, much less is known. Notably, there is the +asymptotic rational rank formula (1.3) for An,d/n by Bordenave, Lelarge and Salez [8]. Recently, building on the +machinery of [8], Ferber et al. [23] have shown that the k-core for k ≥ 3 is non-singular with high probability, thereby +resolving an open conjecture of Vu from 2014. +This work has been inspired by recent advances on the rank of random matrices in the context of random constraint +satisfaction problems, in particular work on the k-XORSAT problem [4, 12, 19, 20, 28] and a model inspired by random +code ensembles [10]. In this context, it is natural to consider the matrices not only over the reals, but as binary matrices +or more generally, matrices over finite fields. +Correspondingly, this article crucially builds on the methodology developed in [4, 10]. However, because of the +symmetry of our model, virtually all core ides have to be developed differently in comparison to [4, 10]. First of all, we +modify the perturbation according to Definition 2.8. While the basic idea of a perturbation as in Definition 2.8 in the +context of random graphical models goes back to information theory [27], it has since been successfully applied to the +study of random inference problems and random factor graphs [11, 13]. The basis for an application to asymmetric +sparse random matrices, in combination with the conceptualisation of linear relations, has been laid out in [10]. In +comparison to this previous application, the perturbation in Definition 2.8 is of a slightly different flavour, since +it cannot be straightforwardly interpreted as the addition of unary factor nodes in the underlying graphical model. +In [10] and the earlier version [4], as well as in results on random factor graphs, the perturbation has proven to be +particularly useful when combined with the Aizenman-Sims-Starr scheme from mathematical physics, which brings us +to our next modification: Instead of combining the pinning operation with the Aizenman-Sims-Starr scheme, we apply +the telescoping argument (2.4) and therefore compare the matrices Tm+1,d/n and Tm,d/n rather than Tn+1,d/(n+1) +and Tn,d/n. This is due to the fact that an application of the Aizenman-Sims-Starr scheme as in [10] would require +knowledge about the event (2.16) that we do not have, and comes at the price of pursuing a different route to characterise +αn,t/n. We therefore introduce frailly frozen variables, which are probably the most essential difference between this +article and the previous work on asymmetric matrices [4, 9, 10]. +Finally, we believe that the methods developed in this article will generalise to broader symmetric matrix structures. +It would also be interesting to see whether the fraction of frozen variables in the unperturbed matrix An,d/n satisfies +an anti-concentration result as its asymmetric counterpart [9], or whether the two models behave differently. In +hindsight, the rank formula Theorem 1.2 gives us some information about the perturbed matrix and (2.16). The proof of +Proposition 2.14 shows that there are essentially two cases: In the first case, the proportion of frailly frozen variables +xn,t/n is approximately zero and the proportion of frozen variables αn,t/n is approximately α⋆(d) or α⋆(d). In the +second case, the proportion of frailly frozen variables xn,t/n is approximately α⋆(d) − α⋆(d) and the proportion of +frozen variables αn,t/n is approximately α⋆(d). From simulations, it seems likely that only the first case corresponds +to the actual asymptotic behaviour of the perturbed matrices under consideration, but we cannot exclude the second +case at present. +3 +Matrix perturbations +In this short section, we prove the two most important properties of the matrix perturbation introduced in +Definition 2.8: In Section 3.1, we construct the coupling from Lemma 2.6, which ensures that w.h.p., for any two +large square matrices that differ by one in their size, their canonical perturbation is based on the same row- and +column-perturbation matrices (compare (2.10)). In Section 3.2, we then prove Proposition 2.10 on the joint deletion of +short proper relations in both the perturbed A and its transpose. +3.1 +Coupling of perturbation matrices: Proof of Lemma 2.6 +To couple the matrices Θr[θr, n1|n2], we couple the locations of their non-zero entries row by row. For a given row +k, the basic idea is to construct a coupling (jk,n1)n1≥1 of uniformly distributed random variables jk,n1 ∼ Unif([n1]) on +increasing integer intervals, such that for any two random variables, P (jk,n0 ̸= jk,n1) = dTV (Unif([n0]), Unif([n1])). +For the overall coupling, we then take the product distribution over the rows. More precisely, let (uk,ℓ)k,ℓ≥1 be an array +of independent random variables such that for all k, ℓ ≥ 1, uk,ℓ is uniformly distributed on [ℓ]. For any n1 ∈ N, set +jk,n1 = max{ℓ ∈ [n1] : uk,ℓ = ℓ}. +(3.1) +Since uk,1 = 1, the set above is nonempty, and it is straightforward to verify that jk,n1 ∼ Unif([n1]). +For any θr, n2 ∈ N, n1 ∈ [n2], let Θr[θr, n1|n2] ∈ Fθr×n2 be the matrix where row k ∈ [θr] has its unique +non-zero entry in column jk,n1. Since the definition of jk,n1 only depends on n1, but not on θr or n2, this coupling +satisfies properties (i) and (ii). +13 + +The rank of sparse symmetric matrices over arbitrary fields +Consider now n0 ≤ n1 ≤ n2. Then Θr[θr, n0|n2] = Θr[θr, n1|n2] if and only if jk,n0 = jk,n1 for all k ∈ [θr], +or equivalently uk,n0+1 < n0 + 1, . . . , uk,n1 < n1. Therefore, +P (Θr[θr, n0|n2] = Θr[θr, n1|n2]) = +n1 +� +k=n0+1 +�k − 1 +k +�θr += +�n0 +n1 +�θr +, +so that the coupling satisfies (iii). +The coupling of {Θc[m1|m2, θc] : θc, m2 ≥ 1, m1 ∈ [m2]} can be constructed along the same lines. +3.2 +Perturbation eliminates most short proper relations: Proof of Proposition 2.10 +Recall the definition of the canonical perturbation from Definition 2.8. In this section, we prove Proposition 2.10, +which ensures that for any δ > 0 and ℓ ∈ N≥2, the canonical perturbation of any (almost) square matrix A, as well +as its transpose, are (δ, ℓ)-free with probability arbitrarily close to one, provided that the matrix dimension and the +perturbation parameter P are chosen large enough. +The main ingredient in the proof of Proposition 2.10 is the following lemma: +Lemma 3.1 ([10, Proposition 2.3]). Let δ > 0 and ℓ ∈ N≥2. Then there exists P ′ = P ′(δ, ℓ) ∈ N such that for any +P ≥ P ′ the following holds: For any matrix A ∈ Fm×n +P +�� +A +Θr[θr, n|n] +� +is (δ, ℓ)-free +� +≥ 1 − δ, +(3.2) +provided that θr ∼ Unif([P]) and is independent of the coupling {Θr[θr, n1|n2] : θr, n2 ≥ 1, n1 ∈ [n2]}. +Remark 3.2. Lemma 3.1 is a minor adaptation of [10, Proposition 2.3]. While the exact wording is for P = P ′(δ, ℓ) +rather than all P ≥ P ′(δ, ℓ), its proof shows that all choices of P > 4ℓ3/δ4 imply (3.2). +■ +Before we prove Proposition 2.10, we observe the following simple consequence of Lemma 3.1: +Corollary 3.3. Let δ > 0 and L ∈ N≥2. Then there exists P ′ = P ′(δ, L) ∈ N such that for any P ≥ P ′ the following +holds: For any matrix A ∈ Fm×n and n1 ∈ [n] +P +�� +A +Θr[θr, n1|n] +� +is (δ, ℓ)-free for 2 ≤ ℓ ≤ L +� +≥ +�n1 +n +�P +− δ, +provided that θr ∼ Unif([P]) and is independent of the coupling {Θr[θr, n1|n2] : θr, n2 ≥ 1, n1 ∈ [n2]}. +Proof. Fix δ > 0 and L ∈ N≥2. For any ℓ ∈ {2, . . . , L}, Lemma 3.1 guarantees the existence of Pℓ = Pℓ(δ/L, ℓ) ∈ N +such that for any P ≥ Pℓ and θr ∼ Unif([P]), +P +�� +A +Θr[θr, n|n] +� +is (δ, ℓ)-free +� +≥ P +�� +A +Θr[θr, n|n] +� +is (δ/L, ℓ)-free +� +≥ 1 − δ/L. +(3.3) +Let P ′ = max2≤ℓ≤L Pℓ. Then for any P ≥ P ′ and θr ∼ Unif([P]), by (3.3) and a union bound, +P +�� +A +Θr[θr, n|n] +� +is (δ, ℓ)-free for 2 ≤ ℓ ≤ L +� +≥ 1 − +L +� +ℓ=2 +P +�� +A +Θr[θr, n|n] +� +is not (δ, ℓ)-free +� +≥ 1 − δ. +(3.4) +By Lemma 2.6 (iii), P (Θr[θr, n|n] = Θr[θr, n1|n]) = E +� +(n1/n)θr� +≥ (n1/n)P . Therefore, +P +�� +A +Θr[θr, n|n] +� +(δ, ℓ)-free for 2 ≤ ℓ ≤ L +� +≤ P +�� +A +Θr[θr, n1|n] +� +(δ, ℓ)-free for 2 ≤ ℓ ≤ L +� ++ 1 − +�n1 +n +�P +. +(3.5) +Combining (3.4) and (3.5) yields the claim. +Proof of Proposition 2.10. Fix δ > 0, L ∈ N≥2 and s ∈ Z. For n, P ∈ N, let θ = (θr, θc) ∼ Unif([P]2) and +A ∈ F(n+s)×n. With the coupling from Lemma 2.6 and A′ := (A +Θc[n + s|n + s, θc]), +A[θ] = +� +A +Θc[n + s|n + s, θc] +Θr[θr, n|n] +0θr×θc +� += +� +A′ +Θr[θr, n|n + θc] +� +. +(3.6) +Conditionally on A′ and θc, because of independence of the row and column perturbations, Θr[θr, n|n + θc] is +distributed as the perturbation in Corollary 3.3 with the ensuing choice of n1 and n. Thus, for any a > 1, if P(δ/a, L) +is chosen large enough, conditioning on A′ and θc in (3.6) yields that for P ≥ P(δ/a, L), +P (A[θ] is (δ, ℓ)-free for 2 ≤ ℓ ≤ L) ≥ P (A[θ] is (δ/a, ℓ)-free for 2 ≤ ℓ ≤ L) ≥ +� +n +n + P +�P +− δ/a. +(3.7) +14 + +The rank of sparse symmetric matrices over arbitrary fields +By an analogous argument, also for P ≥ P(δ/a, L), +P +� +A[θ]T is (δ, ℓ)-free for 2 ≤ ℓ ≤ L +� +≥ +� +n + s +n + s + P +�P +− δ/a. +Since P(B1 ∩ B2) ≥ P(B1) + P(B2) − 1 for any two events B1, B2, we conclude that +P +� +Both A[θ] and A[θ]T are (δ, ℓ)-free for 2 ≤ ℓ ≤ L +� +≥ +� +n + s +n + s + P +�P ++ +� +n +n + P +�P +− 2δ/a − 1. +In particular, +lim sup +P →∞ +lim sup +n→∞ +sup +A∈F(n+s)×n P +� +A[θ] or A[θ]T is not (δ, ℓ)-free for some 2 ≤ ℓ ≤ L +� +≤ 2δ/a. +Since this is upper bound holds for any a > 1, (2.7) follows. +Remark 3.4. It is natural to wonder whether there is a possibility to perturb a symmetric matrix A such that the perturbed +matrix A[θ] is symmetric as well and Proposition 2.10 holds. However, simply choosing Θc[n|n, θc] = Θr[θr, n|n]T +does not have the desired effect: For (3.7) to hold, it is crucial that both the number of rows as well as the columns +of the non-zero indices of Θr[θr, n|n + θc] are chosen uniformly given A′ = (A +Θc[n|n, θc]). Thus, the above +perturbation technique necessarily destroys the matrix symmetry. +■ +4 +Frozen variables: General properties & stability +The principle aim of this section is to derive general properties of the various types of frozen variables as well as +to prove stability of the proportions of types in the transition from Tn,t/n[θ] to Tn+1,t/n[θ]. In this sense, our main +result of this section, Proposition 4.11, asserts that the proportions of the various types remain nearly unchanged when +we grow the matrix from n to n + 1. +4.1 +How the type of a variable encodes rank change under row- and column removal +We first present basic deterministic implications of the type of a variable that are used throughout the article, and +that indicate the significance of the types of Definition 2.13. More specifically, we are ultimately interested in the rank +decrease upon simultaneous removal of row i and column i from a given matrix A ∈ Fm×n. In this section, we prove +that the type of i according to Definition 2.13 completely determines the ensuing rank change. The starting point is the +following lemma on frozen variables: Living up to their name, in Remark 2.4, frozen variables were characterised as +coordinates that take the value zero in any kernel vector. The following lemma shows how the rank of any given matrix +changes, if a column that corresponds to a frozen variable is removed from it: +Lemma 4.1 ([18, Lemma 4.7]). Let A ∈ Fm×n and i ∈ [n]. Then +i ∈ F(A) +⇐⇒ +rk (A) − rk (A ⟨; i⟩) = 1. +Proof. Recall that we denote the ith standard unit vector in F1×n by en(i). While the linear dependencies of column i +of A with the other columns of A may be intricate, attaching en(i) at the bottom of A surely renders column i linearly +independent of all the other columns. Thus +rk +� +A +en(i) +� += 1 + rk (A ⟨; i⟩) . +On the other hand, by Definition 2.3, i is frozen in A if and only if en(i) is in the row span of A, so +rk +� +A +en(i) +� += rk (A) + 1 {i /∈ F(A)} . +The next lemma demonstrates that, generally, column removal and row addition cannot “unfreeze” variables: +Lemma 4.2. Let A ∈ Fm×n, b ∈ Fm×1, c ∈ F1×n and i ∈ [n]. Then +(i) i ∈ F ((A +b)) +=⇒ +i ∈ F(A); +(ii) i ∈ F(A) +=⇒ +i ∈ F +�� +A +c +�� +. +Proof. Both statements immediately follow from the characterisation of frozen variables from Lemma 4.1: variable i is +frozen in A if and only if column i does not lie in the linear span of the other columns of A. +15 + +The rank of sparse symmetric matrices over arbitrary fields +While Lemma 4.2 shows that addition of rows can only enlarge the set of frozen variables, the next lemma studies +the consequences of row removal. Indeed, Lemma 4.3 illustrates that the removal of a row has the same effect as +addition of a unit vector (which effectively forbids to use the corresponding row in representations): +Lemma 4.3. For any matrix A ∈ Fm×n, i ∈ [n] and j ∈ [m], +i ∈ F(A ⟨j;⟩) +⇐⇒ +i ∈ F +�� +A +em(j)T �� +. +Proof. Throughout the proof, we abbreviate B = +� +A +em(j)T � +. +Assume that i ∈ F (A ⟨j;⟩). Since B ⟨j;⟩ only has a zero-column appended at the right in comparison to A ⟨j;⟩, i +is also frozen in B ⟨j;⟩. By Lemma 4.2 (ii), adding a row cannot unfreeze variables, so i ∈ F (B) . +Next, assume that i ∈ F(B) and let y = (y1, . . . , ym) be a representation of {i} in B. Since no row of B apart +from j has a non-zero entry in column n + 1, yj = (yB)n+1 = 0, which implies that y ⟨; j⟩ is a representation of {i} +in A ⟨j;⟩. +We next take a closer look at the frailly frozen variables, which were characterised as those variables that unfreeze +under removal of the identically indexed row (see Definition 2.12 (i)). Since on the other hand, variables can never +freeze under row removal, we obtain the following corollary of Lemma 4.2 (ii), which expresses that the frailly frozen +variables are exactly those variables that are classified differently in the matrix with one appropriately chosen row less +than in the original matrix: +Corollary 4.4. For any A ∈ Fm×n and i ∈ [m ∧ n], +i is frailly frozen in A +⇐⇒ +i ∈ F(A)∆F (A ⟨i;⟩) . +In Definition 2.13, we have claimed that for any matrix A ∈ Fm×n, the set [m ∧ n] can be partitioned into five +types of variables. The next proposition shows that this claim is justified, since any variable in [m ∧ n] is either frailly +frozen, firmly frozen or unfrozen in A, and if it is frailly frozen in A, then it must also be frailly frozen in AT : +Proposition 4.5. Let A ∈ Fm×n and i ∈ [m ∧ n]. Then +i is frailly frozen in A +⇐⇒ +i is frailly frozen in AT . +We prove Proposition 4.5 by means of Lemma 4.1 and the following observation: +Lemma 4.6. For any matrix A ∈ Fm×n, vectors b ∈ Fm×1, c ∈ F1×n and f ∈ F, +rk +� +A +c +� +− rk(A) = 0 +and +rk +� +A +b +c +f +� +− rk (A +b) = 1 +(4.1) +if and only if +rk (A +b) − rk(A) = 0 +and +rk +� +A +b +c +f +� +− rk +� +A +c +� += 1. +(4.2) +Proof of Lemma 4.6. Denote rk +� +A +b +c +f +� +− rk(A) by h and assume that (4.1) holds. Then +h = rk (A +b) − rk(A) + rk +� +A +b +c +f +� +− rk (A +b) ≥ rk +� +A +b +c +f +� +− rk (A +b) = 1, +(4.3) +as well as +h = rk +� +A +b +c +f +� +− rk +� +A +c +� ++ rk +� +A +c +� +− rk(A) ≤ 1 + rk +� +A +c +� +− rk(A) = 1. +(4.4) +Therefore, h = 1, and we must have equality throughout (4.3) and (4.4). (4.2) then follows. The converse implication +can be shown to be true analogously. +Proof of Proposition 4.5. The assertion is an immediate consequence of the characterisation of frozen variables +in terms of rank decrease upon column removal from Lemma 4.1 in combination with Lemma 4.6 applied to +A ⟨i, i⟩ , A ⟨i;⟩ , A ⟨; i⟩ and A, since the rank of a matrix is identical to that of its transpose. +The final result of this section, Lemma 4.7, connects the five variable categories X(A), Y(A), Z(A), U(A), V(A) +from Definition 2.13 to the following rank changes under symmetric row and column removal: +Lemma 4.7. For any A ∈ Fm×n and i ∈ [m ∧ n], +(i) i ∈ Y(A) +⇐⇒ +rk(A) − rk(A ⟨i; i⟩) = 2; +16 + +The rank of sparse symmetric matrices over arbitrary fields +(ii) i ∈ X(A) ∪ U(A) ∪ V(A) +⇐⇒ +rk(A) − rk(A ⟨i; i⟩) = 1; +(iii) i ∈ Z(A) +⇐⇒ +rk(A) − rk(A ⟨i; i⟩) = 0. +Thus, +rk(A) − rk(A ⟨i; i⟩) = 1{i ∈ X(A)} + 2 · 1{i ∈ Y(A)} + 1{i ∈ U(A)} + 1{i ∈ V(A)} += 1 + 1{i ∈ Y(A)} − 1{i ∈ Z(A)}. +(4.5) +Proof. Let i ∈ [m ∧ n]. Lemma 4.1 yields the representation +rk(A) − rk(A ⟨i; i⟩) = rk(A) − rk(A ⟨i;⟩) + rk(A ⟨i;⟩) − rk(A ⟨i; i⟩) = 1 +� +i ∈ F(AT ) +� ++ 1 {i ∈ F(A ⟨i;⟩)} . +(4.6) +Identities (i)-(iii) now follow from (4.6) by an application of Proposition 4.5. +4.2 +Appending a row to a (δ, ℓ)-free matrix +In the present section, we discuss how the rank of a (δ, ℓ)-free matrix A changes upon the attachment of a single +row b with exactly ℓ non-zero entries, which are chosen uniformly from a subset of the columns of A. Recall that in +(2.6), we had observed that for a general vector b to be in the row span of A, it is sufficient that supp (b) ⊆ F(A) and +necessary that either supp (b) ⊆ F(A) or supp (b) forms a proper relation in A. These considerations show that in +the complete absence of “short” proper relations in A, rank stagnation upon attachment of a vector b with ℓ non-zero +entries can be equivalently described by the event that all variables of supp (b) are frozen in A. +Lemma 4.8, which revisits an argument from the proof of [10, Lemma 5.4], shows how to transfer the above +reasoning to matrices with few short proper relations, where the dominant reason for a rank stagnation upon attachment +of a vector should still be the event that all variables in its support are frozen. For convenience of the reader, we revisit +the main step of the argument in [10]. For this, let A ∈ Fm×n and +PRℓ(A) = {I ⊆ [n] : I is a proper relation of A with |I| = ℓ} +and +PR(A) = ∪∞ +ℓ=2 PRℓ(A) +(4.7) +be the set of proper relations of A of size ℓ ≥ 2 as well as the set of all proper relations of A, respectively. +Lemma 4.8 ([10]). Fix δ > 0, ℓ ∈ N≥2 and s ∈ N0. For any sequence ((bn−s,n, bn−s+1,n, . . . , bn,n))n∈N such that +for all n and n1 ∈ [n] \ [n − s − 1], bn1,n ∈ F1×n and supp(bn1,n) is uniformly distributed over all ℓ-subsets of [n1], +sup +m∈{n−s,...,n+s} +n1∈{n−s,...,n} +sup +A∈Fm×n: +A is (δ,ℓ)−free +P (supp (bn1,n) ∈ PRℓ(A)) ≤ δℓ! + on(1), +(4.8) +and +sup +m∈{n−s,...,n+s} +n1∈{n−s,...,n} +sup +A∈Fm×n: +A is (δ,ℓ)−free +�����E +� +rkF +� +A +bn1,n +�� +− rkF(A) − +� +1 − +�|F(A) ∩ [n1]| +n1 +�ℓ������ ≤ δℓ! + on(1). +(4.9) +Proof. Observe that +E +� +rkF +� +A +bn1,n +�� +− rkF(A) = 1 − P (bn1,n is in the span of the rows of A) . +(4.10) +As discussed in the beginning of the subsection, (2.6) gives +P (supp (bn1,n) ⊆ F(A)) ≤P (bn1,n is in the span of the rows of A) +≤P (supp (bn1,n) ∈ PRℓ(A)) + P (supp (bn1,n) ⊆ F(A)) . +(4.11) +For any (δ, ℓ)-free matrix A ∈ Fm×n, |PRℓ(A)| ≤ δnℓ, and therefore +P (supp (bn1,n) ∈ PRℓ(A)) ≤ |PRℓ(A)| +�n1 +ℓ +� +≤ δℓ! +� +n +n − s − ℓ +�ℓ +. +(4.12) +Taking the supremum over all (δ, ℓ)-free matrices A ∈ Fm×n, then m ∈ {n − s, . . . , n + s} and n1 ∈ {n − s, . . . , n} +yields (4.8). To estimate P (supp (bn1,n) ⊆ F(A)), let α(A) = |F(A) ∩ [n1]| /n1 be the proportion of frozen variables +of A among [n1]. Then +��P (supp (bn1,n) ⊆ F(A)) − α(A)ℓ�� = +���� +�n1α(A) +ℓ +���n1 +ℓ +� +− α(A)ℓ +���� = O(1/n) +(4.13) +uniformly in m, n1, A. Combining (4.10) - (4.13) yields (4.9). +17 + +The rank of sparse symmetric matrices over arbitrary fields +4.3 +Stability of types +As outlined in Section 2.5, a central ingredient in our proof strategy is to show that the proportion of frozen +variables in Tn,t/n[θ] is close to that in Tn+1,t/n[θ], which is the core theme of this section. Specifically, we look at +the extended variable types X, Y, Z, U and V from Definition 2.13. For each of these types, we define the share it has +among the variables of a matrix with perturbation, where the artificial row-perturbation columns are not taken into +account: +Definition 4.9 (Proportions of types). +(i) For A ∈ Fn×n and W ∈ {X, Y, Z, U, V}, we use the non-calligraphic +lowercase letter w to denote the proportion of variables i ∈ [n] of the corresponding type: +w(A[θ]) = |W(A[θ]) ∩ [n]| +n +. +(ii) For A ∈ Fn×n, we denote the vector of all proportions by +ζ(A[θ]) = (x(A[θ]), y(A[θ]), z(A[θ]), u(A[θ]), v(A[θ])) . +(iii) For A = Tn,t/n and wn,t/n ∈ {xn,t/n, yn,t/n, zn,t/n, un,t/n, vn,t/n, ζn,t/n}, we simply write +wn,t/n = w +� +Tn,t/n[θ] +� +. +♦ +Remark 4.10 (Summation of proportions). By definition, for any matrix A ∈ Fn×n, +∥ζ(A[θ])∥1 = x(A[θ]) + y(A[θ]) + z(A[θ]) + u(A[θ]) + v(A[θ]) = 1. +(4.14) +Moreover, recall that αn,p and αT +n,p denote the proportions of frozen variables among [n] in Tn,p[θ] and Tn,p[θ]T , +respectively. With the above definitions, +αn,t/n = xn,t/n + yn,t/n + vn,t/n +and +αT +n,t/n = xn,t/n + yn,t/n + un,t/n. +(4.15) +■ +With the notation of Definition 4.9, the main result of the remainder of Section 4 is the following proposition: +Proposition 4.11 (Stability of types). For any d > 0, and w ∈ {x, y, z, u, v, ζ}, +E +��wn,t/n − wn+1,t/n +�� +1 = on,P (1), uniformly in t ∈ [0, d]. +In light of Proposition 4.11 and (4.15), it is tempting to conjecture that the proportions αn,d/n of frozen variables +of Tn,d/n[θ] converge in a suitable sense. Unfortunately, this conjecture turns out to be incorrect, and one of the +implications of our present proof is that αn,d/n does not converge for d > e. Despite this complication, the strictly +weaker statement of Proposition 4.11 is sufficient for our purposes. +The rest of Section 4.3 is organized as follows: In Section 4.3.1, we study the impact of symmetric row- and +column relabelling on the proper relations and variable types of a given matrix. In Section 4.3.2, building on Sections +4.1, 4.2 and 4.3.1, we prove that any fixed variable is unlikely to change from frozen to unfrozen, or the other way +round, under one-step matrix growth of Tn,d/n[θ] and a related matrix. Finally, we present the proof of Proposition 4.11 +in Section 4.3.3. +4.3.1 +Row- and column exchangeability +In the following proofs, exchangeability arguments play an important role. We prepare these arguments in the +current section. Throughout this section, for k ∈ N, let Sk denote the symmetric group of [k]. +Definition 4.12. Let A ∈ Fn×n. For a permutation π ∈ Sn, define the matrix Aπ by setting +Aπ(i, j) = A(π−1(i), π−1(j)), +for i, j ∈ [n]. +(4.16) +♦ +Aπ is the matrix that arises from A through joint relabelling of the rows and columns according to i �→ π−1(i). +Lemma 4.13. For any π ∈ Sn and p ∈ [0, 1], T π +n,p[θ] +d= Tn,p[θ]. +Proof. Recall the definition of Tn,p in (2.3), according to which +Tn,p(i, j) = AN,p(τ(i), τ(j)) +for i, j ∈ [n]. +Hence, +T π +n,p(i, j) = Tn,p(π−1(i), π−1(j)) = AN,p(τ ◦ π−1(i), τ ◦ π−1(j)). +18 + +The rank of sparse symmetric matrices over arbitrary fields +Since τ is a uniform permutation of [N], also τ ◦π−1 is a uniform permutation of [N], where we view π as a permutation +of [N] that leaves {n + 1, . . . , N} fixed. Thus, +T π +n,p +d= Tn,p. +Independence of Tn,p, τ and the row- and column-perturbation matrices now implies that T π +n,p[θ] +d= Tn,p[θ], as +desired. +Corollary 4.14. Let π ∈ Sn and I = {i1, . . . , ik} ⊂ [n]. Setting Iπ = {π(i1), . . . , π(ik)}, +P(I is a proper relation of Tn,p[θ]) = P(Iπ is a proper relation of Tn,p[θ]). +Proof. Note that +I is a proper relation of Tn,p[θ] +⇐⇒ +Iπ is a proper relation of T π +n,p[θ]. +The desired result now follows from Lemma 4.13. +Lemma 4.15. For any A ∈ Fn×n, π ∈ Sn, and W ∈ {X, Y, Z, U, V}, +i ∈ W(A[θ]) +⇐⇒ +π(i) ∈ W(Aπ[θ]). +As a consequence, +ζ(A[θ]) = ζ(Aπ[θ]). +Proof. By the determinantal rank characterisation and the Leibniz determinant formula, the rank of A stays unchanged +under the permutation π. By Lemma 4.1, +i ∈ F(A) ⇐⇒ rk (A) − rk (A ⟨; i⟩) = 1 ⇐⇒ rk (Aπ) − rk (Aπ ⟨; π(i)⟩) = 1 ⇐⇒ π(i) ∈ F(Aπ). +(4.17) +Analogously, +i ∈ F(A ⟨i;⟩) +⇐⇒ +π(i) ∈ F(Aπ ⟨π(i);⟩). +(4.18) +The desired results now follow from (4.17) and (4.18). +In particular, we will make frequent use of the following corollary: +Corollary 4.16. For any i ∈ [n], p ∈ [0, 1] and W ∈ {X, Y, Z, U, V}, +P (i ∈ W (Tn,p[θ])) = E [wn,p] . +Proof. Let π ∈ Sn be the transposition of n + 1 and i. Then Lemmas 4.15 and 4.13 together imply that +P (n + 1 ∈ W (Tn,p[θ])) = P +� +π(n + 1) ∈ W +� +T π +n,p[θ] +�� += P (i ∈ W (Tn,p[θ])) +and therefore +P (n + 1 ∈ W (Tn,p[θ])) = E +� +1 +n + 1 +n+1 +� +i=1 +1 {i ∈ W (Tn,p[θ])} +� += E [wn,p] . +4.3.2 +Freezing and unfreezing under row- and column removal +Building upon the symmetry arguments of Section 4.3.1, we now prove the two main lemmas that are needed to +attack Proposition 4.11, which makes a statement about the expected differences of the various proportions of variable +types in Tn,t/n[θ] and Tn+1,t/n[θ]. Since the type of variable i ∈ [n] with respect to the matrix A ∈ Fn×n is defined +solely in terms of the membership of i in each of the sets F(A), F(AT ), F(A ⟨i;⟩) and F(AT ⟨i;⟩) (see Definitions +2.12 and 2.13) and the matrices Tn,t/n[θ] and Tn+1,t/n[θ] are reasonably alike, it seems like a viable strategy to show +that any given variable is unlikely to change its membership in each of the aforementioned sets of frozen variables in +the transition from Tn,t/n[θ] to Tn+1,t/n[θ], which is precisely what we show. In this sense, Lemma 4.17 shows that +any fixed variable is unlikely to be frozen in exactly one of the matrices Tn,t/n[θ] or Tn+1,t/n[θ]: +Lemma 4.17 (One-step matrix growth, original matrix). Fix d, δ > 0 and L ∈ N≥2. Then for any i ∈ [n], +P +� +i ∈ F +� +Tn,t/n[θ] +� +∆F +� +Tn+1,t/n[θ] +�� +≤ 2(L + 1)!δ + P (Po(d) ≥ L) + on,P (1), uniformly in t ∈ [0, d]. +Conveniently, the pair (Tn,t/n[θ], Tn+1,t/n[θ]) is identically distributed to (Tn,t/n[θ]T , Tn+1,t/n[θ]T ), so it is +enough to work with non-transposed matrices in the above considerations. In the same spirit, Lemma 4.18 shows that +any fixed variable is unlikely to be frozen in exactly one of the matrices Tn,t/n[θ] ⟨i;⟩ or Tn+1,t/n[θ] ⟨i;⟩: +19 + +The rank of sparse symmetric matrices over arbitrary fields +Lemma 4.18 (One-step matrix growth, row-deleted matrix). Fix d, δ > 0 and L ∈ N≥2. Then for any i ∈ [n], +P +� +i ∈ F +� +Tn,t/n[θ] ⟨i;⟩ +� +∆F +� +Tn+1,t/n[θ] ⟨i;⟩ +�� +≤ 2(L + 2)!δ + (L + 2)P (Po (d) ≥ L) + on,P (1), +uniformly in t ∈ [0, d]. +While Lemma 4.18 is structurally similar to Lemma 4.17, its proof proceeds differently. This is due to the fact +that the removal of row i makes this index special, and the exchangeability arguments that are used in the proof of +Lemma 4.17 do not apply directly to the modified setting. We use Lemmas 4.1 and 4.8 to overcome this problem. +Finally, we also prove a third lemma which shows that a small deterministic increase in θ is unlikely to change +whether variable i is frozen or not. Lemma 4.19 is not used in the proof of Proposition 4.11 and will only become +relevant in Section 5, but since it is similar in spirit to the previous two lemmas, we include it here: +Lemma 4.19 (Deterministic perturbation shift). Let µ = (µr, µc) ∈ N2, A ∈ Fm×n and i ∈ [n]. Then +P (i ∈ F (A[θ]) ∆F (A[θ + µ])) ≤ µr + µc +P +. +While the proofs of Lemmas 4.17 and 4.18 heavily depend on the structure of Tn,t/n, the proof of Lemma 4.19 +only uses properties of the perturbation, and thus the result is true for arbitrary matrices. +Two good events. Before we turn to the proofs of Lemmas 4.17 to 4.19, we define two good events that will be +used here and later throughout the article. For p ∈ [0, 1], let +Rn,p = +� +both Tn,p[θ] and Tn,p[θ]T are (δ, ℓ)-free for 2 ≤ ℓ ≤ L +� +, +(4.19) +and +Pn = {Θr[θr, n|n] = Θr[θr, n + 1|n + 1] ⟨; n + 1⟩ , Θc[n|n, θc] = Θc[n + 1|n + 1, θc] ⟨n + 1;⟩} . +(4.20) +Rn,p ensures that the rank increase upon attaching rows and columns can be controlled as in Lemma 4.8, while the +benefit of Pn is that when growing the matrix from n to n + 1, the perturbation stays unchanged. By Proposition 2.10 +and Lemma 2.6, +P (Rn,p) ≥ 1 + on,P (1) +and +P (Pn) = 1 + on(1). +(4.21) +The bound P +� +Rc +n,p +� +≤ on,P (1) holds uniformly in p ∈ [0, 1], since it is based on Proposition 2.10. In the following, +we frequently work on the intersection of Rn,t/n and Pn, which is a sufficiently likely event by (4.21). +We now prove Lemmas 4.17 to 4.19 in their order of appearance: +Proof of Lemma 4.17. Since i can either freeze or unfreeze when the matrix is grown, +P +� +i ∈ F +� +Tn,t/n[θ] +� +∆F +� +Tn+1,t/n[θ] +�� += P +� +i ∈ F +� +Tn,t/n[θ] +� +\F +� +Tn+1,t/n[θ] +�� ++ P +� +i ∈ F +� +Tn+1,t/n[θ] +� +\F +� +Tn,t/n[θ] +�� +. +We bound both cases separately. +(i) Unfreezing: To bound the probability that i is frozen in Tn,t/n[θ], but not in Tn+1,t/n[θ], we first show that on +Pn, +i ∈ F +� +Tn,t/n[θ] +� +\F +� +Tn+1,t/n[θ] +� +=⇒ +{i, n + 1} is a proper relation in Tn+1,t/n[θ]. +(4.22) +Assume that Pn holds and i ∈ F +� +Tn,t/n[θ] +� +\F(Tn+1,t/n[θ]). Then Tn+1,t/n[θ] arises from Tn,t/n[θ] through +the symmetric attachment of a row and a column, which we may break into two steps. By Lemma 4.2, attaching +a row cannot unfreeze i, i.e., +i ∈ F +� +Tn,t/n[θ] +� +=⇒ +i ∈ F +� +Tn+1,t/n[θ] ⟨; n + 1⟩ +� +. +In particular, there exists a representation y of {i} in Tn+1,t/n[θ] ⟨; n + 1⟩. Attaching column n + 1 and using +the representation y on the resulting matrix Tn+1,t/n[θ] yields +{i} ⊆ supp +� +yTn+1,t/n[θ] +� +⊆ {i, n + 1} . +Since i ̸∈ F +� +Tn+1,t/n[θ] +� +by assumption, we conclude that +supp +� +yTn+1,t/n[θ] +� += {i, n + 1} , +which implies that {i, n + 1} is a proper relation in Tn+1,t/n[θ], since the existence of the representation y +ensures that n + 1 cannot be frozen in Tn+1,t/n[θ] without i being frozen in Tn+1,t/n[θ]. This proves (4.22). +20 + +The rank of sparse symmetric matrices over arbitrary fields +The next step is to show that, on the good event Rn+1,t/n, the probability that {i, n + 1} forms a proper relation +in Tn+1,t/n[θ] is small: This is an immediate consequence of Corollary 4.14, which asserts that the probability +to be a proper relation is the same for any pair {i1, i2} for 1 ≤ i1 < i2 ≤ n + 1 and the observation that on +Rn+1,t/n, there are at most δ(n + 1 + P)2 proper relations of length two. Therefore, +P +� +{i, n + 1} is a proper relation in Tn+1,t/n[θ], Rn+1,t/n +� +≤ 2δ + on,P (1), uniformly in t ∈ [0, d] +and thus also +P +� +i ∈ F +� +Tn,t/n[θ] +� +\ F +� +Tn+1,t/n[θ] +� +, Pn, Rn+1,t/n +� +≤ 2δ + on,P (1), uniformly in t ∈ [0, d]. +(4.23) +(ii) Freezing: To bound the probability that i is frozen in Tn+1,t/n[θ], but not in Tn,t/n[θ], we show that on Pn, +i ∈ F +� +Tn+1,t/n[θ] +� +\F +� +Tn,t/n[θ] +� +, +i /∈ supp +� +Tn+1,t/n(n + 1, ) +� +=⇒ +{i} ∪ supp +� +Tn+1,t/n[θ](n + 1, ) +� +is a proper relation in Tn,t/n[θ]. +(4.24) +Assume that Pn holds, i ∈ F(Tn+1,t/n[θ])\F +� +Tn,t/n[θ] +� +and i /∈ supp +� +Tn+1,t/n(n + 1, ) +� +. Then the matrix +Tn,t/n[θ] arises from Tn+1,t/n[θ] through symmetric removal of a column and a row, which we may break into +two steps. By Lemma 4.2, removing a column cannot unfreeze i: +i ∈ F +� +Tn+1,t/n[θ] +� +=⇒ +i ∈ F +� +Tn+1,t/n[θ] ⟨; n + 1⟩ +� +. +This implies in particular that there exists a representation y of {i} in Tn+1,t/n[θ] ⟨; n + 1⟩. If there was a +representation y of {i} in Tn+1,t/n[θ] ⟨; n + 1⟩ with yn+1 = 0, then shortening y to y ⟨; n + 1⟩ would be a +representation of {i} in Tn,t/n[θ], in contrast to our assumption that i is not frozen in Tn,t/n[θ]. Thus, all +representations y of {i} in Tn+1,t/n[θ] ⟨; n + 1⟩ have their (n + 1)st coordinate different from zero. Since we +assume that i /∈ supp +� +Tn+1,t/n(n + 1, ) +� +, we conclude that +supp +� +y ⟨; n + 1⟩ Tn,t/n[θ] +� += {i} ∪ supp +� +Tn+1,t/n[θ](n + 1, ) +� +. +This implies that {i} ∪ supp +� +Tn+1,t/n[θ](n + 1, ) +� +is a proper relation of Tn,t/n[θ], since it contains the +non-frozen variable i. Thus, (4.24) holds. +The next step is to show that on the good event Rn,t/n, the probability that {i} ∪ supp +� +Tn+1,t/n[θ](n + 1, ) +� +forms a proper relation in Tn,t/n[θ] is small. We first upper-bound the probability that row n + 1 has too many +non-zero entries, which is due to the sparsity of the matrix Tn+1,t/n. By [25, Theorem 2.10], we can upper +bound +P +���supp +� +Tn+1,t/n[θ](n + 1, ) +��� ≥ L +� +≤P (Bin (n, t/n) ≥ L) + P (Pc +n) +≤P (Po (t) ≥ L) + P (Pc +n) + on(1) ≤ P (Po (d) ≥ L) + on(1) +uniformly in t ∈ [0, d]. On the other hand, by Corollary 4.14, the probability to be a proper relation in Tn,t/n[θ] +is the same for any subset of [n] of cardinality +��{i} ∪ supp +� +Tn+1,t/n[θ](n + 1, ) +���. If row n + 1 has at most +L − 1 non-zero entries and Rn,t/n holds, then Tn,t/n[θ] is (δ, +��{i} ∪ supp +� +Tn+1,t/n[θ](n + 1, ) +���)-free, and +P +� +{i} ∪ supp +� +Tn+1,t/n[θ](n + 1, ) +� +∈ PR +� +Tn,t/n[θ] +� +, +��{i} ∪ supp +� +Tn+1,t/n[θ](n + 1, ) +��� ≤ L, Rn,t/n +� +≤ L!δ + on,P (1), uniformly in t ∈ [0, d]. +Finally, since +P +� +i ∈ supp +� +Tn+1,t/n(n + 1, ) +�� += P +� +Tn+1,t/n(n + 1, i) = 1 +� += t/n ≤ d/n = on(1) uniformly in t ∈ [0, d], +we conclude from (4.24) that +P +� +i ∈ F +� +Tn+1,t/n[θ] +� +\ F +� +Tn,t/n[θ] +� +, Pn, Rn+1,t/n +� +≤ (L + 1)!δ + P (Po(d) ≥ L) + on,P (1) +(4.25) +uniformly in t ∈ [0, d]. +Combining (4.21), (4.23) and (4.25) finishes the proof of Lemma 4.17. +Proof of Lemma 4.18. Again, we relate a status change of i to the existence of a proper relation: On a sufficiently likely +event Sn, +i ∈ F +� +Tn,t/n[θ] ⟨i;⟩ +� +∆F +� +Tn+1,t/n[θ] ⟨i;⟩ +� +=⇒ +supp +� +Tn,t/n(, i) +� +is a proper relation in Tn,t/n[θ] ⟨i; i⟩T or in Tn+1,t/n[θ] ⟨i; i⟩T . +(4.26) +21 + +The rank of sparse symmetric matrices over arbitrary fields +Definition of Sn. The event that we work on is composed of three parts: First, we define +Sn,1 = Pn ∩ +� +Tn+1,t/n[θ](n + 1, i) = 0, Θr[θr, n|n](, i) = 0θr×1, Θc[n|n, θc](i, k) = 01×θc +� +. +On Sn,1, the non-zero entries of column (or equivalently row) i in all involved matrices are contained in [n], i.e., +supp +� +Tn,t/n(, i) +� += supp +� +Tn,t/n[θ](, i) +� += supp +� +Tn+1,t/n[θ](, i) +� += supp +� +Tn+1,t/n(, i) +� +. +From the construction of the perturbation Lemma 2.6 and the definition of Tn+1,t/n, it is immediate that +P +� +Sc +n,1 +� +≤ +2P +n + 1 + d +n + 2P +n = on(1), uniformly in t ∈ [0, d]. +Next, let +Sn,2 = +� +for all j ∈ supp(Tn,t/n[θ](, i)) : j /∈ F +� +Tn,t/n[θ] ⟨i; i⟩T � +∆F +� +Tn+1,t/n[θ] ⟨i; i⟩T �� +be the event that no element of the support of column i has a different status in Tn+1,t/n[θ] ⟨i; i⟩T than in +Tn,t/n[θ] ⟨i; i⟩T . Since (Tn,t/n[θ] ⟨i; i⟩T , Tn+1,t/n[θ] ⟨i; i⟩T ) conditionally on Sn,1 and +� +Tn−1,t/n[θ], Tn,t/n[θ] +� +conditionally on Pn−1 have the same law, by Lemma 4.17 and Corollary 4.14, +P(Sc +n,2) ≤ LP +� +1 ∈ F +� +Tn−1,t/n[θ] +� +∆F +� +Tn,t/n[θ] +�� ++ P(|supp(Tn,t/n[θ](, i))| > L) + P(Sc +n,1) + P +� +Pc +n−1 +� +≤ (2 + (L + 1)!)Lδ + (L + 1)P (Po(d) ≥ L) + on,P (1), uniformly in t ∈ [0, d]. +Finally, let +Sn,3 = +� +Tn,t/n[θ] ⟨i; i⟩T and Tn+1,t/n[θ] ⟨i; i⟩T are (δ, ℓ)-free for 2 ≤ ℓ ≤ L +� +. +Since (Tn,t/n[θ] ⟨i; i⟩T , Tn+1,t/n[θ] ⟨i; i⟩T ) conditionally on Sn,1 and +� +Tn−1,t/n[θ], Tn,t/n[θ] +� +conditionally on +Pn−1 have the same law, Proposition 2.10 implies that +P(Sc +n,3) = on,P (1), uniformly in t ∈ [0, d]. +We set Sn = Sn,1 ∩ Sn,2 ∩ Sn,3, so that +P(Sc +n) ≤ (2 + (L + 1)!)Lδ + (L + 1)P (Po(d) ≥ L) + on,P (1), uniformly in t ∈ [0, d]. +(4.27) +Proof of implication (4.26). Now suppose that Sn holds and that supp(Tn,t/n(, i)) is neither a proper relation in +Tn,t/n[θ] ⟨i; i⟩T nor in Tn+1,t/n[θ] ⟨i; i⟩T . Since the support may contain frozen variables, there are four cases: +Case 1: supp(Tn,t/n(, i)) neither has a representation in Tn,t/n[θ] ⟨i; i⟩T nor in Tn+1,t/n[θ] ⟨i; i⟩T . +The non-existence of a representation of supp(Tn,t/n ⟨i;⟩ (, i)) in Tn,t/n[θ] ⟨i; i⟩T in particular implies that +column i of Tn,t/n[θ] ⟨i;⟩ is not in the linear span of the other columns of Tn,t/n[θ] ⟨i;⟩. Thus, by Lemma 4.1, +i is frozen in Tn,t/n[θ] ⟨i;⟩. The same reasoning implies that i is frozen in Tn+1,t/n[θ] ⟨i;⟩ as well. +Case 2: supp(Tn,t/n(, i)) has a representation both in Tn,t/n[θ] ⟨i; i⟩T and in Tn+1,t/n[θ] ⟨i; i⟩T . +Since we assume that supp(Tn,t/n(, i)) is neither a proper relation in Tn,t/n[θ] ⟨i; i⟩T +nor in +Tn+1,t/n[θ] ⟨i; i⟩T , all variables in ∅ ̸= supp(Tn,t/n(, i)) must be frozen both in Tn,t/n[θ] ⟨i; i⟩T and in +Tn+1,t/n[θ] ⟨i; i⟩T . In this case, the existence of the respective representations ensures that column i of +Tn,t/n[θ] ⟨i;⟩ is contained in the linear span of the other columns of Tn,t/n[θ] ⟨i;⟩ and that column i of +Tn+1,t/n[θ] ⟨i;⟩ is contained in the linear span of the other columns of Tn+1,t/n[θ] ⟨i;⟩. Thus, by Lemma 4.1, +i is neither frozen in Tn,t/n[θ] ⟨i;⟩ nor in Tn+1,t/n[θ] ⟨i;⟩. +Case 3: supp(Tn,t/n(, i)) has a representation in Tn,t/n[θ] ⟨i; i⟩T , but none in Tn+1,t/n[θ] ⟨i; i⟩T . +Again, all variables in ∅ ̸= supp(Tn,t/n(, i)) must be frozen in Tn,t/n[θ] ⟨i; i⟩T , but there must exist a variable +that is not frozen in Tn+1,t/n[θ] ⟨i; i⟩T . This possibility is excluded by Sn,2. +Case 4: supp(Tn,t/n(, i)) has a representation in Tn+1,t/n[θ] ⟨i; i⟩T , but none in Tn,t/n[θ] ⟨i; i⟩T . +By the same reasoning as in case 3, this cannot happen on Sn,2. +22 + +The rank of sparse symmetric matrices over arbitrary fields +Cases 1 to 4 imply (4.26), which gives +P(i ∈ F +� +Tn,t/n[θ] ⟨i;⟩ +� +∆F +� +Tn+1,t/n[θ] ⟨i;⟩ +� +) +≤P +� +Sn, supp(Tn,t/n(, i)) is a proper relation in Tn,t/n[θ] ⟨i; i⟩T or in Tn+1,t/n[θ] ⟨i; i⟩T � ++ P (Sc +n) . +Finally, since Tn,t/n[θ] ⟨i; i⟩T and Tn+1,t/n[θ] ⟨i; i⟩T are (δ, ℓ)-free for 2 ≤ ℓ ≤ L on Sn, by Lemma 4.8, +P(Sn, supp(Tn,t/n(, i)) is a proper relation in Tn,t/n[θ] ⟨i; i⟩T or in Tn+1,t/n[θ] ⟨i; i⟩T ) +≤2L!δ + P (Po(d) ≥ L) + on(1), uniformly in t ∈ [0, d]. +(4.28) +Combining (4.27) and (4.28) yields the claim. +Proof of Lemma 4.19. The matrix A[θ + µ] arises from A[θ] through the attachment of µr independent unit rows and +µc independent unit columns. We split this row- and column-attachment into two steps. Since A is non-random, +dTV (F(A[θ]), F(A[θ + (0, µc)])) ≤ dTV (θ, θ + (0, µc)) ≤ µc +P , +(4.29) +and +dTV (F(A[θ + (0, µc)]), F(A[θ + µ])) ≤ µr +P . +(4.30) +By Lemma 4.2, increasing the number of columns can only diminish the number of frozen variables among the first n. +Therefore, for i ∈ [n], using (4.29), +P (i ∈ F (A[θ]) ∆F (A[θ + (0, µc)])) = P (i ∈ F (A[θ])) − P (i ∈ F (A[θ + (0, µc)])) ≤ µc +P . +(4.31) +Similarly, also by Lemma 4.2, increasing the number of rows can only enlarge the number of frozen variables among +the first n. Therefore, using (4.30), +P (i ∈ F (A[θ + (0, µc)]) ∆F (A[θ + µ])) = P (i ∈ F (A[θ + µ)])) − P (i ∈ F (A[θ + (0, µc)])) ≤ µr +P . +(4.32) +The claim follows by combining (4.31) and (4.32). +As the final result of this subsection, we note an immediate consequence of Lemma 4.19, that will be used in the +proof of Lemma 5.12 below. Corollary 4.20 shows that for any i ∈ [n], removal of a bounded number of uniformly +chosen rows is unlikely to unfreeze i, even if row i is forbidden to be among the removed rows: +Corollary 4.20 (Random row-removal). For any A ∈ Fm×n, k ∈ [m], i ∈ [n] and a uniformly chosen k-subset +J ⊆ [m]\ {i}, +P (i ∈ F (A[θ]) ∆F (A[θ] ⟨J ;⟩)) ≤ k +P + +� +1 − 1 +m +�k k(k − 1) +2m ++ k +m. +(4.33) +Proof. Let i ∈ [n]. The proof is based on Lemmas 4.3 and 4.19: First, Lemma 4.3 shows that row removal has the +same effect on whether i is frozen or not as addition of a unit column vector. The latter operation then can be treated as +a slight change in the column perturbation, and therefore falls under the scope of Lemma 4.19. +We first replace J by a set obtained from sampling with replacement. Let j′ +1, . . . , j′ +k ∈ [m] be i.i.d. uniform +indices and J ′ = ∪k +s=1 {j′ +s} such that P(J ̸= J ′) = dTV(J , J ′) (i.e., we take an optimal coupling). Then +P (i ∈ F (A[θ]) ∆F (A[θ] ⟨J ;⟩)) ≤ P +� +i ∈ F (A[θ]) ∆F +� +A[θ] +� +J ′; +��� ++ dTV(J , J ′). +(4.34) +Furthermore, an application of Lemma 4.3 to (4.34) gives +P (i ∈ F (A[θ]) ∆F (A[θ] ⟨J ;⟩)) ≤ P(i ∈ F (A[θ]) ∆F (A[θ + (0, k)])) + dTV(J , J ′). +(4.35) +Since dTV(J , J ′) ≤ (1−1/m)kk(k−1)/(2m)+k/m (see [24] in combination with the observation that J ′ samples +from [m] rather than [m] \ {i}, for example), the claim now follows from (4.35) and Lemma 4.19. +23 + +The rank of sparse symmetric matrices over arbitrary fields +4.3.3 +Stability of types: Proof of Proposition 4.11 +With Lemmas 4.17 and 4.18, we are now in the position to prove Proposition 4.11: +Proof of Proposition 4.11. Observe that for any W ∈ {X, Y, Z, U, V}, the sequence (1 +� +i ∈ W +� +Tn,t/n[θ] +�� +− +1 +� +i ∈ W +� +Tn+1,t/n[θ] +�� +)i∈[n] consists of identically distributed random variables: This is a consequence of the fact +that Tn,t/n is a submatrix of Tn+1,t/n, Lemma 4.13 and Lemma 4.15. Therefore, +E +��wn,t/n − wn+1,t/n +�� = 1 +nE +����� +n +� +i=1 +� +1 +� +i ∈ W +� +Tn,t/n[θ] +�� +− 1 +� +i ∈ W +� +Tn+1,t/n[θ] +��� +����� + on(1) +≤ E +��1 +� +1 ∈ W +� +Tn,t/n[θ] +�� +− 1 +� +1 ∈ W +� +Tn+1,t/n[θ] +���� + on(1). +(4.36) +We next bound (4.36) for the different types separately: +Case 1: wn,t/n = xn,t/n. +Corollary 4.4 yields the identity +1 +� +1 ∈ X +� +Tn,t/n[θ] +�� += 1 +� +1 ∈ F +� +Tn,t/n[θ] +� +∆F +� +Tn,t/n[θ] ⟨1;⟩ +�� +. +(4.37) +Using (B1∆B2)∆(B3∆B4) ⊆ (B1∆B3) ∪ (B2∆B4) for any sets B1, B2, B3, B4 and plugging (4.37) into (4.36) +yields the upper bound +P +� +1 ∈ F +� +Tn,t/n[θ] +� +∆F +� +Tn+1,t/n[θ] +�� ++ P +� +1 ∈ F +� +Tn,t/n[θ] ⟨1;⟩ +� +∆F +� +Tn+1,t/n[θ] ⟨1;⟩ +�� ++ on(1) +(4.38) +for E|xn,t/n − xn+1,t/n|. Lemmas 4.17 and 4.18 now imply that +E|xn,t/n − xn+1,t/n| ≤ 4(L + 2)!δ + (L + 3)P (Po (d) ≥ L) + on,P (1), uniformly in t ∈ [0, d]. +(4.39) +In particular, +lim sup +P →∞ +lim sup +n→∞ +sup +N≥n,JN∈SymN(F∗) +sup +t∈[0,d] +E|xn,t/n − xn+1,t/n| ≤ 4(L + 2)!δ + 3LP (Po (d) ≥ L) . +(4.40) +Since the left hand side of (4.40) does not depend on L and δ, we can send δ ↓ 0 followed by L → ∞ to conclude that +lim sup +P →∞ +lim sup +n→∞ +sup +N≥n,JN∈SymN(F∗) +sup +t∈[0,d] +E|xn,t/n − xn+1,t/n| = 0 +(4.41) +or equivalently, E|xn,t/n − xn+1,t/n| = on,P (1) uniformly in t ∈ [0, d]. +Case 2: wn,t/n = yn,t/n. +Definition 2.12 of completely frozen variables and Lemma 4.2 (ii) on row addition yield the identity +1 +� +1 ∈ Y +� +Tn,t/n[θ] +�� += 1 +� +1 ∈ F +� +Tn,t/n[θ] ⟨1;⟩ +� +∩ F +� +Tn,t/n[θ]T ⟨1;⟩ +�� +. +(4.42) +Using (B1 ∩ B2)∆(B3 ∩ B4) ⊆ (B1∆B3) ∪ (B2∆B4) for any sets B1, B2, B3, B4 and plugging (4.42) into (4.36) +yields the upper bound +P +� +1 ∈ F +� +Tn,t/n[θ] ⟨1;⟩ +� +∆F +� +Tn+1,t/n[θ] ⟨1;⟩ +�� ++ P +� +1 ∈ F(Tn,t/n[θ]T ⟨1;⟩)∆F(Tn+1,t/n[θ]T ⟨1;⟩) +� ++ on(1) +(4.43) +for E|yn,t/n − yn+1,t/n|. Lemma 4.18 then yields +E|yn,t/n − yn+1,t/n| ≤ 4(L + 2)!δ + 2(L + 2)P (Po (d) ≥ L) + on,P (1), uniformly in t ∈ [0, d]. +Now the same limiting argument as in Case 1 yields +E|yn,t/n − yn+1,t/n| = on,P (1), uniformly in t ∈ [0, d]. +Case 3: wn,t/n = zn,t/n. +Definition 2.3 of frozen variables yields the identity +1 +� +1 ∈ Z +� +Tn,t/n[θ] +�� += 1 +� +1 /∈ F +� +Tn,t/n[θ] +� +∪ F +� +Tn,t/n[θ]T �� +. +(4.44) +Using (B1 ∪ B2)∆(B3 ∪ B4) ⊆ (B1∆B3) ∪ (B2∆B4) for any sets B1, B2, B3, B4 and plugging (4.44) into (4.36) +yields the upper bound +P +� +1 ∈ F +� +Tn,t/n[θ] +� +∆F +� +Tn+1,t/n[θ] +�� ++ P +� +1 ∈ F +� +Tn,t/n[θ]T � +∆F +� +Tn+1,t/n[θ]T �� ++ on(1) +(4.45) +for E|zn,t/n − zn+1,t/n|. Lemma 4.17 then yields +E|zn,t/n − zn+1,t/n| ≤ 4(L + 1)!δ + 2P (Po(d) ≥ L) + on,P (1), uniformly in t ∈ [0, d]. +24 + +The rank of sparse symmetric matrices over arbitrary fields +Now the same limiting argument as in Case 1 yields +E|zn,t/n − zn+1,t/n| = on,P (1), uniformly in t ∈ [0, d]. +Case 4: wn,t/n = un,t/n. +Definition 2.3 of frozen variables and Lemma 4.2 (ii) on row addition yield the identity +1 +� +1 ∈ U +� +Tn,t/n[θ] +�� += 1 +� +1 ∈ F +� +Tn,t/n[θ]T ⟨1;⟩ +� +\ F +� +Tn,t/n[θ] +�� +. +(4.46) +Using (B1 \ B2)∆(B3 \ B4) ⊆ (B1∆B3) ∪ (B2∆B4) for any sets B1, B2, B3, B4 and plugging (4.46) into (4.36) +yields the upper bound +P +� +1 ∈ F +� +Tn,t/n[θ] +� +∆F +� +Tn+1,t/n[θ] +�� ++ P +� +1 ∈ F(Tn,t/n[θ]T ⟨1;⟩)∆F(Tn+1,t/n[θ]T ⟨1;⟩) +� ++ on(1). +(4.47) +for E +��un,t/n − un+1,t/n +��. Lemmas 4.17 and 4.18 then give +E|un,t/n − un+1,t/n| ≤ 4(L + 2)!δ + (L + 3)P (Po (d) ≥ L) + on,P (1), uniformly in t ∈ [0, d]. +Now the same limiting argument as in Case 1 yields +E|un,t/n − un+1,t/n| = on,P,(1), uniformly in t ∈ [0, d]. +Case 5: wn,t/n = vn,t/n. +This is completely analogous to Case 4. +Case 6: wn,t/n = ζn,t/n. +This is an immediate consequence of Cases 1-5. +5 +Type fixed point equations +As laid out in detail in Section 2.5, for our lower bound on the rank to be tight, we need further means to restrict +the potential values of the proportion αn,t/n of frozen variables. In this section, with the help of the stability properties +of the types that we derived in Section 4, we derive asymptotic fixed point equations for the proportions of finer types +yn,t/n, un,t/n and vt,t/n in Tn,t/n[θ], as well as a lower bound for zn,t/n. Correspondingly, the single main result of +this section is Proposition 5.1 below. +The proof of the characterisations in Proposition 5.1 is based on a detailed analysis of the connection of the type +of variable n + 1 in the larger matrix Tn+1,t/n[θ] to the types of the non-zero entries of row n + 1 in the smaller +matrix Tn,t/n[θ]. In this way, we can relate the proportions of types to certain functions of other proportions, that +simply correspond to the choices of the non-zero entries of row n + 1 and therefore comparatively easy to evaluate, see +Section 5.2.2. Of course, the details are considerably more involved, but indeed, this proof scheme is quite similar to +the deduction of the heuristic fixed point equation in Section 2.5, where we relate the type of the new coordinate to the +types of its neighbours by the combination of eqs. (2.5) and (2.6). +5.1 +Section overview +The main goal of this section is to derive the following fixed point equations for the types from Definition 4.9: +Proposition 5.1 (Type fixed point equations). For any n ≥ 0 and d > 0, +yn,t/n = 1 − φt +� +xn,t/n + yn,t/n + un,t/n +� +− φt +� +xn,t/n + yn,t/n + vn,t/n +� ++ φt +� +xn,t/n + yn,t/n +� ++ ¯oP(1); +(5.1) +un,t/n = φt +� +xn,t/n + yn,t/n + un,t/n +� +− φt +� +xn,t/n + yn,t/n +� ++ ¯oP(1); +(5.2) +vn,t/n = φt +� +xn,t/n + yn,t/n + vn,t/n +� +− φt +� +xn,t/n + yn,t/n +� ++ ¯oP(1); +(5.3) +zn,t/n ≥ φt +� +yn,t/n +� ++ ¯oP(1). +(5.4) +For the proof of Proposition 5.1, in whose course we also work with a more general matrix model, we give names +to the functions on the right hand sides of (5.1) to (5.3). We use the following suggestive notation: +Definition 5.2 (Type functions). Let G denote the set of non-decreasing functions g : [0, 1] → [0, 1] and ∆4 be the +four-dimensional standard simplex. We then define the following three functions Y, U, V : ∆4 × G → [0, 1] by setting: +(i) Y (ζ, g) = 1 − g(x + y + u) − g(x + y + v) + g(x + y) for (ζ, g) ∈ ∆4 × G; +(ii) U (ζ, g) = g(x + y + u) − g(x + y) for (ζ, g) ∈ ∆4 × G; +(iii) V (ζ, g) = g(x + y + v) − g(x + y) for (ζ, g) ∈ ∆4 × G. +♦ +25 + +The rank of sparse symmetric matrices over arbitrary fields +The proof of Proposition 5.1 is split into two main parts: Lemma 5.3 and Lemma 5.4. First, Lemma 5.3 reduces the +approximation of the types through the type functions to the separate approximation of conditional type probabilities in +a larger matrix through the type functions and approximation of ζn+1,t/n through ζn,t/n: +Lemma 5.3. Let t ∈ [0, d] with d > 0. For any W ∈ {Y, U, V } and K ∈ Z≥2, +E +��wn,t/n − W +� +ζn,t/n, φt +��� ≤E +��P +� +n + 1 ∈ W +� +Tn+1,t/n[θ] +� ��ζn,t/n +� +− W +� +ζn,t/n, φt +��� +(5.5) ++ (5 + 8K2)E∥ζn,t/n − ζn+1,t/n∥∞ + 4 − 4(1 − 1/K)5 + 10 (1 + 2d) /K, +and +E +�� +zn,t/n − φt +� +yn,t/n +��−� +≤E +�� +P +� +n + 1 ∈ Z +� +Tn+1,t/n[θ] +� ��ζn,t/n +� +− φt +� +yn,t/n +��−� +(5.6) ++ (5 + 8K2)E∥ζn,t/n − ζn+1,t/n∥∞ + 4 − 4(1 − 1/K)5 + 10 (1 + 2d) /K. +The extensive proof of Lemma 5.3 is deferred to Appendix C, since it is separate from our main proof strategy and +rather technical. Since thanks to Proposition 4.11, we have good control over the differences E∥ζn,t/n − ζn+1,t/n∥∞ +already, it only remains to take care of the conditional probabilities in Lemma 5.3. This is exactly what the second +Lemma 5.4 does by illustrating the probabilistic interpretation of the type functions: It shows that the type functions +(resp. φt) are approximations (resp. lower bounds) of the probabilities that column n + 1 in Tn+1,t/n[θ] has the type +associated with the function (resp. type Z), conditionally on ζn,t/n: +Lemma 5.4 (Conditional probabilities and type functions). For any W ∈ {Y, U, V } and d > 0, +P +� +n + 1 ∈ W +� +Tn+1,t/n[θ] +� ��ζn,t/n +� +− W +� +ζn,t/n, φt +� += ¯oP(1), +(5.7) +while +P +� +n + 1 ∈ Z +� +Tn+1,t/n[θ] +� ��ζn,t/n +� +− φt +� +yn,t/n +� +≥ ¯oP(1). +(5.8) +The proof of Lemma 5.4 is presented in Section 5.2. With Lemmas 5.3 and 5.4 in hand, we are ready to prove +Proposition 5.1: +Proof of Proposition 5.1 subject to Lemmas 5.3 and 5.4. Let W ∈ {Y, U, V } and K ∈ N≥2. By Lemma 5.3, an +application of Proposition 4.11 and (5.7) to the right hand side of (5.5) together with the fact ∥ · ∥∞ ≤ ∥ · ∥1 gives +E +��wn,t/n − W +� +ζn,t/n, φt +��� ≤ on,P (1) + 4 − 4(1 − 1/K)5 + 10(1 + 2d)/K +uniformly in t ∈ [0, d]. +(5.9) +Since (5.9) holds true for any K ≥ 2, and the left hand side does not depend on K, (5.9) implies that +lim sup +P →∞ +lim sup +n→∞ +sup +N≥n,JN∈SymN(F∗) +sup +0≤t≤d +E +��wn,t/n − W +� +ζn,t/n, φt +��� = 0, +which gives (5.1) to (5.3). +Analogously, by Lemma 5.3, an application of Proposition 4.11 and (5.6) to the right hand side of (5.8) together +with the fact ∥ · ∥∞ ≤ ∥ · ∥1 gives +E +�� +zn,t/n − φt +� +yn,t/n +��−� +≤ on,P (1) + 4 − 4(1 − 1/K)5 + 10(1 + 2d)/K +uniformly in t ∈ [0, d]. +(5.10) +As before , (5.10) implies that +� +zn,t/n − φt +� +yn,t/n +��− = ¯oP(1), i.e., zn,t/n − φt +� +yn,t/n +� +≥ ¯oP(1). +It thus only remains to prove Lemma 5.4. +5.2 +Conditional probabilities and type functions: Proof of Lemma 5.4 +As outlined in Section 5.1, the only ingredient in the proof of Proposition 5.1 that is still lacking a proof is +Lemma 5.4. This gap is closed in the current section. In addition, we prove a more general version of Lemma 5.4, +which holds true for arbitrary square matrices and more general types of symmetric row- and column-attachment. We +believe that this extension will prove useful in future applications of our strategy. +More precisely, for any positive integer n, let A ∈ Fn×n be an arbitrary square matrix and let h be an integer-valued +random variable with probability generating function ψ. Given h, let h ∈ F1×n be a random vector whose non-zero +entries are chosen uniformly at random from the set +�[n] +h +� +. Throughout this section, we write +Ah = +� +A +hT +h +0 +� +to denote the matrix A after symmetric row- and column-attachment of h. Finally, we omit the explicit dependence +of the proportions on the underlying matrix and simply write x for x(A[θ]) throughout Section 5.2. The quantities +y, z, u, v and ζ are defined analogously. +The main result of this section is the following generalised version of Lemma 5.4: +26 + +The rank of sparse symmetric matrices over arbitrary fields +Proposition 5.5 (Approximating the type probabilities for column n + 1). For any A ∈ Fn×n, L ∈ N≥2, δ > 0 and +W ∈ {Y, U, V }, +E +��P +� +n + 1 ∈ W +� +Ah[θ] +� ��ζ +� +− W(ζ, ψ) +�� ≤ 6L!δ + 7P (h ≥ L) + on,P (1), +(5.11) +and +E +�� +P +� +n + 1 ∈ Z +� +Ah[θ] +� ��ζ +� +− ψ (y) +�−� +≤ 2P (h ≥ L) + on,P (1). +(5.12) +Remark 5.6 (Error terms). We emphasize that the error terms in Proposition 5.5 and the rest of this section are uniform +in A and ψ. In the current more general setting, it becomes evident that the distribution of the type of the new column +n + 1 only depends on A through the proportions of the types in A[θ]. +■ +Lemma 5.4 is now a direct consequence of Proposition 5.5: +Proof of Lemma 5.4 subject to Proposition 5.5. In the set-up of Proposition 5.5, let A = Tn,t/n and h = Tn+1,t/n(n+ +1, ) ⟨; n + 1⟩, such that ψ becomes the probability generating function of a Bin(n, t/n)-variable. We first look at (5.7). +For W ∈ {Y, U, V }, by the triangle inequality, +E +��P +� +n + 1 ∈ W +� +Tn+1,t/n[θ] +� ��ζn,t/n +� +− W +� +ζn,t/n, φt +��� +≤ E +��P +� +n + 1 ∈ W +� +Tn+1,t/n[θ] +� ��ζn,t/n +� +− W +� +ζn,t/n, ψ +��� + E +��W +� +ζn,t/n, ψ +� +− W +� +ζn,t/n, φt +��� . +(5.13) +By [25, Theorem 2.10], +sup +r∈[0,1] +|ψ(r) − φt(r)| ≤ +∞ +� +k=0 +|P (Bin(n, t/n) = k) − P (Po (t) = k)| = dTV (Bin (n, t/n) , Po (t)) ≤ d2/n. +(5.14) +Equation (5.7) then follows from Proposition 5.5, (5.13) and (5.14). Inequality (5.8) follows anagolously. +In the remainder of Section 5.2, we carry out the proof of Proposition 5.5. For this, we first introduce five type +events in Section 5.2.1 that establish a connection between the type of n + 1 and the types of supp (h) in the underlying +matrix Ah[θ]. Since the non-zero coordinates of h are chosen uniformly given h, we can then estimate the probabilities +of the type events in Section 5.2.2, and complete the proof of Proposition 5.5 in Section 5.2.3. +5.2.1 +Type events +As announced, in this subsection, we introduce a number of “type” events that are solely defined in terms of +supp (h) and A[θ]. These events capture the main causes for variable n + 1 to belong to a particular set W +� +Ah[θ] +� +in +terms of whether all variables in supp (h) are frozen with respect to A[θ] or A[θ]T . In this sense, we show that the +probability that n + 1 does not have a certain type on its matching type event is small in Lemmas 5.11 and 5.12 below. +As in Section 4.3.2, throughout this section, we will frequently work on two good events. The first event Pn +will be the same as in (4.20), since it only involves the perturbation and ensures that the perturbations in A[θ] and +Ah[θ] agree. By (4.21), P (Pn) = 1 + on(1). Secondly, and analogously to the definition of (4.19), we define an event +R that is used to make our target matrix (δ, ℓ)-free, so that we can apply Lemma 4.8. More precisely, we denote by +R = R(δ, L) the good event that both A[θ] and A[θ]T are (δ, ℓ)-free for 2 ≤ ℓ ≤ L. Proposition 2.10 gives that +P (R) ≥ 1 + on,P (1). +(5.15) +Before we introduce the actual type events in Definition 5.9, we first define two basic events that are used to decide +whether variable n + 1 is firmly frozen in Ah[θ]. +Definition 5.7 (Basic events). Given A ∈ Fn×n and h as above, we define the following events: +F = {supp (h) ⊆ F (A[θ])}, +(5.16) +Ftr = {supp (h) ⊆ F +� +A[θ]T � +}. +(5.17) +♦ +The following preparatory lemma then shows that the main reason for the new variable n + 1 to be firmly frozen in +Ah[θ] is that not all of the variables in supp (h) are frozen in A[θ]T , and thus the event Fc +tr. This observation is later +used to characterise the other possible types of n + 1 in terms of the support of h. +Lemma 5.8. For any δ > 0, L ∈ N≥2 and A ∈ Fn×n, +P +� +n + 1 is firmly frozen in Ah[θ], Ftr +� += on(1), +(5.18) +and +P +� +n + 1 is not firmly frozen in Ah[θ], Fc +tr +� +≤ L!δ + P (h ≥ L) + on,P (1). +(5.19) +27 + +The rank of sparse symmetric matrices over arbitrary fields +Proof. +(i) We first show (5.18). By definition, n + 1 is firmly frozen in Ah[θ] if and only if it is frozen in +Ah[θ] ⟨n + 1;⟩. On the good event Pn, removal of row n + 1 leaves us with the matrix A[θ] plus the +additional column (h +01×θr)T . By Lemma 4.1, +n + 1 is firmly frozen in Ah[θ], Pn =⇒ (h +01×θr)T cannot be linearly combined by the columns of A[θ]. +=⇒ supp (h) ̸⊆ F +� +A[θ]T � +. +On the other hand, on Ftr, supp (h) ⊆ F +� +A[θ]T � +. Therefore, +P +� +n + 1 is firmly frozen in Ah[θ], Ftr +� += P +� +n + 1 is firmly frozen in Ah[θ], Ftr, Pn +� ++ on(1) = on(1), +as required. +(ii) We next prove (5.19). By definition, if n + 1 is not firmly frozen in Ah[θ], it is not frozen in Ah[θ] ⟨n + 1;⟩. +On the good event Pn, removal of row n + 1 leaves us with the matrix A[θ] plus the additional column +(h +01×θr)T . Since n + 1 is not frozen in this matrix, Lemma 4.1 gives that +n + 1 is not firmly frozen in Ah[θ], Pn =⇒ (h +01×θr)T can be lin. combined by the columns of A[θ]. +On the other hand, on Fc +tr, supp (h) ̸⊆ F +� +A[θ]T � +. This implies that both supp (h) and supp (h) \F +� +A[θ]T � +are non-empty. +If additionally, (h +01×θr)T can be linearly combined by the columns of A[θ], +supp (h) \F +� +A[θ]T � +is a relation of A[θ]T . Hence, by Definition 2.3 (iii), +n + 1 is not firmly frozen in Ah[θ], Fc +tr, Pn +=⇒ +supp (h) is a proper relation of A[θ]T . +By Lemma 4.8 and (5.15), +P +� +n + 1 is not firmly frozen in Ah[θ], Fc +tr, Pn +� +≤ P +� +supp (h) is a proper relation in A[θ]T � +≤ L!δ + P (h ≥ L) + on,P (1), +as required. +With Lemma 5.8, we are now in the position to characterise the type of variable n + 1 in terms of the role of the +variables in supp (h) in A[θ] and A[θ]T through the following events. +Definition 5.9 (Type events). With the notation of Definition 5.7, let +Y = Fc ∩ Fc +tr, +U = Fc ∩ Ftr, +V = F ∩ Fc +tr, +XZ = F ∩ Ftr +and +Z◦ = {supp (h) ⊆ Y (A[θ])}. +♦ +Remark 5.10. By construction, the four events Y, U, V, XZ are pairwise disjoint, and their union Y ⊎ U ⊎ V ⊎ XZ +gives the whole sample space. +■ +In the following two Lemmas 5.11 and 5.12, we first show that for each choice of W ∈ {Y, U, V, XZ}, the +probability that n + 1 does not have the type corresponding to W on W is small and then that the probability that +n + 1 /∈ Z(Ah[θ]) on Z◦ is small. This offers an almost complete description of the type of n + 1 in terms of the +events in Definition 5.9. Lemma 5.11 deals with the simpler cases W ∈ {Y, U, V, XZ}, where the type events are +intersections of basic events. +Lemma 5.11. For any δ > 0, L ∈ N≥2, A ∈ Fn×n and W ∈ {Y, U, V }, +P +� +n + 1 /∈ W +� +Ah[θ] +� +, W +� +≤ 2L!δ + 2P (h ≥ L) + on,P (1), +(5.20) +as well as +P +� +n + 1 ̸∈ X +� +Ah[θ] +� +∪ Z +� +Ah[θ] +� +, XZ +� += on(1). +(5.21) +28 + +The rank of sparse symmetric matrices over arbitrary fields +Proof. We show the claim for each of the possible variable types separately. +Completely frozen variables - (5.20) for W = Y : By definition, if n + 1 is not completely frozen in Ah[θ], then it +is not firmly frozen in Ah[θ] or not firmly frozen in Ah[θ]T . Since Lemma 5.8 also applies to Ah[θ]T , a union bound +gives +P +� +n + 1 ̸∈ Y +� +Ah[θ] +� +, Y +� +≤ P +� +n + 1 not firmly frozen in Ah[θ], Fc +tr +� ++ P +� +n + 1 not firmly frozen in Ah[θ]T , Fc� +≤ 2L!δ + 2P (h ≥ L) + on,P (1). +One-sided firmly frozen variables - (5.20) for W ∈ {U, V }: If n + 1 ̸∈ U +� +Ah[θ] +� +, then, by definition, either n + 1 +is not firmly frozen in Ah[θ]T , or, if this is not the case, it is frozen in Ah[θ] and firmly frozen in Ah[θ]T . In the latter +case, the symmetry of frailly frozen variables under transposition (see Proposition 4.5) implies that n + 1 is also firmly +frozen in Ah[θ]. We conclude that if n + 1 ̸∈ U +� +Ah[θ] +� +, then either n + 1 is not firmly frozen in Ah[θ]T or n + 1 is +firmly frozen in Ah[θ]. Again, by a union bound and Lemma 5.8, +P +� +n + 1 ̸∈ U +� +Ah[θ] +� +, U +� +≤ P +� +n + 1 not firmly frozen in Ah[θ]T , Fc� ++ P +� +n + 1 firmly frozen in Ah[θ], Ftr +� +≤ L!δ + P (h ≥ L) + on,P (1). +The claim for W = V follows analogously. +Frailly frozen or two-sided non-frozen variables - (5.21): If n + 1 ̸∈ X +� +Ah[θ] +� +∪ Z +� +Ah[θ] +� +, then by definition, +n + 1 is firmly frozen in Ah[θ] or Ah[θ]T . By a union bound and Lemma 5.8, +P +� +n + 1 ̸∈ X +� +Ah[θ] +� +∪ Z +� +Ah[θ] +� +, XZ +� +≤ P +� +n + 1 firmly frozen in Ah[θ], Ftr +� ++ P +� +n + 1 firmly frozen in Ah[θ]T , F +� += on(1). +We have the following analogous lemma for the event Z◦: +Lemma 5.12. For any L ∈ N≥2 and A ∈ Fn×n, +P +� +n + 1 ̸∈ Z +� +Ah[θ] +� +, Z◦ +� +≤ P (h ≥ L) + on,P (1). +Proof. The first (and main) step is to prove that on the intersection of Z◦ with a sufficiently likely event, the (n + 1)st +row in Ah[θ] can be linearly combined by the other rows of Ah[θ], from which it follows through Lemma 4.1 that +n + 1 is not frozen in Ah[θ]T . +On Z◦ ∩ Pn, Ah[θ](n + 1, ) = (h 01×(θc+1)) and all variables in supp (h) = supp(Ah[θ](n + 1, )) are firmly +frozen in A[θ]. Ideally, to derive the desired linear combination of Ah[θ](n + 1, ) by the other rows of Ah[θ], we +would like to take one representation for each i ∈ supp (h), and then simply sum over the representations. Alas, the +matrix Ah[θ] has one more column than A[θ], and it is not clear that for the existing representations, also the entries of +column n + 1 sum to zero. Therefore, we are looking for representations of i ∈ supp (h) that expressly do not use one +of the rows in supp (h), if such representations exist. +In fact, on Z◦, since any i ∈ supp (h) is firmly frozen in A[θ], there exists a representation of i that does not use +row i. To take care of the other rows corresponding to elements of supp (h), we define the event +C = {for all i ∈ supp (h), i /∈ F (A[θ] ⟨supp (h) ;⟩) ∆F (A[θ] ⟨i;⟩)} . +The event C is sufficiently likely for our purposes, as +P (Cc) ≤P (h ≥ L) + +� +i∈[n] +L−1 +� +k=2 +L +n P (i ∈ F (A[θ] ⟨supp (h) ;⟩) ∆F (A[θ] ⟨i;⟩) |i ∈ supp (h) , h = k) +· P (i ∈ supp (h) h = k) +≤P (h ≥ L) + L2 +P + on(1). +Here, in the last step, we have used Corollary 4.20, which states that for any i ∈ [n] and k ≤ L, +P (i ∈ F (A[θ] ⟨supp (h) ;⟩) ∆F (A[θ] ⟨i;⟩) |i ∈ supp (h) , h = k) ≤ L +P + on(1). +By design, on the event C ∩ Z◦, any i ∈ supp (h) is frozen in A[θ] ⟨supp (h) ;⟩. In particular, there exists a +representation of {i} in A[θ] ⟨supp (h) ;⟩. On the good event Pn, each such representation can be extended to a +representation b = (b1, . . . , bn+θr) of {i} in Ah[θ] ⟨n + 1;⟩ such that +bAh[θ] ⟨n + 1;⟩ = en+θc(i) +and +bk = 0 for k ∈ supp (h). +(5.22) +29 + +The rank of sparse symmetric matrices over arbitrary fields +Thus, on the event Z◦ ∩ C ∩ Pn, any i ∈ supp (h) is frozen in Ah[θ] ⟨n + 1;⟩. We conclude that the (n + 1)st row +in Ah[θ] can be linearly combined by the other rows of Ah[θ] (this is also true if supp (h) = ∅). Therefore, by +Lemma 4.1, n + 1 is not frozen in Ah[θ]T , which only leaves the possibility n + 1 ∈ V +� +Ah[θ] +� +∪ Z +� +Ah[θ] +� +on +Z◦ ∩ C ∩ Pn. +On the other hand, since Z◦ ⊆ Ftr, by (5.18), n + 1 cannot be firmly frozen in Ah[θ] on the event Z◦ ∩ C ∩ Pn. +Therefore, n + 1 ∈ Z +� +Ah[θ] +� +and we arrive at +P +� +n + 1 ̸∈ Z +� +Ah[θ] +� +, Z◦, C +� += on(1), +i.e., +P +� +n + 1 ̸∈ Z +� +Ah[θ] +� +, Z◦ +� +≤ P (Cc) + on(1) ≤ P (h ≥ L) + L2 +P + on(1) = P (h ≥ L) + on,P (1). +This yields the claim. +5.2.2 +Probabilities of type events +In Section 5.2.1, we have related the type of n + 1 in Ah[θ] to the occurrence of a bunch of type events, which +are formulated in terms of supp (h). We now approximate the conditional probabilities of the type events through the +corresponding functions Y, U, V from Definition 5.2 and ψ. In this way, we build the connection between the event +� +n + 1 ∈ W +� +Ah[θ] +�� +and W(ζ, ψ). For the current section, recall that we use boldface letters w to abbreviate the +proportions w(A[θ]). We then show that conditionally on the vector ζ, for any W ∈ {Y, U, V }, the function W(ζ, ψ) +is a good approximation of the probability of W, while ψ is a good approximation of Z◦. Since the type events are +defined solely in terms of the membership of supp (h) in the sets W(A[θ]) and h is chosen independently of A[θ], this +basically reduces to a comparison between drawing supp (h) with and without replacement. +Lemma 5.13. For any L ∈ N≥2, W ∈ {Y, U, V }, +|P (W|ζ) − W(ζ, ψ)| ≤ P (h ≥ L) + on,P (1) +(5.23) +and +|P (Z◦|ζ) − ψ (y)| ≤ P (h ≥ L) + on,P (1). +(5.24) +Proof. +(i) We first prove (5.23) for W = Y . +Recall from Definition 5.9 that Y = Fc ∩ Fc +tr. By the inclusion-exclusion principle, +P (Y) = P (Fc) + P (Fc +tr) − P (Fc ∪ Fc +tr) = 1 − P (F) − P (Ftr) + P (F ∩ Ftr) . +(5.25) +Moreover, by Definitions 5.7 and 2.13, +(a) F coincides with the event that {supp (h) ⊆ X (A[θ]) ∪ Y (A[θ]) ∪ V (A[θ])}, +(b) Ftr coincides with the event that {supp (h) ⊆ X (A[θ]) ∪ Y (A[θ]) ∪ U (A[θ])} and +(c) F ∩ Ftr coincides with the event that {supp (h) ⊆ X (A[θ]) ∪ Y (A[θ])}. +Given the number of non-zero entries h of h, the positions of these non-zero entries are chosen uniformly at +random from all h-subsets of [n], and independently of A[θ]. Moreover, by [24], for any k ≥ 0, +����� +�(x+y+v)n +n +� +�n +k +� +− (x + y + v)k + +�(x+y+u)n +n +� +�n +k +� +− (x + y + u)k − +�(x+y)n +n +� +�n +k +� ++ (x + y)k +����� ≤ 3k(k − 1) +2n +. +(5.26) +Thus +|P (Y|ζ) − Y (ζ, ψ)| +≤ +∞ +� +k=0 +P (h = k) +����� +�(x+y+v)n +n +� +�n +k +� +− (x + y + v)k + +�(x+y+u)n +n +� +�n +k +� +− (x + y + u)k − +�(x+y)n +n +� +�n +k +� ++ (x + y)k +����� +≤ P (h ≥ L) + 3L(L − 1) +2n += P (h ≥ L) + on,P (1). +(ii) We next prove (5.23) for W = U. +Recall from Definition 5.9 that U = Fc ∩ Ftr. Moreover, +(a) Fc coincides with the event that {supp (h) ∩ (U (A[θ]) ∪ Z (A[θ])) ̸= ∅} and +(b) Ftr coincides with the event that {supp (h) ⊆ X (A[θ]) ∪ Y (A[θ]) ∪ U (A[θ])}. +30 + +The rank of sparse symmetric matrices over arbitrary fields +In other words, U coincides with the event that supp (h) is a subset of X (A[θ]) ∪ Y (A[θ]) ∪ U (A[θ]), but +not of X (A[θ]) ∪ Y (A[θ]). As before, using the total variation estimate between sampling with and without +replacement of [24], for any k ≥ 0, +����� +�(x+y+u)n +n +� +�n +k +� +− +�(x+y)n +n +� +�n +k +� +− +� +(x + y + u)k − (x + y)k� +����� ≤ 2k(k − 1) +2n +. +(5.27) +Thus, +|P (U|ζ) − U(ζ, ψ)| = +∞ +� +k=0 +P (h = k) +����� +�(x+y+u)n +n +� +�n +k +� +− +�(x+y)n +n +� +�n +k +� +− +� +(x + y + u)k − (x + y)k� +����� +≤ P (h ≥ L) + L(L − 1) +n += P (h ≥ L) + on,P (1). +(iii) By symmetry, (5.23) for W = V follows as in (ii). +(iv) We finally prove (5.24). +Given the number of non-zero entries h of h, the positions of these non-zero entries are chosen uniformly at +random from all h-subsets of [n], and independently of A[θ]. Thus, conditionally on h and the proportions +of types ζ in A[θ], the event Z◦ holds if and only if all of these h positions are chosen from the set Y(A[θ]). +Therefore, +P (Z◦|ζ, h) = +�ny +h +� +�n +h +� . +(5.28) +On the other hand, by [24], for any fixed k ≥ 0, +����� +�ny +k +� +�n +k +� − yk +����� ≤ k(k − 1) +2n +. +(5.29) +Therefore, for any L ∈ N≥2, +|P (Z◦|ζ) − ψ(y)| ≤ +∞ +� +k=0 +P (h = k) +����� +�ny +k +� +�n +k +� − yk +����� ≤ P (h ≥ L) + sup +0≤k≤L +����� +�ny +k +� +�n +k +� − yk +����� +≤ P (h ≥ L) + L(L − 1) +2n += P (h ≥ L) + on,P (1). +5.2.3 +Approximating the type probabilities for column n + 1: Proof of Proposition 5.5 +With the results of the previous two subsections, we are now in the position to prove Proposition 5.5. +Proof of Proposition 5.5. At least one-sided firmly frozen variables - proof of (5.11): For W ∈ {Y, U, V }, by the +triangle inequality, +E +��P +� +n + 1 ∈ W +� +Ah[θ] +� ��ζ +� +− W(ζ, ψ) +�� ≤ E +��P +� +n + 1 ∈ W +� +Ah[θ] +� ��ζ +� +− P (W|ζ) +�� + E |P (W|ζ) − W(ζ, ψ)| . +(5.30) +We bound both summands on the right hand side of (5.30) separately, beginning with the first. By conditional Jensen’s +inequality and the tower property, +E +���P +� +n + 1 ∈ W +� +Ah[θ] +� ���ζ +� +− P (W|ζ) +��� ≤ E +��1 +� +n + 1 ∈ W +� +Ah[θ] +�� +− 1W +�� +(5.31) +≤ P +� +n + 1 ∈ W +� +Ah[θ] +� +, Wc� ++ P +� +n + 1 ̸∈ W +� +Ah[θ] +� +, W +� +. +Now, let IW = {Y, U, V, XZ} \ {W}. Since the type events apart from Z◦ are pairwise disjoint (see Remark 5.10), +Wc = +� +W′∈IW +W′. +(5.32) +Thus, with (5.32) and using the abbreviation XZ +� +Ah[θ] +� +to denote the union X +� +Ah[θ] +� +∪ Z +� +Ah[θ] +� +, we obtain +P +� +n + 1 ∈ W +� +Ah[θ] +� +, Wc� +≤ +� +W ′∈IW +P +� +n + 1 ∈ W +� +Ah[θ] +� +, W′� +≤ +� +W ′∈IW +P +� +n + 1 /∈ W′ � +Ah[θ] +� +, W′� +. +(5.33) +31 + +The rank of sparse symmetric matrices over arbitrary fields +Plugging (5.33) into (5.31) and using Lemma 5.11 on all four summands yields +E +���P +� +n + 1 ∈ W +� +Ah[θ] +� ���ζ +� +− P (W|ζ) +��� ≤ 6L!δ + 6P(h ≥ L) + on,P (1). +(5.34) +Finally, the upper bound on the second summand E |P (W|ζ) − W(ζ, ψ)| on the right hand side of (5.30) follows +immediately from (5.23) in Lemma 5.13. Plugging the two bounds (5.23) and (5.34) into (5.30) gives (5.11). +Nowhere frozen variables - proof of (5.12): Since (a + b)− ≤ a− + b− and a− ≤ |a|, +E +�� +P +� +n + 1 ∈ Z +� +Ah[θ] +� ��ζ +� +− ψ (y) +�−� +≤E +�� +P +� +n + 1 ∈ Z +� +Ah[θ] +� ��ζ +� +− P (Z◦|ζ) +�−� ++ E |P (Z◦|ζ) − ψ (y)| +≤P +� +n + 1 ̸∈ Z +� +Ah[θ] +� +, Z◦ +� ++ E |P (Z◦|ζ) − ψ (y)| . +Equation (5.12) now follows from Lemma 5.12 and (5.24) in Lemma 5.13. +6 +Analysis of the rank-difference +In Section 2, we have reduced the lower bound of Theorem 2.2 to Propositions 2.11, 2.14 and Lemma 2.15. This +section is devoted to the proof of those three results, of which Proposition 2.14 requires the most efforts. Our starting +points here are Proposition 5.1, the fixed point equations for the proportions of frozen types, and (4.14): +yn,t/n = 1 − φt(xn,t/n + yn,t/n + un,t/n) − φt(xn,t/n + yn,t/n + vn,t/n) + φt(xn,t/n + yn,t/n) + ¯oP(1); (6.1) +un,t/n = φt(xn,t/n + yn,t/n + un,t/n) − φt(xn,t/n + yn,t/n) + ¯oP(1); +(6.2) +vn,t/n = φt(xn,t/n + yn,t/n + vn,t/n) − φt(xn,t/n + yn,t/n) + ¯oP(1); +(6.3) +zn,t/n ≥ φt(yn,t/n) + ¯oP(1); +(6.4) +xn,t/n + yn,t/n + zn,t/n + un,t/n + vn,t/n = 1. +(6.5) +The combination of eqs. (6.1), (6.2) and (6.5) gives that +xn,t/n + zn,t/n = 1 − yn,t/n − un,t/n − vn,t/n = φt +� +xn,t/n + yn,t/n +� ++ ¯oP(1); +(6.6) +Equations (6.1) to (6.6), as well as Proposition 4.11, are the main results from the previous sections and the proofs in +this section highly depend on them. +6.1 +The rank increase: Proof of Proposition 2.11 +Recall the function ht : [0, 1] → R, ht (α) = α + 1 − φt (α) from (2.13) as well as Proposition 2.11 from +Section 2.5: +Proposition 2.11 (The rank increase). For any d > 0, +E +� +rkF +� +Tn+1,t/n[θ] +� +− rkF +� +Tn,t/n[θ] +�� += E +� +ht +� +αn,t/n +�� ++ on,P (1), uniformly in t ∈ [0, d]. +(2.14) +Proof. Recall the good event Pn from (4.20). On Pn, the matrix Tn,t/n[θ] arises from the matrix Tn+1,t/n[θ] through +removal of the (n + 1)st row and column, and therefore, Lemma 4.7 gives the following representation of their rank +difference in terms of the type of n + 1: +rkF +� +Tn+1,t/n[θ] +� +− rkF +� +Tn,t/n[θ] +� +=1 +� +n + 1 ∈ X +� +Tn+1,t/n[θ] +�� ++ 2 · 1 +� +n + 1 ∈ Y +� +Tn+1,t/n[θ] +�� ++ 1 +� +n + 1 ∈ U +� +Tn+1,t/n[θ] +�� ++ 1 +� +n + 1 ∈ V +� +Tn+1,t/n[θ] +�� +. +On the other hand, in any case, +��rkF +� +Tn+1,t/n[θ] +� +− rkF +� +Tn,t/n[θ] +��� ≤ 2P + 2, since both matrices can be obtained +from Tn,t/n by adding at most P + 1 rows and at most P + 1 columns. By (4.21), the above equation holds with high +probability. Hence, +E +� +rkF +� +Tn+1,t/n[θ] +� +− rkF +� +Tn,t/n[θ] +�� += P +� +n + 1 ∈ X +� +Tn+1,t/n[θ] +�� ++ 2 · P +� +n + 1 ∈ Y +� +Tn+1,t/n[θ] +�� ++ P +� +n + 1 ∈ U +� +Tn+1,t/n[θ] +�� ++ P +� +n + 1 ∈ V +� +Tn+1,t/n[θ] +�� ++ on(1). +(6.7) +On the other hand, by Corollary 4.16, for any W ∈ {X, Y, Z, U, V} and any i ∈ [n + 1], P(i ∈ W +� +Tn+1,t/n[θ] +� +) = +E[wn+1,t/n], and Proposition 4.11 shows that +E +� +wn+1,t/n +� += E +� +wn,t/n +� ++ on,P (1), uniformly in t ∈ [0, d]. +32 + +The rank of sparse symmetric matrices over arbitrary fields +Therefore, (6.7) reduces to +E +� +rkF +� +Tn+1,t/n[θ] +� +− rkF +� +Tn,t/n[θ] +�� += E +� +xn,t/n + 2yn,t/n + un,t/n + vn,t/n +� ++ on,P (1), +uniformly in t ∈ [0, d]. Since αn,t/n = xn,t/n + yn,t/n + vn,t/n as observed in in (4.15), the combination of eqs. (6.1) +and (6.2) gives that +E +� +rkF +� +Tn+1,t/n[θ] +� +− rkF +� +Tn,t/n[θ] +�� +=E +�� +xn,t/n + yn,t/n + vn,t/n +� ++ +� +yn,t/n + un,t/n +�� ++ on,P (1) +=E +� +αn,t/n + 1 − φt +� +αn,t/n +�� ++ on,P (1) = E +� +ht +� +αn,t/n +�� ++ on,P (1), +uniformly in t ∈ [0, d], as desired. +6.2 +Lower bound on the rank increase: Proof of Proposition 2.14 +In this section, we prove Proposition 2.14: +Proposition 2.14 (Lower bound on the rank increase). For any d > 0, +ht +� +αn,t/n +� +≥ ht (α⋆(t)) + ¯oP(1). +(2.19) +The proof of Proposition 2.14 heavily depends on the properties of the function Gt defined in (2.17) and its zeroes: +Recall that Gt : [0, 1] → R, +Gt(α) = α + φt (1 − φt (α)) − 1 +and α⋆(t) and α⋆(t) were defined as the smallest and the largest zeroes of Gt in [0, 1], respectively. Moreover, α0(t) +denotes the unique zero of the increasing function Ξt : [0, 1] → R, Ξt(α) = α + φt(α) − 1, which is also always a +zero of Gt (see Lemma A.1). With this terminology, we note the following properties of Gt and its zeroes: +Lemma 6.1 (Useful properties of Gt and its zeroes; see [9, Section 3]). +1. For t ∈ [0, e], Gt is strictly increasing and has a unique zero: α⋆(t) = α0(t) = α⋆(t). +2. For t ∈ (e, ∞), Gt has exactly three distinct zeroes α⋆(t) < α0(t) < α⋆(t), and α0(t) ≥ 1 − ln t/t. +3. For all t ≥ 0, α⋆(t) = 1 − φt (α⋆(t)) and α⋆(t) = 1 − φt (α⋆(t)). +4. For t ∈ (e, ∞), Gt is positive on (α⋆(t), α0(t)) ∪ (α⋆(t), 1] and negative on [0, α⋆(t)) ∪ (α0(t), α⋆(t)). +Moreover, Gt is strictly increasing on [α⋆(t), 1]. +5. For t ̸= e, Gt and G′ +t have no common zero. For t = e, their unique common zero is given by α0(e) = 1−1/e. +6. For all t > 0 and α ∈ [0, 1] \ {α⋆(t), α⋆(t)}, Rt(α⋆(t)) = Rt(α⋆(t)) < Rt(α). +7. The functions t �→ α⋆(t), t �→ α0(t) and t �→ α⋆(t) are differentiable on [0, ∞) with continuous derivatives +on (0, e) ∪ (e, ∞). +8. Let (bn,P,N,JN,t)n,P,N∈Z+,JN∈SymN(F∗),t∈[0,d] ⊆ [0, 1] be an arbitrary family of random variables. If +Gt(bn,P,N,JN,t) = ¯oP(1), then also +min {|bn,P,N,JN,t − α⋆(t)| , |bn,P,N,JN,t − α0(t)| , |bn,P,N,JN,t − α⋆(t)|} = ¯oP(1). +We emphasize that a large part of Lemma 6.1 is covered by the results of [9, Section 3]. On the other hand, several +of the specific properties that we need only arise in proofs, and are correspondingly difficult to cite. For the sake of +completeness and easy reference, we therefore give a proof of all properties that we need in Appendix A. +As a last preparation for the proof of Proposition 2.14, we prove two short lemmas on the evaluation of ht +at specific points: The first lemma shows that α⋆(t) and α⋆(t) minimize ht among the zeroes of Gt. It is a direct +consequence of Lemma 6.1: +Lemma 6.2. For any t ≥ 0, +ht (α⋆(t)) = ht (α⋆(t)) ≤ ht (α0(t)) . +(6.8) +Proof. By items 1 and 2 in Lemma 6.1, +α⋆(t) = α0(t) = α⋆(t) for t ≤ e +and +α⋆(t) > α0(t) ≥ 1 − ln t/t for t > e. +Taking the derivative of ht w.r.t. α, we have h′ +t(α) = 1 − tφt(α) = 1 − tet(α−1), so +α �→ ht(α) is a strictly increasing function on [0, 1 − ln t/t] and a strictly decreasing function on [1 − ln t/t, 1] +(6.9) +and thus ht (α⋆(t)) ≤ ht (α0(t)). It thus only remains to show that ht (α⋆(t)) = ht (α⋆(t)). By item 3 in Lemma 6.1, +1 − α⋆(t) = φt(α⋆(t)) and 1 − α⋆(t) = φt(α⋆(t)). It now directly follows that +ht (α⋆(t)) = α⋆(t) + 1 − φt (α⋆(t)) = α⋆(t) + α⋆(t) = α⋆(t) + 1 − φt (α⋆(t)) = ht (α⋆(t)) . +33 + +The rank of sparse symmetric matrices over arbitrary fields +The second lemma is a consequence of the type fixed point equations (6.1) to (6.6): +Lemma 6.3. For any d > 0, +ht +� +xn,t/n + yn,t/n + un,t/n +� += ht +� +xn,t/n + yn,t/n + vn,t/n +� ++ ¯oP(1) = ht +� +xn,t/n + yn,t/n +� ++ ¯oP(1) +≤ ht +� +yn,t/n +� ++ ¯oP(1). +(6.10) +Proof. The first and second equalities in (6.10) follow directly from eqs. (6.2) and (6.3). +On the other hand, the combination of eqs. (6.4) and (6.6) gives +ht +� +xn,t/n + yn,t/n +� += xn,t/n + yn,t/n + 1 − φt +� +xn,t/n + yn,t/n +� += yn,t/n + 1 − zn,t/n + ¯oP(1) +≤ yn,t/n + 1 − φt +� +yn,t/n +� ++ ¯oP(1) = ht +� +yn,t/n +� ++ ¯oP(1), +and thus the last inequality in (6.10) follows. +With Lemmas 6.2 and 6.3 in hand, we are finally in the position to prove Proposition 2.14. +Proof of Proposition 2.14. Define +¯τn = 1 +� +h′ +t(xn,t/n + yn,t/n) ≥ 0 +� +and +¯ηn = 1 − ¯τn = 1 +� +h′ +t(xn,t/n + yn,t/n) < 0 +� +. +(6.11) +Since ¯τn + ¯ηn = 1, we divide equation (2.19) into two parts as follows: +¯τn +� +ht (α⋆(t)) − ht +� +αn,t/n +�� +≤ ¯oP(1), +(6.12) +and +¯ηn +� +ht (α⋆(t)) − ht +� +αn,t/n +�� +≤ ¯oP(1). +(6.13) +In the absence of the error terms ¯oP(1) and under the assumption that ¯τn ≡ 1 (or ¯ηn ≡ 1), the proof of Proposition 2.14 +would amount to an analytic treatment of the properties of ht. Unfortunately, we have to deal with the error terms and +both cases. In the ensuing argument, we therefore fall back upon Taylor’s Theorem with Lagrange Remainder, item 8 in +Lemma 6.1 and the following two facts: +(i) For any function g and υ ∈ {0, 1}, if υa = υb, then υg(a) = υg(b); +(ii) For a family of differentiable functions (gt)t∈[0,d], if there exists a uniform bound b such that +supt∈[0,d],s∈[0,1] |g′ +t(s)| ≤ b, then gt(a′ +t) = gt(at) + ¯oP(1) for any random variables at, a′ +t ∈ [0, 1], +a′ +t = at + ¯oP(1) ∈ [0, 1] and t ∈ [0, d], since |gt(a′ +t) − gt(at)| ≤ b |a′ +t − at|. +1. Proof of (6.12). By definition of ¯τn, ¯τnh′ +t +� +xn,t/n + yn,t/n +� +≥ 0. +Fix ε ∈ (0, 1/d) and let b = +inft∈[εd,d] +� +t2e−t� +> 0, such that supα∈[0,1] h′′ +t (α) ≤ −b for t ∈ [εd, d]. Then by Taylor’s Theorem +with Lagrange remainder, for t ∈ [εd, d], +¯τnht(yn,t/n) ≤ ¯τn +� +ht +� +xn,t/n + yn,t/n +� +− h′ +t +� +xn,t/n + yn,t/n +� +xn,t/n − b +2x2 +n,t/n +� +≤ ¯τnht +� +xn,t/n + yn,t/n +� +− ¯τn1{t ∈ [εd, d]} b +2x2 +n,t/n. +Analogously, let c = 1 − εd > 0 such that infα∈[0,1] h′ +t(α) ≥ c for t ∈ [0, εd). Then by Taylor’s Theorem +with Lagrange remainder, for t ∈ [0, εd), +¯τnht(yn,t/n) ≤¯τn +� +ht +� +xn,t/n + yn,t/n +� +− cxn,t/n +� +. +On the other hand, (6.10) shows that for all t ∈ [0, d], +¯τnht(yn,t/n) ≥ ¯τnht +� +xn,t/n + yn,t/n +� ++ ¯oP(1). +(6.14) +Since xn,t/n ∈ [0, 1], (6.14) implies that +min +� b +2, c +� +¯τnx2 +n,t/n ≤ ¯τn1{t ∈ [εd, d]} b +2x2 +n,t/n + ¯τn1{t ∈ [0, εd)}cxn,t/n ≤ ¯oP(1), +and we conclude that that ¯τnx2 +n,t/n = ¯oP(1). The Cauchy-Schwarz inequality E[¯τnxn,t/n] ≤ E[¯τnx2 +n,t/n]1/2 +then yields +¯τnxn,t/n = ¯oP(1). +(6.15) +34 + +The rank of sparse symmetric matrices over arbitrary fields +Since αn,t/n = xn,t/n + yn,t/n + vn,t/n and αT +n,t/n = xn,t/n + yn,t/n + un,t/n, (6.15) in combination +with (6.1) and (6.3) implies that +¯τnαn,t/n = ¯τn +� +yn,t/n + vn,t/n + ¯oP(1) +� += ¯τn +� +1 − φt +� +xn,t/n + yn,t/n + un,t/n +� ++ ¯oP(1) +� += ¯τn(1 − φt(αT +n,t/n) + ¯oP(1)). +Analogously, +(6.15) +in +combination +with +(6.1) +and +(6.2) +implies +that +¯τnαT +n,t/n += +¯τn +� +1 − φt +� +αn,t/n +� ++ ¯oP(1) +� +. Hence, +¯τnαn,t/n = ¯τn(1 − φt(αT +n,t/n) + ¯oP(1)) = ¯τn(1 − φt +� +1 − φt +� +αn,t/n +�� ++ ¯oP(1)), +i.e., +¯τnGt +� +αn,t/n +� += ¯oP(1). +(6.16) +Let βn,t/n = ¯τnαn,t/n + ¯ηnα⋆(t). Since Gt(α⋆(t)) = 0, (6.16) implies that +Gt(βn,t/n) = ¯τnGt +� +αn,t/n +� ++ ¯ηnGt (α⋆(t)) = ¯oP(1). +(6.17) +Hence, item 8 in Lemma 6.1 implies that +min +���βn,t/n − α⋆(t) +�� , +��βn,t/n − α0(t) +�� , +��βn,t/n − α⋆(t) +��� += ¯oP(1). +By Lemma 6.2, ht (α⋆(t)) = ht (α⋆(t)) ≤ ht (α0(t)), so +ht (α⋆(t)) ≤ ht +� +βn,t/n +� ++ ¯oP(1) = ¯τnht +� +αn,t/n +� ++ ¯ηnht (α⋆(t)) + ¯oP(1), +and (6.12) follows immediately. +2. Proof of (6.13). By definition of ¯ηn, ¯ηnh′ +t +� +xn,t/n + yn,t/n +� +< 0. Since the function ht is strictly increasing +on [0, 1 − ln(t)/t], this implies that ¯ηn +� +xn,t/n + yn,t/n +� +≥ ¯ηn (1 − ln t/t), so that +¯ηnαn,t/n ≥ ¯ηn +� +xn,t/n + yn,t/n +� +≥ ¯ηn (1 − ln t/t) . +(6.18) +Another application of Taylor’s Theorem with Lagrange remainder to (6.2) and (6.3) as in the argument leading +to (6.15) yields that ¯ηnun,t/n = ¯oP(1) and ¯ηnvn,t/n = ¯oP(1). Hence, by (6.1), +¯ηnyn,t/n = ¯ηn(1 − φt(xn,t/n + yn,t/n) + ¯oP(1)) and ¯ηnαn,t/n = ¯ηn(xn,t/n + yn,t/n + ¯oP(1)). (6.19) +Let +β′ +n,t/n = ¯ηn1 +� +αn,t/n > α⋆(t) +� +αn,t/n + +� +1 − ¯ηn1 +� +αn,t/n > α⋆(t) +�� +α⋆(t). +Then β′ +n,t/n ≥ α⋆(t). Since Gt(α⋆(t)) = 0, by eqs. (6.4), (6.5) and (6.19), +Gt(β′ +n,t/n) =¯ηn1 +� +αn,t/n > α⋆(t) +� � +αn,t/n + φt +� +1 − φt +� +αn,t/n +�� +− 1 +� +≤¯ηn1 +� +αn,t/n > α⋆(t) +� � +xn,t/n + yn,t/n + zn,t/n − 1 + ¯oP(1) +� +≤ ¯oP(1). +On the other hand, by item 4 in Lemma 6.1, Gt is strictly increasing on [α⋆(t), 1]. Hence +Gt(β′ +n,t/n) ≥ Gt (α⋆(t)) = 0, +so that +Gt(β′ +n,t/n) = ¯oP(1). +Then the combination of item 8 in Lemma 6.1 and β′ +n,t/n ≥ α⋆(t) yields that β′ +n,t/n = α⋆(t) + ¯oP(1), which +leads to +¯ηnαn,t/n ≤ ¯ηnβ′ +n,t/n = ¯ηn (α⋆(t) + ¯oP(1)) . +(6.20) +Hence, by (6.9), (6.18) and (6.20), +¯ηnht +� +αn,t/n +� +≥ ¯ηnht +� +β′ +n,t/n +� += ¯ηn (ht (α⋆(t)) + ¯oP(1)) , +and thus (6.13) holds. +The combination of eqs. (6.12) and (6.13) gives (2.19). +35 + +The rank of sparse symmetric matrices over arbitrary fields +6.3 +Integral evaluation: Proof of Lemma 2.15 +In this section, we prove Lemma 2.15 from Section 2.8: +Lemma 2.15 (Integral evaluation). For any d ≥ 0, +� d +0 +ht (α⋆(t)) dt = d · Rd (α⋆(d)) . +Proof. Throughout the proof, we use the abbreviations +q(d) = +� d +0 +ht(α⋆(t)) dt = +� d +0 +(α⋆(t) − φt (α⋆(t)) + 1) dt +and +r(d) = d · Rd(α⋆(d)) = 2d − dφd (1 − φd(α⋆(d))) − dφd (α⋆(d)) − d2φd (α⋆(d)) (1 − α⋆(d)) . +We have q(0) = r(0) = 0. Moreover, by item 7 in Lemma 6.1, the function t �→ α⋆(t) is continuous on [0, ∞), which +then transfers to the functions d �→ q(d) and d �→ r(d). In order to prove that q(d) = r(d) for all d ≥ 0, it is thus +sufficient to certify that q′(d) = r′(d) for all d ∈ (0, e) ∪ (e, ∞). +Recall that the derivative of d �→ α⋆(d) is continuous on (0, e) ∪ (e, ∞) by item 7 in Lemma 6.1. To derive an +expression for r′(d), we compute the partial derivatives of the function (d, α) �→ φd (1 − φd(α)) at (d, α⋆(d)) for +d ̸= e, where we use that α⋆(d) is a zero of Gd, i.e., φd (1 − φd (α⋆(d))) = 1 − α⋆(d): +∂ +∂dφd (1 − φd(α)) +��� +α=α⋆(d) = − φd (α⋆(d)) φd (1 − φd (α⋆(d))) (1 − d (1 − α⋆(d))) += − φd (α⋆(d)) (1 − α⋆(d)) (1 − d (1 − α⋆(d))) +as well as +∂ +∂αφd (1 − φd(α)) +��� +α=α⋆(d) = −d2φd (α⋆(d)) φd (1 − φd (α⋆(d))) = −d2φd (α⋆(d)) (1 − α⋆(d)). +Hence, +r′(d) =2 − φd (1 − φd(α⋆(d))) − d ∂ +∂dφd (1 − φd(α)) +��� +α=α⋆(d) − d ∂ +∂αφd (1 − φd(α)) +��� +α=α⋆(d) +dα⋆(d) +dd +(6.21) +− φd (α⋆(d)) − dφd (α⋆(d)) +� +α⋆(d) − 1 + ddα⋆(d) +dd +� +− 2dφd (α⋆(d)) (1 − α⋆(d)) +− d2φd (α⋆(d)) +� +α⋆(d) − 1 + ddα⋆(d) +dd +� +(1 − α⋆(d)) + d2φd (α⋆(d)) dα⋆(d) +dd +. +Substituting the two partial derivatives of (d, α) �→ φd (1 − φd(α)) into (6.21), we see that the sum of the terms with +dα⋆(d)/dd on the right hand side vanishes and +r′(d) = 2 − φd (1 − φd(α⋆(d))) − φd (α⋆(d)) . +On the other hand, q′(d) = α⋆(d) − φd (α⋆(d)) + 1 = r′(d) since φd (1 − φd (α⋆(d))) = 1 − α⋆(d). Thus +r(d) = q(d) for all d ≥ 0, as desired. +Acknowledgement. +This research was supported by the European Union’s Horizon 2020 research and innovation +programme under the Marie Skłodowska-Curie grant agreement no. 945045, and by the NWO Gravitation project +NETWORKS under grant no. 024.002.003. +References +[1] L. Addario-Berry and L. Eslava. Hitting time theorems for random matrices. Combinatorics, Probability and +Computing, 23(5):635–669, 2014. +[2] M. Aizenman, R. Sims, and S. L. Starr. Extended variational principle for the Sherrington-Kirkpatrick spin-glass +model. Physical Review B, 68(21):214403, 2003. +[3] J. Aronson, A. Frieze, and B. G. Pittel. Maximum matchings in sparse random graphs: Karp–Sipser revisited. +Random Structures & Algorithms, 12(2):111–177, 1998. +[4] P. Ayre, A. Coja-Oghlan, P. Gao, and N. Müller. 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Characteristic vectors of bordered matrices with infinite dimensions. Annals of Mathematics, 62(3), +1955. +37 + +The rank of sparse symmetric matrices over arbitrary fields +Appendices +A +Useful properties of the functions Gt and Rt +For t ≥ 0, recall the rank function Rt : [0, 1] → R, +Rt(α) = 2 − φt (1 − φt(α)) − (1 + t(1 − α))φt(α), +defined in (1.2), as well as Gt : [0, 1] → R, +Gt(α) = α + φt (1 − φt(α)) − 1, +defined in (2.17). The following lemma shows that for all t ≥ 0, Gt has at least one zero α0(t), that additionally +satisfies the equation α0(t) = 1 − φt(α0(t)): +Lemma A.1 (See [9, Section 3]). For any t ≥ 0, the function Ξt : [0, 1] → R, Ξt(α) = α + φt(α) − 1, has a unique +zero α0(t). Moreover, +Gt(α0(t)) = 0. +(A.1) +Proof. We have Ξ′ +t(α) = 1 + tφt(α) > 0 as well as Ξt(0) = e−t − 1 ≤ 0 and Ξt(1) = 1 > 0. Thus, Ξt has a unique +zero α0 ∈ [0, 1]. Moreover, +Gt(α0(t)) = α0(t) − 1 + φt(1 − φt(α0(t))) = −φt(α0(t)) + φt(α0(t)) = 0. +Recall that α⋆(t) and α⋆(t) denote the smallest and largest zero of Gt, respectively. We now prove Lemma 6.1: +Lemma 6.1 (Useful properties of Gt and its zeroes; see [9, Section 3]). +1. For t ∈ [0, e], Gt is strictly increasing and has a unique zero: α⋆(t) = α0(t) = α⋆(t). +2. For t ∈ (e, ∞), Gt has exactly three distinct zeroes α⋆(t) < α0(t) < α⋆(t), and α0(t) ≥ 1 − ln t/t. +3. For all t ≥ 0, α⋆(t) = 1 − φt (α⋆(t)) and α⋆(t) = 1 − φt (α⋆(t)). +4. For t ∈ (e, ∞), Gt is positive on (α⋆(t), α0(t)) ∪ (α⋆(t), 1] and negative on [0, α⋆(t)) ∪ (α0(t), α⋆(t)). +Moreover, Gt is strictly increasing on [α⋆(t), 1]. +5. For t ̸= e, Gt and G′ +t have no common zero. For t = e, their unique common zero is given by α0(e) = 1−1/e. +6. For all t > 0 and α ∈ [0, 1] \ {α⋆(t), α⋆(t)}, Rt(α⋆(t)) = Rt(α⋆(t)) < Rt(α). +7. The functions t �→ α⋆(t), t �→ α0(t) and t �→ α⋆(t) are differentiable on [0, ∞) with continuous derivatives +on (0, e) ∪ (e, ∞). +8. Let (bn,P,N,JN,t)n,P,N∈Z+,JN∈SymN(F∗),t∈[0,d] ⊆ [0, 1] be an arbitrary family of random variables. If +Gt(bn,P,N,JN,t) = ¯oP(1), then also +min {|bn,P,N,JN,t − α⋆(t)| , |bn,P,N,JN,t − α0(t)| , |bn,P,N,JN,t − α⋆(t)|} = ¯oP(1). +Proof. +1. First observe that G0(α) = α, so G0 is strictly increasing with a unique zero in α = 0. +Taking the first and second derivative of α �→ Gt, we have +G′ +t(α) = 1 − t2φt(α)φt (1 − φt(α)) +and +G′′ +t (α) = t3φt(α)φt (1 − φt (α)) (tφt(α) − 1). +Since t > 0, G′′ +t (α) < 0 precisely when α < 1 − ln t/t, and G′′ +t (α) > 0 precisely when α > 1 − ln t/t. Thus, +for any t > 0, the first derivative α �→ G′ +t(α) is strictly decreasing on [0, 1 − ln t/t) and strictly increasing +on (1 − ln t/t, 1]. When t ∈ (0, 1], 1 − ln t/t ≥ 1 and for all α ∈ [0, 1], G′ +t(α) ≥ G′ +t(1) = 1 − t2e−t > 0. +When t ∈ (1, e], 1 − ln t/t ∈ (0, 1) and for all α ∈ [0, 1], +G′ +t(α) ≥ G′ +t(1 − ln t/t) = 1 − t/e. +(A.2) +We conclude that for all t ∈ [0, e) and α ∈ [0, 1], G′ +t(α) > 0 and that Gt is strictly increasing. In this case, Gt +has at most one zero, which is given by α0(t) from Lemma A.1. +For t = e, G′ +e has exactly one zero in α = 1 − 1/e and is positive otherwise. Thus Ge is strictly increasing +and has a unique zero, which is given by α0(e) = 1 − 1/e from Lemma A.1. +2. & 3. Suppose that t ∈ (e, ∞), so that 1 − ln t/t ∈ (0, 1). We first show that Gt has at most three zeroes. The proof +of item 1 shows that G′ +t is strictly decreasing on [0, 1 − ln t/t) and strictly increasing on (1 − ln t/t, 1] with +G′ +t(1 − ln t/t) = 1 − t/e < 0. Moreover, +G′ +t(0) = 1 − t2e−te−te−t ≥ 1 − t2e−t = G′ +t(1) > 0, +38 + +The rank of sparse symmetric matrices over arbitrary fields +where we have used that et/2 = eet/2−1 > e (1 + t/2 − 1) > t for t > e in the last step. The intermediate +value theorem now implies that G′ +t has exactly two zeroes in [0, 1], which we denote by α1(t) < α2(t). By +the above, +α1(t) < 1 − ln t/t < α2(t). +This implies for Gt that +Gt is strictly increasing on [0, α1(t)) ∪ (α2(t), 1] and strictly decreasing on (α1(t), α2(t)). +(A.3) +By the intermediate value theorem and (A.3), Gt has at most three zeroes in [0, 1]. +We next argue that for t > e, Gt has exactly three zeroes. For this, observe that +Gt(0) = e−te−t − 1 < 0 +and +Gt(1) = e−t > 0, +Gt(1 − ln t/t) = e−1 − ln t/t = e−1 � +1 − ln t/eln t−1� +> e−1 (1 − ln t/ (1 + ln t − 1)) = 0, +and +Gt(1 − 1/t) = −1/t + e−te−1 = −1/t + 1/ +� +eete−1−1� +< −1/t + 1/ +� +e +� +1 + te−1 − 1 +�� += 0. +By the intermediate value theorem, Gt has at least one zero in each of the intervals (0, 1 − ln t/t), (1 − +ln t/t, 1 − 1/t) and (1 − 1/t, 1). It follows that Gt has exactly three zeroes. +We next show that for t > e, α0(t) is neither the largest nor the smallest zero, such that the zeroes α⋆(t), α⋆(t) +and α0(t) are distinct. Let t ≥ 0. If ˙α is any zero of Gt, then ˙α = 1 − φt(1 − φt( ˙α)). This implies that +Gt(1 − φt( ˙α)) = 1 − φt( ˙α) + φt(1 − φt(1 − φt( ˙α))) − 1 = 1 − φt( ˙α) + φt( ˙α) − 1 = 0. +Therefore, 1 − φt( ˙α) is also a zero of Gt, for any t ≥ 0. +For t > e, let αM(t) be the zero of Gt that is contained in the non-empty interval (1 − ln t/t, 1 − 1/t). Since +Gt has exactly the three zeros α⋆(t) < αM(t) < α⋆(t), by the above, 1 − φt(α⋆(t)) < 1 − φt(αM(t)) < +1 − φt(α⋆(t)) are also three distinct zeroes of Gt(α). Thus we must have that +α⋆(t) = 1 − φt(α⋆(t)), +αM(t) = 1 − φt(αM(t)) +and +α⋆(t) = 1 − φt(α⋆(t)), +(A.4) +which implies that αM(t) = α0(t) (see Lemma A.1), as well as α0(t) ∈ (1 − ln t/t, 1 − 1/t). In particular, +the proof of item 2 is now complete. +Moreover, (A.4) proves item 3 for t > e. For t ∈ [0, e] on the other hand, item 3 follows from the fact that +α⋆(t) = α0(t) = α⋆(t) and Lemma A.1. +4. As shown in the proof of item 2, for t > e, α⋆(t) ∈ (0, 1 − ln t/t), α0(t) ∈ (1 − ln t/t, 1 − 1/t) and +α⋆(t) ∈ (1 − 1/t, 1) with Gt(0) < 0, Gt(1 − ln t/t) > 0, Gt(1 − 1/t) < 0 and Gt(1) > 0. This implies +the first part of the claim. Moreover, recall the two zeroes α1(t) < α2(t) of G′ +t as well as observation (A.3). +Since there can be at most one zero in each of the intervals [0, α1(t)], [α1(t), α2(t)] and [α2(t), 1] and Gt has +exactly three zeroes, we must have that α⋆(t) ∈ [0, α1(t)], α0(t) ∈ [α1(t), α2(t)] and α⋆(t) ∈ [α2(t), 1]. The +second part of item 4 now follows from (A.3). +5. Assume that ¯α(t) ∈ [0, 1] is a common zero of Gt and G′ +t. Then ¯α(t) < 1, since Gt(1) = e−t. Let +b = φt(¯α(t))/(1 − ¯α(t)) > 0. We distinguish the following two cases: +i) If b = 1, then ¯α(t) = 1−φt(¯α(t)). Since ¯α(t) is a zero of G′ +t, 0 = G′ +t(¯α(t)) = 1−t2φ2 +t(¯α(t)), i.e. t > 0 +and φt(¯α(t)) = 1/t. By definition of φt, φt(¯α(t)) = et(¯α(t)−1), so ¯α(t) = 1 − ln t/t. On the other hand, +¯α(t) = 1 − φt(¯α(t)) = 1 − 1/t. This is only possible for t = e. In this case, ¯α(e) = 1 − 1/e = α0(e), +and indeed, Ge(α0(e)) = G′ +e(α0(e)) = 0. +ii) If b ̸= 1, then +φt(¯α(t))/b = 1 − ¯α(t) = φt(1 − φt(¯α(t))) = e−tφt(¯α(t)) = +� +et(¯α(t)−1)�b += φt(¯α(t))b > 0, +where we have used Gt(¯α(t)) = 0 in the second step. Therefore, +φt(¯α(t)) = b−1/(b−1) +and +1 − ¯α(t) = b−1φt(¯α(t)) = b−b/(b−1). +(A.5) +Hence, by definition of φt and (A.5), +t = (ln φt(¯α(t))) /(¯α(t) − 1) = bb/(b−1) ln b/(b − 1). +(A.6) +Since also G′ +t(¯α(t)) = 0, we have that +0 = 1 − t2φt(¯α(t))φt (1 − φt(¯α(t))) +Gt(¯α(t))=0 += 1 − t2φt(¯α(t))(1 − ¯α(t)) = 1 − b (ln b)2 /(b − 1)2, +39 + +The rank of sparse symmetric matrices over arbitrary fields +where we have used (A.5) and (A.6) in the last step. Thus (b − 1)2 − b (ln b)2 = 0. Hence (b − 1)b−1/2 − +ln b = 0 since b − 1 and ln b have the same sign. +Let l(c) = (c − 1)c−1/2 − ln c for c > 0. Then l(b) = 0 and l(1) = 0. Taking the derivative of l, we have +l′(c) = c−1/2/2 + c−3/2/2 − 1/c ≥ 0, +with equality only if c = 1. Therefore, l is strictly increasing on (0, ∞). Since b ̸= 1, we conclude that +l(b) ̸= 0, which is a contradiction. +6. By item 3, +Rt(α⋆(t)) = 2 − φt (1 − φt(α⋆(t))) − (1 + t(1 − α⋆(t)))φt(α⋆(t)) += 2 − φt (α⋆(t)) − (1 + tφt (α⋆(t)))φt(α⋆(t)) += 2 − φt (α⋆(t)) − φt (α⋆(t)) − tφt (α⋆(t)) φt (α⋆(t)) . +Analogously, Rt(α⋆(t)) = 2 − φt (α⋆(t)) − φt (α⋆(t)) − tφt (α⋆(t)) φt (α⋆(t)), so Rt(α⋆(t)) = Rt(α⋆(t)). +One the other hand, +R′ +t(α) = t2φt (α) Gt(α), +so for t > 0, the sign of R′ +t(α) is equal to the sign of Gt(α) for α ∈ [0, 1]. +i) For t ≤ e, as shown under item 1, Gt is strictly increasing with a unique zero in α⋆(t) = α⋆(t). Therefore, +Rt obtains its unique minimum in α = α⋆(t) = α⋆(t). +ii) For t > e, item 4 shows that Rt is strictly decreasing on (0, α⋆(t))∪(α0(t), α⋆(t)) and strictly increasing +on (α⋆(t), α0(t)) ∪ (α⋆(t), 1). Therefore, Rt attains its minimum either in α⋆(t) or in α⋆(t). +We conclude that for all α /∈ {α⋆(t), α⋆(t)}, Rt(α) > Rt(α⋆(t)) = Rt(α⋆(t)). +7. We apply the implicit function theorem. Consider the two-variable function ˜G : R≥0 × [0, 1] → R, +˜G(t, α) = Gt(α) = α − 1 + e−tet(α−1). +˜G has continuous partial derivatives and thus is differentiable on R>0 × (0, 1). By Items 1 and 2, for any +t0 ≥ 0, +˜G(t0, α⋆(t0)) = 0, +˜G(t0, α0(t0)) = 0 +and +˜G(t0, α⋆(t0)) = 0, +and, by item 5, for t0 ̸= e, the partial derivative ∂α ˜G does not vanish in the respective zeroes: +∂α ˜G(t0, α⋆(t0)) ̸= 0, +∂α ˜G(t0, α0(t0)) ̸= 0 +and +∂α ˜G(t0, α⋆(t0)) ̸= 0. +For t0 ∈ (0, e) ∪ (e, ∞), the implicit function theorem provides the existence of continuously differentiable +functions t �→ β⋆(t), t �→ β0(t) and t �→ β⋆(t) defined on an open set t0 ∈ T ⊂ [0, ∞) such that +β⋆(t0) = α⋆(t0), +β0(t0) = α0(t0), +β⋆(t0) = α⋆(t0), +(A.7) +and +˜G(t, β⋆(t)) = ˜G(t, β0(t)) = ˜G(t, β⋆(t)) = 0 +for all t ∈ T. +(A.8) +Assume now that t0 ∈ (0, e). In this case, (A.8) and item 1 imply that on T ∩(0, e), β⋆, β0 and β⋆ are identical +to α0, since for any t ∈ (0, e), Gt has exactly one zero α0(t). Therefore, for any t0 ∈ (0, e), the function +t �→ α0(t) is continuously differentiable in t0. +Let now t0 ∈ (e, ∞). Since in this case, the three zeroes of Gt are distinct by Item 2, (A.7) implies that +β⋆(t0) < β0(t0) < β⋆(t0). +Since β⋆(t), β0(t) and β⋆(t) are continuous functions, we can further restrict T such that T ⊂ (e, ∞) and +β⋆(t) < β0(t) < β⋆(t) for all t ∈ T. Then the combination of (A.8) and item 2 gives that on T, +α⋆(t) = β⋆(t), +α0(t) = β0(t) +and +α⋆(t) = β⋆(t). +Therefore, for any t0 ∈ (e, ∞), the functions t �→ α⋆(t), t �→ α0(t) and t �→ α⋆(t) are continuously +differentiable in t0. +We finally consider continuity of t �→ α⋆(t) in the point t = e. Let a := lim supt→e α⋆(t) ∈ [0, 1] and suppose +that a ̸= α⋆(e). Since α⋆(e) is the only zero of Ge, ˜G(e, a) ̸= 0. As ˜G is a continuous function, there exists +δ > 0 such that for all (t, α) ∈ Uδ := [e−δ, e+δ]×[a+δ, a−δ], | ˜G(t, α)− ˜G(e, a)| ≤ | ˜G(e, a)|. In particular, +˜G(t, α) ̸= 0 for all (t, α) ∈ Uδ. On the other hand, by definition of a, there exists tδ ∈ [e − δ, e + δ] \ {e} with +α⋆(tδ) > a − δ, such that (tδ, α⋆(tδ)) ∈ Uδ. But ˜G(tδ, α⋆(tδ)) = 0, which gives the desired contradiction. +We conclude that lim supt→e α⋆(t) = α⋆(e). +Analogously, it can be shown that lim inft→e α⋆(t) = α⋆(e) and therefore, t �→ α⋆(t) is continuous in t = e. +Similarly, one can show that t �→ α⋆(t) is continuous in t = 0. The results for α0(t) and α⋆(t) follow along +the same lines. +40 + +The rank of sparse symmetric matrices over arbitrary fields +8. As in the proof of item 7, we use ˜G to denote the two-variable function (t, α) �→ Gt(α). For ε ≥ 0, let +Uε = {(t, α) ∈ [0, d] × [0, 1] : | ˜G(t, α)| ≤ ε} ⊆ R2, +such that +U0 = {(t, α) ∈ [0, d] × [0, 1] : α ∈ {α⋆(t), α0(t), α⋆(t)}}. +For b ∈ R2 and A ⊂ R2, let d (b, A) := infa∈A ∥a − b∥2. We next argue that +lim +ε→0 sup +x∈Uε +d (x, U0) =: lim +ε→0 ∆ε = 0. +(A.9) +Indeed, suppose that (A.9) does not hold. Then there exist δ > 0, εn ↓ 0 and xn ∈ Uεn such that for all n ≥ 1, +d (xn, U0) ≥ δ. +As a uniformly bounded sequence, (xn)n≥1 has a convergent subsequence (xnk)k≥1 with limit x∗, and +d (x∗, U0) = lim +k→∞ d (xnk, U0) ≥ δ. +However, since ˜G is continuous, | ˜G(x∗)| = limk→∞ | ˜G (xnk) | ≤ limk→∞ εnk = 0, i.e., d (x∗, U0) = 0, +which is a contradiction. Therefore, (A.9) holds. +Recall +that +we +assume +that +Gt(bn,P,N,JN,t) += +¯oP(1) +for +a +family +of +random +variables +(bn,P,N,JN,t)n,P,N∈Z+,JN∈SymN(F∗),t∈[0,d] ⊆ [0, 1]. This is equivalent to +lim inf +P →∞ lim inf +n→∞ +inf +N≥n,JN∈SymN(F∗) inf +t∈[0,d] P (|Gt(bn,P,N,JN,t)| ≤ ε) = 1 +for all ε > 0, +since for any ε +> +0 and B +:= +supn,P,N∈N,JN∈SymN(F∗),t∈[0,d] |Gt(bn,P,N,JN,t)| +< +∞, +εP (|Gt(bn,P,N,JN,t)| > ε) ≤ E |Gt(bn,P,N,JN,t)| ≤ ε + BP (|Gt(bn,P,N,JN,t)| > ε) . +We conclude that for any ε > 0, +lim inf +P →∞ lim inf +n→∞ +inf +N≥n,JN∈SymN(F∗) inf +t∈[0,d] P +� +(t, bn,P,N,SymN(F∗),t) ∈ Uε +� += 1. +The definition of ∆ε then implies that for all ε > 0, +lim inf +P →∞ lim inf +n→∞ +inf +N≥n,JN∈SymN(F∗) inf +t∈[0,d] P +� +d +� +(t, bn,P,N,SymN(F∗),t), U0 +� +≤ ∆ε +� += 1. +(A.10) +On the event {d ((t, bn,P,N,JN,t), U0) ≤ ∆ε}, there exists (˜t, ˜α) ∈ U0 with |t − ˜t| ≤ ∆ε and |bn,P,N,JN,t − +˜α| ≤ ∆ε. We next argue that for ε chosen small enough and thus ˜t close to t, also the value of ˜α ∈ +{α⋆(˜t), α0(˜t), α⋆(˜t)} is close to one of α⋆(t), α0(t) or α⋆(t): Observe that since the functions s �→ α⋆(s), +s �→ α0(s) and s �→ α⋆(s) are uniformly continuous on [0, d] by item 7, for any υ > 0, there exists ω > 0 +such that for any t1, t2 ∈ [0, d] with |t1 − t2| ≤ ω, +max {|α0(t1) − α0(t2)| , |α⋆(t1) − α⋆(t2)| , |α⋆(t1) − α⋆(t2)|} ≤ υ/2. +(A.11) +As limε↓0 ∆ε = 0, we can choose ε such that ∆ε < min {υ/2, ω}, such that in particular +��t − ˜t +�� ≤ ω +and +|bn,P,N,JN,t − ˜α| ≤ υ/2. +(A.12) +Then by equations (A.11) and (A.12), +max +���α0(t) − α0(˜t) +�� , +��α⋆(t) − α⋆(˜t) +�� , +��α⋆(t) − α⋆(˜t) +��� +≤ υ/2. +Combining the above inequality with (A.12), we conclude that +Mn,P,N,JN,t := min {|bn,P,N,JN,t − α0(t)| , |bn,P,N,JN,t − α⋆(t)| , |bn,P,N,JN,t − α⋆(t)|} ≤ υ. +Hence, for any υ > 0 and ε sufficiently small, by (A.10), +lim inf +P →∞ lim inf +n→∞ +inf +N≥n, +JN∈SymN(F∗) +inf +t∈[0,d] P (Mn,P,N,JN,t ≤ υ) +≥ lim inf +P →∞ lim inf +n→∞ +inf +N≥n, +JN∈SymN(F∗) +inf +t∈[0,d] P (d ((t, bn,P,N,JN,t), U0) ≤ ∆ε) = 1, +which, by the above equivalent characterisation of ¯oP(1)-convergence, gives the claim: Mn,P,N,JN,t = ¯oP(1). +41 + +The rank of sparse symmetric matrices over arbitrary fields +B +Upper bound via leaf-removal: Derivation of Theorem 2.1 +In this section, we briefly explain how to derive Theorem 2.1 from known results about the Karp-Sipser core of +sparse Erd˝os-Rényi random graphs [3, 26]. To obtain the Karp-Sipser core of a graph G, we iteratively remove vertices +of degree one and their unique neighbors from G, until only isolated vertices and a subgraph of minimum degree at +least two remain. We call the number of isolated vertices in the reduced graph IKS(G). The derivation of Theorem 2.1 +rests on the following result: +Theorem B.1 ([3, 26]). For any d > 0, +IKS(Gn,d/n) +n +P +−→ γ⋆ + γ⋆ + γ⋆γ⋆ +d +− 1, +n → ∞. +Here, γ⋆ is the smallest root of the equation x = d exp(−d exp(−x)) and γ⋆ = d exp(−γ⋆). +Theorem B.1 is of importance in our setting since crucially, removal of degree-one vertices and their neighbours +does not change the nullity of the corresponding adjacency matrix (see [6]). Therefore, irrespective of the field or the +matrix entries, +rkF +� +An,d/n +� +n += 1 − nulF(An,d/n) +n +≤ 1 − IKS(Gn,d/n) +n +a.s. +and Theorem B.1 thus implies that for any d, ε > 0 and any field F, +lim +n→∞ P +� +sup +Jn∈Symn(F∗) +rkF +� +An,d/n +� +n +≤ 2 − γ⋆ + γ⋆ + γ⋆γ⋆ +d ++ ε +� += 1. +Thus, it remains to relate the limit from Theorem B.1 to the rank function Rd. +For this, observe that any root x⋆ of the equation x = d exp(−d exp(−x)) satisfies x⋆ ∈ (0, d) as well as +Gd(1 − x⋆/d) = 0. Items 1 and 2 in Lemma 6.1 thus imply that +γ⋆ = d(1 − α⋆) +and +γ⋆ = d(1 − α⋆). +Moreover, item 3 in Lemma 6.1 as well as Gd(α⋆) = 0 give that +2 − γ⋆ + γ⋆ + γ⋆γ⋆ +d += 2 − d(1 − α⋆) + d(1 − α⋆) + d2(1 − α⋆)(1 − α⋆) +d += 2 − φd(1 − φd(α⋆)) − φd(α⋆) − dφd(α⋆)(1 − α⋆) = Rd(α⋆) = min +α∈[0,1] Rd(α). +Here, in the last step, we have used Lemma 6.1, part 6. This concludes the derivation of Theorem 2.1 from Theorem B.1. +C +Difference approximation via conditional expectations: proof of Lemma 5.3 +For a differentiable function f : Rk �→ R, let ∇f be the gradient of f. We prove the following more general +version of Lemma 5.3: +Proposition C.1. Fix a dimension k ∈ N and K > 1. Let Z1, Z2 and X be defined on the same probability space with +convex codomains RZ1, RZ2 ⊂ Rk and RX ⊂ R respectively, such that RX is bounded. Then for any differentiable +functions f, g : Rk → R, +E |f(Z2) − g(Z2)| +≤ +� +sup +ζ∈RZ1 +|f(ζ)| + sup +x∈RX +|x| +� � +4K2E∥Z1 − Z2∥∞ + 2 − 2(1 − 1/K)k� ++ k sup +ζ∈RZ2 +∥∇f(ζ)∥∞ E∥Z1 − Z2∥∞ ++ E |f(Z1) − E [X|Z1]| + E |E [X|Z2] − g(Z2)| + 2k +K +� +sup +ζ∈RZ1 +∥∇f(ζ)∥∞ + +sup +ζ∈RZ2 +∥∇g(ζ)∥∞ +� +(C.1) +and +E +� +(f(Z2) − g(Z2))−� +≤ +� +sup +ζ∈RZ1 +|f(ζ)| + sup +x∈RX +|x| +� � +4K2E∥Z1 − Z2∥∞ + 2 − 2(1 − 1/K)k� ++ k sup +ζ∈RZ2 +∥∇f(ζ)∥∞ E∥Z1 − Z2∥∞ ++ E +� +(f(Z1) − E [X|Z1])−� ++ E +� +(E [X|Z2] − g(Z2))−� ++ 2k +K +� +sup +ζ∈RZ1 +∥∇f(ζ)∥∞ + +sup +ζ∈RZ2 +∥∇g(ζ)∥∞ +� +. +(C.2) +42 + +The rank of sparse symmetric matrices over arbitrary fields +The philosophy behind Proposition C.1 is that for sufficiently nice functions f and g, good control over E∥Z1 − +Z2∥∞, E |E [X|Z1] − f(Z1)| and E |E [X|Z2] − g(Z2)| allows to bound the difference of f(Z2) and g(Z2) in +expectation. Indeed, Proposition C.1 is designed to deal with situations where Z1 and Z2 are close, and it might be +helpful to keep this in mind during the following proofs. +To prove Proposition C.1, we partition Rk into small hypercubes: For K > 0, let Zk/K = +� +q ∈ Rk : Kq ∈ Zk� +. +We define the k-dimensional, half-open hypercube with side-length r > 0 and center s = (s1, s2, . . . , sk) ∈ Rk as +Ds(r) = {(t1, t2, . . . , tk) : ti − si ∈ [−r/2, r/2), i = 1, 2, . . . , k} . +The following lemma shows that if Z1 and Z2 are close, they are likely to be found within the same box, after the +application of a random uniform translation. This random translation ensures that the rather arbitrary random variables +Z1, Z2 do not always take values in the boundary of the partitioning hypercubes. +Lemma C.2. Fix a dimension k ∈ N and hypercube edge length 1/K > 0. Then for any two random vectors +Z1, Z2 ∈ Rk that are defined on the same probability space, an independently and uniformly chosen “shift” vector +ξK ∈ (0, 1/K]k and 0 < ε < 1/K, +� +q∈Zk/K +E [|1{Z1 − ξK ∈ Dq (1/K)} − 1{Z2 − ξK ∈ Dq (1/K)}|] ≤ 4 +ε E∥Z1 − Z2∥∞ + 2 − 2(1 − Kε)k. +(C.3) +Proof. For the sake of brevity, we omit the range of summation from � +q∈Zk/K throughout this proof. +Fix any hypercube Dq (1/K) and let j ∈ {1, 2}. For Zj − ξK to fall into Dq (1/K) and Z3−j − ξK to fall into a +distinct box, one of the following two cases must happen: +(a) Zj − ξK is in the “inner part” Dq (1/K − ε) of the box, but Z3−j − ξK /∈ Dq (1/K) (“separation”), or +(b) Zj − ξK is in the “ε-boundary” Dq (1/K) \ Dq (1/K − ε) of the box (“boundary”). +We call the separation event S(j) +q , and the boundary event B(j) +q , which yields the almost sure upper bound +|1{Z1 − ξK ∈ Dq (1/K)} − 1{Z2 − ξK ∈ Dq (1/K)}| ≤ 1S(1) +q ++ 1B(1) +q ++ 1S(2) +q ++ 1B(2) +q . +(C.4) +It thus remains to upper bound the right hand side of (C.4) in expectation and then sum over q ∈ Zk/K. +Separation: Deterministically, for j ∈ {1, 2}, +1S(j) +q +≤ 1{Zj − ξK ∈ Dq (1/K − ε) , ∥Z2 − Z1∥∞ ≥ ε/2} ≤ 1{Zj − ξK ∈ Dq (1/K − ε)}2 +ε ∥Z2 − Z1∥∞. +(C.5) +Summing over q ∈ Zk/K in (C.5) and taking expectation gives +E +�� +1S(j) +q +� +≤ 2 +εE [∥Z2 − Z1∥∞] . +(C.6) +Boundary: This is the case where the benefit of the random translation ξK becomes apparent. Again, let j ∈ {1, 2} +and fix q ∈ Zk/K. Conditionally on Zj, the random variable Zj − ξK − q is uniformly distributed over the box +�k +i=1[(Zj)i − qi − 1/K, (Zj)i − qi). Therefore, +P +� +B(j) +q +��Zj +� += Kkλ +� k +� +i=1 +[(Zj)i − qi − 1/K, (Zj)i − qi) ∩ (D0 (1/K) \ D0 (1/K − ε)) +� +, +(C.7) +where λ denotes the k-dimensional Lebesgue measure. +Now, since also the boxes �k +i=1[(Zj)i − qi − 1/K, (Zj)i − qi), q ∈ Zk/K, partition Rk, (C.7) further yields that +E +�� +1B(j) +q +� += E +�� +P +� +B(j) +q +��Zj +�� += Kkλ (D0 (1/K) \ D0 (1/K − ε)) = 1 − (1 − Kε)k. +(C.8) +The claim now follows from summing (C.4) over q ∈ Zk/K, (C.6) and (C.8). +We next turn to the proof of Proposition C.1. +Proof of Proposition C.1. Again, for brevity, we omit the range of summation from � +q∈Zk/K throughout the proof. +As in Lemma C.2, let ξK ∈ (0, 1/K]k be a uniformly chosen “shift” vector that is independent of (Z1, Z2, X). We +first distinguish the possible hypercube-locations for Z2 − ξK and apply the tower property to get +E |f(Z2) − g(Z2)| = +� +E [E [|f(Z2) − g(Z2)| 1{Z2 − ξK ∈ Dq (1/K)}|ξK]] . +(C.9) +43 + +The rank of sparse symmetric matrices over arbitrary fields +Given ξK, on the event {Z2 − ξK ∈ Dq (1/K)}, Z2 is located in the hypercube Dq+ξK (1/K) of sidelength +1/K. Since the hypercubes are small and f, g are continuous, the values of f and g should not fluctuate too much on +Dq+ξK (1/K). More precisely, let t ∈ Dq+ξK (1/K) ∩ RZ2 be arbitrary. If Dq+ξK (1/K) ∩ RZ2 = ∅, let t = 0. +Then by the mean value theorem, +E [|f(Z2) − f(t)| 1{Z2 − ξK ∈ Dq (1/K)}|ξK] ≤ k +K +sup +ζ∈RZ2 +∥∇f(ζ)∥∞ P (Z2 − ξK ∈ Dq (1/K) |ξK) , +and +E [|g(Z2) − g(t)| 1{Z2 − ξK ∈ Dq (1/K)}|ξK] ≤ k +K +sup +ζ∈RZ2 +∥∇g(ζ)∥∞ P (Z2 − ξK ∈ Dq (1/K) |ξK) . +In the last two displays, both sides are zero if Dq+ξK (1/K) ∩ RZ2 = ∅. By the triangle inequality, we get +E [|f(Z2) − g(Z2)| 1{Z2 − ξK ∈ Dq (1/K)}|ξK] +(C.10) +≤ +� +|f(t) − g(t)| + k +K +sup +ζ∈RZ2 +(∥∇f(ζ)∥∞ + ∥∇g(ζ)∥∞) +� +P (Z2 − ξK ∈ Dq (1/K) |ξK) +(C.11) +≤ |E [(f(Z2) − g(Z2)) 1{Z2 − ξK ∈ Dq (1/K)}|ξK]| +(C.12) ++ 2k +K +sup +ζ∈RZ2 +(∥∇f(ζ)∥∞ + ∥∇g(ζ)∥∞) P (Z2 − ξK ∈ Dq (1/K) |ξK) , +(C.13) +where now the modulus is outside of the expectation in (C.12) in comparison to (C.10). Summing (C.12) over q ∈ Zk/K +and applying the triangle inequality together yield that +� +|E [(f(Z2) − g(Z2)) 1{Z2 − ξK ∈ Dq (1/K)}|ξK]| +(C.14) +≤ +� +E [|f(Z2) − f(Z1)| 1{Z2 − ξK ∈ Dq (1/K)}|ξK] +(C.15) ++ +� +E [|f(Z1)| |1{Z2 − ξK ∈ Dq (1/K)} − 1{Z1 − ξK ∈ Dq (1/K)}| |ξK] +(C.16) ++ +� +|E [f(Z1)1{Z1 − ξK ∈ Dq (1/K)} − g(Z2)1{Z2 − ξK ∈ Dq (1/K)}|ξK]| . +(C.17) +For (C.15), since � 1{Z2 − ξK ∈ Dq (1/K)} = 1, again the mean value theorem implies that +� +E [|f(Z2) − f(Z1)| 1{Z2 − ξK ∈ Dq (1/K)}|ξK] ≤ E |f(Z2) − f(Z1)| ≤ k sup +ζ∈RZ2 +∥∇f(ζ)∥∞ E∥Z1 − Z2∥∞. +(C.18) +Taking expectation in (C.16), then an application of Lemma C.2 gives that +� +E [|f(Z1)| |1{Z2 − ξK ∈ Dq (1/K)} − 1{Z1 − ξK ∈ Dq (1/K)}|] +≤ +sup +ζ∈RZ1 +|f(ζ)| +�4 +εE∥Z1 − Z2∥∞ + 2 − 2(1 − Kε)k +� +. +(C.19) +Finally, using the triangle inequality once more, (C.17) can again be divided into three sub-parts as follows: +� +|E [f(Z1)1{Z1 − ξK ∈ Dq (1/K)} − g(Z2)1{Z2 − ξK ∈ Dq (1/K)}|ξK]| +(C.20) +≤ +� +E [|f(Z1) − E [X|Z1]| 1{Z1 − ξK ∈ Dq (1/K)}|ξK] +(C.21) ++ +� +E [|E [X|Z2] − g(Z2)| 1{Z2 − ξK ∈ Dq (1/K)}|ξK] +(C.22) ++ +� +|E [E [X|Z1] 1{Z1 − ξK ∈ Dq (1/K)}|ξK] − E [E [X|Z2] 1{Z2 − ξK ∈ Dq (1/K)}|ξK]| . +(C.23) +Since � 1{Z2 − ξK ∈ Dq (1/K)} = 1 and ξK and (Z1, Z2, X) are independent, (C.21) and (C.22) reduce to +� +E [|f(Z1) − E [X|Z1]| 1{Z1 − ξK ∈ Dq (1/K)}|ξK] = E [|f(Z1) − E [X|Z1]|] , +(C.24) +and +� +E [|E [X|Z2] − g(Z2)| 1{Z2 − ξK ∈ Dq (1/K)}|ξK] = E [|E [X|Z2] − g(Z2)|] . +(C.25) +44 + +The rank of sparse symmetric matrices over arbitrary fields +Let now i ∈ {1, 2}. Again, since ξK and (Z1, Z2, X) are independent, each expectation in (C.23) can be simplified as +E [E [X|Zi] 1{Zi − ξK ∈ Dq (1/K)}|ξK] = E [E [X|Zi, ξK] 1{Zi − ξK ∈ Dq (1/K)}|ξK] += E [X · 1{Zi − ξK ∈ Dq (1/K)}|ξK] . +(C.26) +Plugging identity (C.26) into (C.23) and the triangle inequality yield +� +|E [E [X|Z1] 1{Z1 − ξK ∈ Dq (1/K)}|ξK] − E [E [X|Z2] 1{Z2 − ξK ∈ Dq (1/K)}|ξK]| +≤ +� +E [|X| |1{Z1 − ξK ∈ Dq (1/K)} − 1{Z2 − ξK ∈ Dq (1/K)}| |ξK] . +(C.27) +Now again, by Lemma C.2, +� +E [E [|X| |1{Z1 − ξK ∈ Dq (1/K)} − 1{Z2 − ξK ∈ Dq (1/K)}| |ξK]] +≤ sup +x∈RX +|x| +�4 +εE∥Z1 − Z2∥∞ + 2 − 2(1 − Kε)k +� +. +(C.28) +(C.1) now follows by combining the bounds (C.9) – (C.28) and the choice ε = 1/K2. +The proof of (C.2) follows along the same lines, since the triangle inequality (a + b)− ≤ a− + b− and Jensen’s +inequality (E [a])− ≤ E[(a)−] hold for the negative part, as well as a− ≤ |a|. Indeed, the only difference between the +proofs is that we replace all absolute values |·| in eqs. (C.9) to (C.12), (C.14), (C.17), (C.20), (C.22) and (C.25) by the +corresponding negative parts, while we keep the absolute values in all other bounds. +Proof of Lemma 5.3. Lemma 5.3 is an immediate consequence of Proposition C.1: In the notation of Proposition C.1, +let k = 5 and fix any K > 1. We choose Z1 = ζn+1,t/n, Z2 = ζn,t/n and X = 1 +� +n + 1 ∈ W +� +Tn+1,t/n[θ] +�� +for +W ∈ {Y, U, V} with codomains RZ1 = RZ2 = [0, 1]5 and RX = [0, 1], respectively. Next, let f : Rk → R be the +projection onto the coordinate of ζ corresponding to w ∈ {y, u, v}, i.e. f(ζ) = f((x, y, z, u, v)) = w, and g : Rk → R, +g(ζ) = W (ζ, φt). Then (5.5) follows from (C.1) by checking that +(i) E |f(Z2) − g(Z2)| = E +��wn,t/n − W(ζn,t/n, φt) +��; +(ii) E [X|Z1] = wn+1,t/n = f(Z1) by Lemma 4.15; +(iii) supζ∈[0,1]5 |f(ζ)| = 1; +(iv) supx∈[0,1] |x| = 1; +(v) supζ∈[0,1]5 ∥∇f(ζ)∥∞ = 1; +(vi) supζ∈[0,1]5 ∥∇g(ζ)∥∞ ≤ 2d. +Analogously, +(5.6) follows from (C.2) by choosing f (ζ) += +z, +g (ζ) += +φt (y) and X += +1 +� +n + 1 ∈ Z +� +Tn+1,t/n[θ] +�� +, while the other parameters are as in the derivation of (5.5). +45 + diff --git a/aNFOT4oBgHgl3EQf_jRR/content/tmp_files/load_file.txt b/aNFOT4oBgHgl3EQf_jRR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0055f293faef7e84884dc368d575b1e97676091b --- /dev/null +++ b/aNFOT4oBgHgl3EQf_jRR/content/tmp_files/load_file.txt @@ -0,0 +1,2645 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf,len=2644 +page_content='THE RANK OF SPARSE SYMMETRIC MATRICES OVER ARBITRARY FIELDS Remco van der Hofstad, Noela Müller, Haodong Zhu January 31, 2023 ABSTRACT Let F be an arbitrary field and (Gn,d/n)n be a sequence of sparse weighted Erd˝os-Rényi random graphs on n vertices with edge probability d/n, where weights from F \\ {0} are assigned to the edges according to a fixed matrix Jn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We show that the normalised rank of the adjacency matrix of (Gn,d/n)n converges in probability to a constant, and derive the limiting expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Our result shows that for the general class of sparse symmetric matrices under consideration, the asymptotics of the normalised rank are independent of the edge weights and even the field, in the sense that the limiting constant for the general case coincides with the one previously established for adjacency matrices of sparse (non-weighted) Erd˝os-Rényi matrices over R from [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Our proof, which is purely combinatorial in its nature, is based on an intricate extension of the novel perturbation approach from [10] to the symmetric setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Keywords Rank · Random matrix · Erd˝os-Rényi graph 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 Background and motivation The study of matrices with random entries, going back to the 1950’s [29], is an important and lively field of modern probability and combinatorics with close ties to a multitude of other scientific disciplines such as theoretical physics, mathematical statistics, computer science, neuroscience or machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Up to this day, the theory of random matrices has developed into a mature field and advanced to a very precise understanding of classical models such as Gaussian Ensembles, Bernoulli matrices or Wishart matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Moreover, in the last decade, there has been a burst of progress in the theoretical understanding of random matrices which appear naturally in the study of random graphs, such as their adjacency matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Especially the adjacency matrix of the classical Erd˝os-Rényi random graph model and its spectral properties have attracted a great deal of attention [7, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The Erd˝os-Rényi graph Gn,pn = ({1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , n}, En), which is arguably the simplest random graph model, is a graph on n vertices, where each edge is present independently with probability pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Its adjacency matrix An,pn is a symmetric n × n-matrix with entries An,pn(i, j) = 1 {{i, j} ∈ En}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In particular, it is a symmetric Bernoulli matrix, which, depending on the limiting behaviour of the edge probability pn, displays different asymptotic behaviour: Results by Costello, Tao and Vu [15] and later by Basak and Rudelson [5] have shown that there is a sharp transition in the invertibility of the adjacency matrix around ln n/n + k(n)/n, for a function k(n) that tends slowly to infinity: When pn > ln n/n + k(n)/n, with high probability (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=') the adjacency matrix is nonsingular, while it is singular w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' for pn < ln n/n − k(n)/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Following this threshold result, a natural question is to determine the rank of the adjacency matrix An,pn when pn is small enough such that the matrix is singular w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In the regime where pn ∈ [c ln n/n, 1/2] for c > 1/2, Costello and Vu [17] show that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=', the rank of An,pn is exactly equal to n minus the number of isolated vertices in the underlying Erd˝os-Rényi random graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' They extend their result to c > 0 and arbitrary deterministic non-zero entries (instead of 1) in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This result shows that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=', the rank only depends on the structure of the graph, regardless of the precise value of the nonzero entries of the adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Finally, when pn = d/n for fixed d > 0, Bordenave, Lelarge and Salez [8] derive an asymptotic rank formula for An,d/n (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 1For an event B, 1{B} denotes the indicator function of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' When appropriate, we also use 1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12978v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='CO] 30 Jan 2023 The rank of sparse symmetric matrices over arbitrary fields While all these results naturally consider the rank of the adjacency matrix An,pn over R (or equivalently, Q), we will be interested in the rank of An,pn over arbitrary fields F in the sparse regime where pn = d/n (interpreting a 1-entry as the multiplicative identity of the field, and a 0-entry as its additive identity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Moreover, inspired by [16], we consider the more general class of matrices where the non-zero entries of An,pn are arbitrary deterministic non-zero elements of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Our main Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 shows that even under this vast generalisation, the asymptotic rank formula of Bordenave, Lelarge and Salez still remains valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This result suggests that the rank indeed only depends on the positions of the non-zero entries of the adjacency values, which is reflected in our proof strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Indeed, thanks to observations of Bauer and Golinelli [6], there is a by now well-known and purely combinatorial upper bound on the asymptotic rank of An,d/n, which is based on the Karp-Sipser algorithm for finding large matchings [26]: Start with Gn,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' At each step of the algorithm, recursively, a vertex of degree one along with its unique neighbor is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The process stops once only isolated vertices and vertices of degree at least two, the so-called Karp-Sipser core, are left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' It is straightforward to check that this “leaf-removal” leaves the nullity of the graph invariant (for a proof, see [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since the nullity of the reduced graph is apparently lower bounded by its number of isolated vertices, this number of isolated vertices provides an upper bound on the rank of the original graph that is completely oblivious to the field or the precise values of the non-zero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Karp and Sipser [26] also derive a formula for the asymptotic number of isolated vertices in the reduced graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Moreover, for d ≤ e, all but a vanishing proportion of vertices become isolated after running the Karp-Sipser algorithm on Gn,d/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thus, for d ≤ e, the question is already completely settled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' However, when d > e, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=', the Karp-Sipser core is not negligible, which complicates matters significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since there is already the rank formula of [8] in the sparse case, a natural take on the problem of the missing lower bound would be to turn to the proof methods of Bordenave, Lelarge and Salez and adapt them to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' However, their analysis makes heavy use of spectral properties of real symmetric matrices, so to the best of our knowledge, there is no possibility to follow their approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, inspired by insights from statistical physics, Coja-Oghlan, Ergür, Gao, Hetterich and Rolvien [10] found a new combinatorial approach to derive an asymptotic rank formula for a broad class of asymmetric sparse random matrices, generalising earlier results by Cooper, Frieze and Pegden for F2 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Correspondingly, the results of [10] are valid over any field, regardless of the distribution of the non-zero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' However, their approach cannot straightforwardly be applied to symmetric random matrices, since these retain much less independence among the positions of their non-zero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Indeed, the authors note that “an intriguing question for future research is to extend the techniques from the present paper to symmetric random matrices.” In this paper, we build on several of the core concepts of [10] to develop a corresponding combinatorial approach towards rank formulas for sparse symmetric matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' As in [10], instead of investigating the rank of An,d/n directly, we work with a perturbed version of An,d/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Moreover, as in [10], we use a telescoping argument to lower bound the expected rank and relate the rank difference of matrices whose sizes differ by one to so-called “frozen” variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' However, the symmetry of our matrices poses serious obstructions to any attempt to literally follow in the footsteps of [10], and we therefore introduce quite a number of changes and adaptations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' These changes allow us to give a precise characterization of the rank increase when we add a row and a column, and therefore to show that the asymptotic behavior of the rank of a broad class of random matrices, whose non-zero entries are prescribed by the adjacency structure of a sparse Erd˝os-Rényi random graph, over any field F, is indeed the same as the rank of the simple 0/1-adjacency matrix of Gn,d/n over the field R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This paper is organised as follows: In Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2, we introduce our precise model and main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' A proof overview, together with the most important intermediate steps, can be found in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Section 3 collects results on our matrix perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In Section 4, we investigate various properties of the different variable (or vertex) types introduced earlier, and their relation to the rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We then derive the fixed point equations for the asymptotic proportions of some of the different types in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Section 6 uses these fixed point equations to derive the desired lower bound on the asymptotic rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In Appendix A, we provide important properties of the various functions related to the rank formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Appendix B explains how to derive an upper bound on the normalised rank from results on the Karp-Sipser leaf-removal algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Finally, Appendix C contains a proposition which is used to compare different conditional expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 (Notation for random variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Throughout the article, we use bold letters to indicate random variables and regular letters to indicate deterministic quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ■ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 Main results Let F be an arbitrary field and F∗ := F \\ {0} its multiplicative group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For a general matrix A ∈ Fm×n, rkF(A) specifically denotes the rank of A over F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' the dimension of the linear subspace of Fn spanned by the columns of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Moreover, we use Symn(F∗) for the set of all symmetric n × n matrices with entries in F∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In the present article, we study adjacency matrices of sparse Erd˝os-Rényi random graphs with arbitrary non-zero edge weights over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' To define the precise model, let (Jn)n≥1 be any deterministic sequence of “template” matrices such that for all n ≥ 1, Jn ∈ Symn(F∗), and (q(i, j))i,j≥1 be an array of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' uniform random variables in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For 2 The rank of sparse symmetric matrices over arbitrary fields p ∈ [0, 1], we then define the matrix An,p by setting An,p(i, j) = � � � 1{q(i, j) < p}Jn(i, j), i < j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 1{q(j, i) < p}Jn(j, i), i > j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 0, i = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1) An,p can be alternatively regarded as the adjacency matrix of a weighted Erd˝os-Rényi random graph on the vertex set [n], where each potential edge {i, j} is present independently with probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' If it is present, it is assigned edge weight Jn(i, j) = Jn(j, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The construction (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1) also incorporates a natural coupling of the positions of the nonzero entries of the matrices An,p for all choices of n and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In the important special case where Jn(i, j) ≡ 1 for all i, j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , n}, An,p coincides with the adjacency matrix of an unweighted Erd˝os-Rényi graph with n vertices and edge probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' An asymptotic rank formula for this model over F = R in the regime where p = d/n was given by Bordenave, Lelarge and Salez in [8]: For any d > 0, let φd : [0, 1] → R, φd(α) := exp(d(α − 1)) be the probability generating function of a Poisson random variable with parameter d and Rd : [0, 1] → R be defined by setting Rd(α) = 2 − φd (1 − φd(α)) − (1 + d(1 − α))φd(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2) Bordenave, Lelarge and Salez [8] then show that for any d > 0, in the coupling given above, lim n→∞ 1 n rkR � An,d/n � = min α∈[0,1] Rd(α) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3) The article [8] also provides asymptotic rank formulas for the adjacency matrices of any sequence of random graphs that converges locally to a rooted Galton-Watson tree whose degree distribution has a finite second moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For general fields F, of course, rkF(An,d/n) need not be identical to rkR(An,d/n) (even in the case where Jn(i, j) ≡ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For example, if F = Fp is the finite field with p elements, then generally only the upper bound rkFp(An,d/n) ≤ rkR(An,d/n) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Moreover, the proof of the rank formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3) is based on the rank-nullity theorem and the fact that nulF(An,d/n) is identical to the dimension of the eigenspace of A corresponding to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since for real symmetric matrices, the geometric and algebraic multiplicities of all eigenvalues coincide, the dimension of the eigenspace of A corresponding to 0 can be studied through an associated spectral measure in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, for symmetric matrices over Fp, there is no reason to assume the matrix to be diagonalisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Pursuing a purely combinatorial approach that does not rely on the analysis of a spectral measure, our main result generalises the asymptotic rank formula of [8] to arbitrary fields F and general non-zero entries: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any d > 0 and any field F, rkF � An,d/n � /n converges in probability to minα∈[0,1] Rd(α) uniformly in (Jn)n≥1 in the sense that for any ε > 0, lim n→∞ sup Jn∈Symn(F∗) P ����� 1 n rkF � An,d/n � − min α∈[0,1] Rd(α) ���� ≥ ε � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4) Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 (Almost sure convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In the case where Jn(i, j) ≡ 1 and one is interested in convergence of the sequence (An,d/n)n≥1 of adjacency matrices of a sparse Erd˝os-Rényi random graph, the convergence in probability can easily be lifted to almost sure convergence by a standard martingale argument as given in [8, Appendix 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ■ In line with previous results on the rank of sparse random asymmetric matrices [10], Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 illustrates that (within the specified framework) the rank formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4) solely depends on d, but not on the field F or the choice of the sequence (Jn)n≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 2 Proof overview On the following pages, we present an overview of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' After fixing some notation, we first reduce the uniform convergence in probability in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4) to an upper bound in probability and a lower bound in expectation in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' While the upper bound is based on the leaf-removal algorithm and the results of [3, 26], the lower bound constitutes the main contribution of our article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' To lower bound the expected rank of An,d/n, we transform it to a “symmetrised” matrix and grow the modified matrix from εn to n step by step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' An essential ingredient in the quantification of the described one-step rank change are the powerful techniques developed in [10], which allow us to focus on the positions of the nonzero entries in the target matrix rather than their precise values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Finally, the rank formula follows by interpreting the sum of the lower bounds as the Riemann sum of an integral, which is analytically tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 Notation This section can be used as a reference for recurring notation that is used throughout the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 3 The rank of sparse symmetric matrices over arbitrary fields Sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We write [ℓ] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , ℓ} and denote the cardinality of a set B by |B|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For two sets B1 and B2, we denote their symmetric difference as B1∆B2 and use ⊎i∈IBi to indicate the union over pairwise disjoint sets (Bi)i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' If B is a set and ℓ ≤ |B|, we write �B ℓ � for the collection of ℓ-subsets of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Real numbers and fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For a, b ∈ R, we write a∨b = max {a, b} and a∧b = min {a, b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' F is reserved to denote a generic field, and F∗ = F \\ {0} its multiplicative group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Vectors and matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For A ∈ Fm×n, we denote its transpose by AT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For a vector b = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , bn) ∈ F1×n, we let supp(b) = supp(bT ) = {i ∈ [n]: bi ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We denote by en(i) the ith standard unit vector in F1×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For s = (s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , sℓ) ∈ R1×ℓ, define ∥s∥∞ = supi∈[ℓ] |si| and ∥s∥k = (�ℓ i=1 |si|k)1/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For A ∈ Fm×n, we denote (i) the ith row of A by A(i, ) and the jth column of A by A(, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (ii) the matrix obtained by removing rows ℓ1, ℓ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , ℓs and columns ℓ′ 1, ℓ′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , ℓ′ t from A by A ⟨ℓ1, ℓ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , ℓs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ℓ′ 1, ℓ′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , ℓ′ t⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By a slight abuse of indexing, the ith row in the diminished matrix A ⟨ℓ1, ℓ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , ℓs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ℓ′ 1, ℓ′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , ℓ′ t⟩ refers to the row vector A(i, ) ⟨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ℓ′ 1, ℓ′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , ℓ′ t⟩, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=', the ith row of A (mi- nus the entries corresponding to columns ℓ′ 1, ℓ′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , ℓ′ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We use an analogous convention for columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For a function f : Ω → R, we denote by f + its positive and by f − its negative part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' f +(x) = 0∨f(x) and f −(x) = 0 ∨ (−f(x)) for x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For a finite set B, we write Unif(B) to denote a discrete uniform random variable on B, Bin (n, p) to denote a binomial random variable with n trials and success probability p and Po (d) to denote a Poisson variable with parameter d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For two random variables X, Y taking values in (Ω, G), we denote the total variation distance between X and Y as dTV(X, Y ) = sup B∈G |P (X ∈ B) − P (Y ∈ B)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Notions of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Throughout the article, the order in which limits are taken matters significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For families of real numbers (an,P,N,JN )n,P,N∈Z+,JN∈SymN(F∗), we write (i) an,P,N,JN = on(1) ⇐⇒ For all P ≥ 1 : limn→∞ supN≥n,JN∈SymN(F∗) |an,P,N,JN | = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (ii) an,P,N,JN = on,P (1) ⇐⇒ lim supP →∞ lim supn→∞ supN≥n,JN∈SymN(F∗) |an,P,N,JN | = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Given a family of real numbers (cn,P,N,JN,t)n,P,N∈Z+,JN∈SymN(F∗),t∈[0,d], we say that (i) cn,P,N,JN,t = on(1) uniformly in t ∈ [0, d] ⇐⇒ supt∈[0,d] cn,P,N,JN,t = on(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (ii) cn,P,N,JN,t = on,P (1) uniformly in t ∈ [0, d] ⇐⇒ supt∈[0,d] cn,P,N,JN,t = on,P (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For a family of uniformly bounded random variables (bn,P,N,JN,t)n,P,N∈Z+,JN∈SymN(F∗),t∈[0,d], we write (i) bn,P,N,JN,t = ¯oP(1) ⇐⇒ E |bn,P,N,JN,t| = on,P (1) uniformly in t ∈ [0, d];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (ii) bn,P,N,JN,t ≥ ¯oP(1) ⇐⇒ (bn,P,N,JN,t)− = ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (iii) bn,P,N,JN,t ≤ ¯oP(1) ⇐⇒ (bn,P,N,JN,t)+ = ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For a family of events (Bn,P,N,JN,t)n,P,N∈Z+,JN∈SymN(F∗),t∈[0,d], we say that Bn,P,N,JN,t occurs w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' if P (Bn,P,N,JN,t) = 1 + on,P (1) uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We extend the above notions of convergence to families of numbers and events that only depend on subsets of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For example, for a family of real numbers (cn,P )n,P ∈Z+, by treating it as constant on the unspecified parameters, we write cn,P = on,P (1) whenever lim supP →∞ limn→∞ cn,P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 Deduction of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 from suitable upper and lower bounds Our main result, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2, is a statement about convergence in probability of the normalised rank sequence rkF � An,d/n � /n that holds uniformly in (Jn)n≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In this section, we show how Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 readily follows from the following upper bound in probability and the subsequent lower bound in expectation: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 (Upper bound in probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let d > 0 and F be any field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then for any ε > 0, lim n→∞ P � sup Jn∈Symn(F∗) rkF � An,d/n � n ≤ min α∈[0,1] Rd(α) + ε � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1) 4 The rank of sparse symmetric matrices over arbitrary fields Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 (Lower bound in expectation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any d > 0 and any field F, lim inf n→∞ inf Jn∈Symn(F∗) E � rkF � An,d/n � n � ≥ min α∈[0,1] Rd(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2) While Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 straightforwardly follows from the fact that the nullity of an adjacency matrix remains invariant under “leaf-removal” (see [6]) and the results of [26]2, the derivation of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 is the main contribution of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The central steps towards (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2) are laid out in the remainder of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' With Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 in hand, we are in the position to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2: Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 subject to Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let sn = sn(Jn) = rkF � An,d/n � n − min α∈[0,1] Rd(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then |sn| ≤ 1 + ��minα∈[0,1] Rd(α) ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, for any ε > 0, > 0 lim sup n→∞ sup Jn∈Symn(F∗) E � s+ n � ≤ lim sup n→∞ E � sup Jn∈Symn(F∗) s+ n � ≤ ε + � 1 + ���� min α∈[0,1] Rd(α) ���� � lim sup n→∞ P � sup Jn∈Symn(F∗) s+ n ≥ ε � = ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since ε can be chosen arbitrarily small, we conclude that lim supn→∞ supJn∈Symn(F∗) E [s+ n ] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2, lim infn→∞ infJn∈Symn(F∗) E [sn] ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since sn = s+ n − s− n , lim sup n→∞ sup Jn∈Symn(F∗) E � s− n � ≤ lim sup n→∞ sup Jn∈Symn(F∗) E � s+ n � − lim inf n→∞ inf Jn∈Symn(F∗) E [sn] ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' As a consequence, lim supn→∞ supJn∈Symn(F∗) E [|sn|] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The uniform convergence in probability now follows from Markov’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We conclude that it remains to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 and outline the main steps in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 The lower bound: Building the matrix Instead of proving Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 for the sequence (An,d/n)n≥1 directly, we work with a “symmetrised” version that possesses a suitable form of joint row and column exchangeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' To define the auxiliary matrices, fix a number N ∈ N≥1 and let τ be a uniform permutation of [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For n ∈ [N], define the matrix T (N) n,p ∈ Fn×n by setting T (N) n,p (i, j) = � � � 1{q(τ(i), τ(j)) < p}JN(τ(i), τ(j)), i < j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 1{q(τ(j), τ(i)) < p}JN(τ(j), τ(i)), i > j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 0, i = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3) For any N ∈ N≥1, this construction yields N matrices T (N) 1,p , T (N) 2,p , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , T (N) N,p of growing dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Specifically, we have T (N) N,p (i, j) = AN,p(τ(i), τ(j)) and rkF � T (N) N,p � = rkF (AN,p), so that Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 would follow from the lower bound lim inf n→∞ inf Jn∈Symn(F∗) E � 1 n rkF � T (n) n,d/n �� ≥ min α∈[0,1] Rd(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' However, for technical reasons that will become apparent later, we actually show the stronger statement lim inf n→∞ inf N≥n inf JN∈SymN(F∗) E � 1 n rkF � T (N) n,d/n �� ≥ min α∈[0,1] Rd(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Correspondingly, in the following, we focus on the derivation of a lower bound on E[rkF(T (N) n,d/n)]/n for N ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Nonetheless, for a lighter notation, we omit the superscript N in the matrices below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The basic idea in this derivation is 2See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 5 The rank of sparse symmetric matrices over arbitrary fields rather simple: Fix a small number ε ∈ (0, 1) and trace the rank change when the matrix Tεn,d/n is grown to Tn,d/n step by step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then, by a telescoping sum, 1 nE � rkF � Tn,d/n �� ≥ 1 n n−1 � m=εn � E � rkF � Tm+1,d/n �� − E � rkF � Tm,d/n ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4) The last expression thus reduces the problem of lower bounding E[rkF � Tn,d/n � ]/n to lower bounding �n−1 m=εn � E � rkF � Tm+1,d/n �� − E � rkF � Tm,d/n ��� /n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since this bound is based on a comparison of the two matrices Tm+1,d/n and Tm,d/n whose sizes differ by one, our approach might superficially resemble the Aizenman-Sims-Starr scheme from mathematical physics, which had previously found its application in the study of the rank of random matrices in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The Aizenman-Sims-Starr scheme, whose basic idea is to compare a system of n variables to a system of n + 1 variables and to study the influence on the (n + 1)st variable, has originally been developed to tackle the Sherrington-Kirkpatrick spin glass model [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' However, our approach cannot straightforwardly be interpreted as a cavity computation for the original matrix sequence, since we do not (directly or indirectly) compare two matrices of the form An,d/n and An−1,d/(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Instead, we compare matrices Tm,d/n and Tm+1,d/n whose sizes differ by one, but who are of a purely auxiliary nature and do not represent copies of the original matrix model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4 Taming linear relations While a comparison of the rather similar matrices Tm+1,d/n and Tm,d/n might look innocuous at first glance, obtaining good control over the ensuing rank change is not a simple task, since it requires detailed knowledge of the intricate linear dependencies of the matrix Tm,d/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Following and extending core ideas of [10], this section collects the main tools that are necessary to deal with these relations and to accurately describe the change in rank from Tm,d/n to Tm+1,d/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The following definition from [10] contains a collection of terminology that will turn out useful in the coming considerations on linear dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 (Linear relations: [10, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let A ∈ Fm×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (i) A set ∅ ̸= I ⊆ [n] is a relation of A if there exists a row vector y ∈ F1×m such that ∅ ̸= supp(yA) ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' If furthermore supp(yA) = I, then we call y a representation of I in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (ii) If I = {i} is a relation of A, then we call i a frozen variable in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let F(A) be the set of all frozen variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (iii) A relation I ⊆ [n] is a proper relation of A if I\\F(A) is a relation of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (iv) For δ > 0, ℓ ≥ 2, we say that A is (δ, ℓ)-free if there are no more than δnℓ proper relations I ⊆ [n] of size |I| = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ♦ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4 (Frozen variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (i) The terminology frozen variable refers to the role that the corresponding coordinate plays in the kernel of A: Frozen variables are exactly those coordinates that are invariably 0 in all vectors of kerF(A) (see [10, Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (ii) In Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 (also in [18, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7]), we will see yet another convenient characterization of frozen variables in terms of column removal as follows: i ∈ F(A) ⇐⇒ rkF (A) − rkF (A ⟨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5) ■ Let A ∈ Fm×n be any matrix and b ∈ F1×n be a non-zero row vector, and suppose that we want to attach b to A and characterise the ensuing rank change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This is a simpler operation than what we actually need (attaching both a row and a column), but still instructive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In terms of frozen variables and proper relations, we can say the following about the rank increase of attaching b to A: If all variables of supp (b) are frozen, then surely b lies in the linear span of the rows of A, since it can be linearly combined using the representations of its non-zero coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, if b is contained in the linear span of the rows of A, then because of the existence of a linear combination, either all variables of supp (b) are frozen or they form a proper relation of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' As a consequence, we have the following key implications: supp (b) ⊆ F(A) =⇒ b is in the span of the rows of A =⇒ supp (b) ⊆ F(A) or supp (b) is a proper relation of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6) These implications are useful for our purposes since the concept of a relation only takes into account the locations of non-zero entries, but not their entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' However, unfortunately, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6) does not come in form of an equivalence, since supp (b) being a proper relation of A does not imply that b lies in the span of the rows of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 6 The rank of sparse symmetric matrices over arbitrary fields To remedy this issue, based on ideas from [10], we use a matrix perturbation that greatly reduces the overall number of short proper linear relations in the resulting matrix, such that morally, an equivalence of the form “supp (b) ⊆ F(A) ⇐⇒ b is in the span of the rows of A” holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' While the perturbation from [10] is based on the attachment of unit rows, we will augment this definition by the attachment of unit columns to account for the symmetry of our matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The basic idea is that the attachment of unit rows at the bottom of a given matrix A can eliminate short proper relations in the augmented matrix, while the attachment of unit columns to the left of A can eliminate short proper relations in its transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The details of the perturbation are considerably more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We split its definition into two main parts, since it involves two stages of randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In the first definition, we present the basic row and column attachment matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Their non-zero entries may be confined to fixed initial segments of the column set [n] and row set [m], respectively: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5 (Perturbation matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (i) Let θr, n1, n2 ∈ N with n1 ≤ n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The row-perturbation matrix Θr[θr, n1|n2] ∈ {0, 1}θr×n2 with parameters θr, n1, n2 is defined by setting exactly one entry in each of its θr rows equal to 1, where the choice of this entry is uniform among the first n1 out of its n2 columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' More precisely, the unique 1-entry of row k ∈ [θr] is in column jk, where j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , jθr ∈ [n1] are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' uniformly distributed random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (ii) Let θc, m1, m2 ∈ N with m1 ≤ m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The column-perturbation matrix Θc[m1|m2, θc] ∈ {0, 1}m2×θc with parameters θc, m1, m2 is defined by setting exactly one entry in each of its θc columns equal to 1, where the choice of this entry is uniform among the first m1 out of its m2 rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' More precisely, the unique 1-entry of column k ∈ [θc] is in row ik, where i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , iθc ∈ [m1] are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' uniformly distributed random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ♦ Figure 1: Schematic representation of the row-perturbation matrix Θr[θr, n1|n2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For A ∈ Fm×n and a row perturbation matrix Θr[θr, n1|n] with non-zero column-coordinates j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , jθr ∈ [n1] of its θr rows, consider the perturbed matrix A′ := � A Θr[θr, n1|n] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then in A′, the non-zero columns j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , jθr ∈ [n1] of Θr[θr, n1|n] are part of the set of frozen variables: Since js is the index of the only non-zero entry in the (m + s)th row, the Boolean row vector em+θr(m + s) is a representation of js in A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In this sense, one can view the attachment of Θr[θr, n1|n] at the bottom of a matrix as explicitly freezing the variables corresponding to non-zero columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, appending Θc[m1|m, θc] to the right of A has quite a contrary and more subtle effect upon the set of frozen variables: In a sense, additional columns have the same impact as row removals and therefore can “unfreeze” coordinates (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 for a proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The necessity of column perturbation matrices constitutes the main difference to the previously employed perturbation from [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Before we introduce a second level of randomness to the perturbation, in the next lemma, we construct a coupling of the row-perturbation matrices Θr[θr, n1|n2] for all possible sizes θr × n2 and subsets of freezable coordinates [n1] ⊆ [n2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The benefit of this coupling is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' First, perturbation matrices of increasing size, but with fixed subset of freezable coordinates, will be nested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Second, the probability that matrices of fixed dimension, but with different subsets of freezable coordinates, disagree, can be bounded explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This coupling ensures that with high probability, we can apply the same perturbation to both Tm,d/n and Tm+1,d/n, and still get the desired properties: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6 (Coupling of perturbation matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' There is a coupling of the family {Θr[θr, n1|n2] : θr, n2 ≥ 1, n1 ∈ [n2]} with the following properties: (i) For any θr, n2 ≥ 1 and n1 ∈ [n2], Θr[θr, n1|n2 + 1] ⟨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' n2 + 1⟩ = Θr[θr, n1|n2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (ii) For any θr, n2 ≥ 1 and n1 ∈ [n2], Θr[θr + 1, n1|n2] ⟨θr + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ = Θr[θr, n1|n2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (iii) For any θr, n2 ≥ 1 and n0 ≤ n1 ≤ n2, P (Θr[θr, n0|n2] = Θr[θr, n1|n2]) = (n0/n1)θr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Similarly, there is a coupling with analogous properties for the family {Θc[m1|m2, θc] : θc, m2 ≥ 1, m1 ∈ [m2]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 7 n1 Or[0r, ni|n2] = n2The rank of sparse symmetric matrices over arbitrary fields Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' From now on, we assume that the perturbation families {Θr[θr, n1|n2] : θr, n2 ≥ 1, n1 ∈ [n2]} and {Θc[m1|m2, θc] : θc, m2 ≥ 1, m1 ∈ [m2]} are coupled as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6 and independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ■ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6 is proved in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Based on the ensembles {Θr[θr, n1|n2] : θr, n2 ≥ 1, n1 ∈ [n2]} and {Θc[m1|m2, θc] : θc, m2 ≥ 1, m1 ∈ [m2]} from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6, we finally introduce the central perturbation of this article: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8 (Canonical perturbation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For A ∈ Fm×n and θ = (θr, θc) ∈ N2, we write A[θ] = � A Θc[m|m, θc] Θr[θr, n|n] 0θr×θc � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For the canonical choice θ = (θr, θc) ∼ Unif([P]2), where P ∈ N is fixed and θ is independent of the couplings {Θr[θr, n1|n2] : θr, n2 ≥ 1, n1 ∈ [n2]} and {Θc[m1|m2, θc] : θc, m2 ≥ 1, m1 ∈ [m2]}, we simply write A[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ♦ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In the rest of this paper, θ always denotes a random vector chosen uniformly at random from [P]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' It is important to keep in mind that the random vector θ is always understood to depend on the parameter P, even though this is omitted from the notation (in line with the notation in [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ■ As advertised earlier, perturbation typically greatly reduces the number of short proper relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The next proposition shows that, for any fixed L ∈ N≥2, A[θ] and A[θ]T are w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (δ, ℓ)-free for all 2 ≤ ℓ ≤ L (observe that even if A is symmetric, the perturbed matrix A[θ] generally is not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This makes the matrices A[θ] and A[θ]T much more convenient to study in comparison to A: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10 (Perturbation eliminates most short proper relations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Fix δ > 0, L ∈ N≥2 and s ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then sup A∈F(n+s)×n P � A[θ] or A[θ]T is not (δ, ℓ)-free for some 2 ≤ ℓ ≤ L � = on,P (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7) The proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10 is given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10 is the symmetric version of [10, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' It is remarkable in the sense that it shows that the simple perturbation of attaching a bounded number of unit rows and columns eliminates a large proportion of short proper relations both column- and row-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Rather than lower bounding E � rkF � Tn,d/n �� /n as indicated in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4), in the next subsections, we will outline how to lower bound the expected normalised rank of the perturbed matrix Tn,d/n[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This also gives a lower bound for E � rkF � Tn,d/n �� /n, since if we add a row or a column to a matrix, its rank stays unchanged or increases by 1, and therefore, rk(A) ≤ rk(A[θ]) ≤ rk(A) + θr + θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8) Thus, as long as θr, θc are bounded random variables, all results on the asymptotic rank of the perturbed matrices transfer to the unperturbed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5 Rank increase for the perturbed matrix and obstructions due to symmetry At this point, our strategy rests on lower bounding the differences E � rkF � Tm+1,d/n[θ] �� − E � rkF � Tm,d/n[θ] �� for m ≥ εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since m grows linearly in n, a reparametrisation yields the more convenient expression E � rkF � Tn+1,t/n[θ] �� − E � rkF � Tn,t/n[θ] �� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9) where now t ∈ [εd, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' While all the perturbed matrices use the same vector θ that fixes the dimensions of the perturba- tion, the positions of the non-zero entries in the perturbation part may change from matrix to matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Conveniently, this does not happen frequently, since thanks to the coupling from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6, with high probability, Tn+1,t/n[θ] ⟨n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' n + 1⟩ = Tn,t/n[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10) On the event (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10), observation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5) on column removal and frozen variables implies that rkF � Tn+1,t/n[θ] � − rkF � Tn,t/n[θ] � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11) =1{n + 1 ∈ F(Tn+1,t/n[θ]T )} + 1{n + 1 ∈ F(Tn+1,t/n[θ] ⟨n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩)}, where we first remove the (n + 1)st row and then the (n + 1)st column to go from Tn+1,t/n[θ] to Tn,t/n[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In the analysis of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11), both the benefits of working with the matrix Tn,t/n and then its perturbation Tn,t/n[θ] become apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We next explain how these ideas can be used effectively in the evaluation of the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 8 The rank of sparse symmetric matrices over arbitrary fields For p ∈ (0, 1), let αn,p and αT n,p be the proportions of frozen variables i ∈ [n] in Tn,p[θ] and Tn,p[θ]T , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By the distributional invariance of the matrix Tn+1,t/n[θ] under joint row- and column-relabelling3, conditionally on αT n+1,t/n, the probability that n + 1 is frozen in Tn+1,t/n[θ]T is simply given by P(n + 1 ∈ F(Tn+1,t/n[θ]T ) | αT n+1,t/n) = αT n+1,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This provides a simple expression for the first indicator in r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We next consider the second indicator that n + 1 is frozen in Tn+1,t/n[θ] ⟨n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Again by observation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5), this event is the same as the event the (n + 1)st column of Tn+1,t/n[θ] ⟨n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ lies in the span of the columns of Tn,t/n[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Considering the transposed matrix, this translates to the event that the (n + 1)st row of Tn+1,t/n[θ] ⟨n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩T lies in the span of the rows of Tn,t/n[θ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By this chain of equivalences, we have turned the original event into one that we can handle very well thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6) and the perturbation: Since the perturbation effectively excludes the possibility that the non-zero components of the (n + 1)st row of Tn+1,t/n[θ] ⟨n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩T form a proper relation, the event in question roughly corresponds to the event that all the non-zero components of the (n + 1)st row of Tn+1,t/n[θ] ⟨n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩T are frozen in Tn,t/n[θ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For a lighter notation, we abbreviate b := Tn+1,t/n[θ](n+1, ), so that b is the (n+1)st row of Tn+1,t/n[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since the positions of the non-zero entries of b are chosen uniformly at random and independently of Tn,t/n[θ], conditionally on αn,t/n and |supp (b) |, the probability that all the non-zero components of the (n+1)st row of Tn+1,t/n[θ] ⟨n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩T are frozen in Tn,t/n[θ]T should be close to 1 − α|supp(b)| n,t/n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, |supp (b) | asymptotically follows a Po (t) distribution, so that after taking expectation with respect to |supp (b) |, we arrive at the approximation P(n + 1 ∈ F(Tn+1,t/n[θ] ⟨n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩)|αT n,t/n) ≈ 1 − φt(αT n,t/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In the above, recall that φt is the probability generating function of a Po (t) variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thus, on a heuristic level, E � rkF � Tn+1,t/n[θ] � |αT n+1,t/n � − E � rkF � Tn,t/n[θ] � |αT n,t/n � ≈ αT n+1,t/n + 1 − φt � αT n,t/n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12) This expression has two flaws: First of all, rather than depending on one random variable, it depends on both αT n,t/n and αT n+1,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Secondly, even though we trace the rank change in Tn,t/n[θ], the left hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12) comes in terms of the proportions in the transposed matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Fortunately, in Section 4, we will show that in expectation, the difference αn+1,t/n − αn,t/n is small, which allows us to reduce the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12) to one parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, αT n+1,t/n and αn+1,t/n are identically distributed, so the second problem is solved as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' With ht : [0, 1] → R, ht (α) := α + 1 − φt (α) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13) we have thus heuristically derived the following result: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 (The rank increase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any d > 0, E � rkF � Tn+1,t/n[θ] � − rkF � Tn,t/n[θ] �� = E � ht � αn,t/n �� + on,P (1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14) We give a full proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14) in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 lays the basis for the targeted lower bound on E[rkF(Tn,d/n)]/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In view of the rank formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3), it might be tempting to just take the minimum over all α ∈ [0, 1] on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Unfortunately, this is not sufficient to arrive at (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3), and we need means to restrict the potential values of αn,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' It thus “only” remains to get our hands on αn,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' With (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14) in mind, it is natural to suspect that αn,t/n converges and to try to calculate its limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' However, the situation is not that simple, and based on results for a similar class of asymmetric sparse matrices [9], it is not reasonable to expect αn,t/n to stabilise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Instead, our strategy will be to derive an asymptotic fixed point equation for αn,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The ensuing characterisation will finally allow us to make the connection to the rank formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' To motivate the desired equation for αn,t/n, we again take a look at the evaluation of the second indicator in the derivation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12) above: P(n + 1 ∈ F(Tn+1,t/n[θ] ⟨n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩)|αn,t/n, αT n,t/n) ≈ 1 − φt(αT n,t/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since the matrix Tn+1,t/n[θ] ⟨n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ is rather similar to Tn,t/n[θ], one might make the bold assumption that P(n + 1 ∈ F(Tn+1,t/n[θ] ⟨n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩)|αn,t/n, αT n,t/n) ≈ P(n + 1 ∈ F(Tn+1,t/n[θ])|αn,t/n, αT n,t/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 3For the precise arguments, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 9 The rank of sparse symmetric matrices over arbitrary fields On the other hand, P(n + 1 ∈ F(Tn+1,t/n[θ])|αn+1,t/n, αT n+1,t/n) ≈ αn+1,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Based on the previous assumption, we can again argue that αn+1,t/n ≈ αn,t/n and use a handy proposition on the comparison of conditional expectations4 to conclude that 1 − φt(αT n,t/n) ≈ αn,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Along the same lines, we can conclude that 1 − φt(αn,t/n) ≈ αT n,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Combining the two approximations, we heuristically deduce that αn,t/n should approximately satisfy the equation αn,t/n ≈ 1 − φt(1 − φt(αn,t/n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15) While (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15) is surely based on a plausible line of arguments, crucially, the very first step in its derivation might have been too bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Indeed, this approximation was in essence based on the assumption that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=', for any fixed i ∈ [n], i ̸∈ F(Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩)∆F(Tn,t/n[θ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='16) Does (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='16) hold w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We believe so5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Sadly, we cannot prove it, and therefore (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15) is just a conjecture at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Nevertheless, the heuristic approximation illustrates the pivotal role of events of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='16) for symmetric matrices, which motivates a more fine-grained description of frozen variables as introduced in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This description will finally allow us to find another, more indirect route towards (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15), while still, the belief in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='16) lies at the heart of the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6 Frozen variables revisited As discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5, we cannot prove that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=', removal of row i from Tn,t/n[θ] does not unfreeze i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' To keep track of those “problematic” variables where removal of row i unfreezes variable i, we now give a name to them: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12 (Frailly, firmly and completely frozen variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any matrix A ∈ Fm×n and i ∈ [m ∧ n], we say that (i) i is frailly frozen in A if i ∈ F(A)\\F (A ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩)6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (ii) i is firmly frozen in A if i ∈ F (A ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (iii) i is completely frozen in A if i is firmly frozen in both A and AT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ♦ In addition, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 we show that variables which are frailly frozen in A are also frailly frozen in the transpose AT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' So indeed, we can partition the set of coordinates into five disjoint sets as follows: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (Typecasting of variables) For any matrix A ∈ Fm×n, we partition the set [m ∧ n] into (i) the set X(A) of frailly frozen variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (ii) the set Y(A) of completely frozen variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (iii) the set Z(A) of variables that are neither frozen in A or AT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (iv) the set U(A) of variables that are not frozen in A and firmly frozen in AT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (v) the set V(A) of variables that are firmly frozen in A and not frozen in AT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For each i ∈ [m ∧ n], we refer to the category it belongs to with respect to the above partition as its type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ♦ This distinction between different types of frozen variables is a chief ingredient in our calculation of the lower bound, and the main difference with respect to the preceding works [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Notably, it allows us to extend core ideas of these articles to symmetric matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For example, with the terminology of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13, we can now express the rank increase of interest alternatively as rkF � Tn+1,t/n[θ] � − rkF � Tn,t/n[θ] � = 1{n + 1 ∈ X(Tn+1,t/n[θ])} + 2 · 1{n + 1 ∈ Y(Tn+1,t/n[θ])} + 1{n + 1 ∈ U(Tn+1,t/n[θ])} + 1{n + 1 ∈ V(Tn+1,t/n[θ])} (for a proof of this identity, see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Returning to the discussion at the end of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5, the typecasting allows us to the derive fixed point equations not for αn,t/n, but for some of the proportions of the finer types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thereby, we gain a better understanding of the proportion of frailly frozen variables and of αn,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' And, what is more, these fixed point equations provide enough information to derive the desired lower bound on ht(αn,t/n) as given below in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14, and therefore to bypass (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='16), which is precisely what we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The derivation of the fixed point equations is the content of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 4See Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 5Our belief is underpinned by the fact that removal of row i has the same effect as attachment of a unit column (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3), which is akin to a pinning operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 6This is equivalent to what we need, see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 10 The rank of sparse symmetric matrices over arbitrary fields 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7 The heuristic fixed point equation and its connection to Rd(α) Let us return to the heuristic fixed point equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15), which suggests that only zeroes of the function Gd : [0, 1] �→ R, Gd(α) := α + φd (1 − φd (α)) − 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17) constitute viable candidates for αn,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' And indeed, for any d ≥ 0, Gd has at least one zero: If α0(d) ∈ [0, 1] is such that α0(d) = 1 − φd (α0(d))7, then Gd(α0(d)) = α0(d) + φd (1 − φd (α0(d))) − 1 = −φd(α) + φd(α) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='18) Unfortunately, for some d ≥ 0, Gd has more zeroes: Let α⋆(d) and α⋆(d) denote the smallest and largest zeroes of Gd(α) in [0, 1], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The existence of α⋆(d) and α⋆(d) is guaranteed by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' A detailed analysis of the function Gd, its zeroes and relation to the function 1 − Rd is carried out in [9], where the asymmetric counterpart of An,p with all non-zero entries being identical to 1 was studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In [9], the authors show that Gd has at most the three zeroes α⋆(d) ≤ α0(d) ≤ α⋆(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' From the analysis of the finer types as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6, it will become apparent that in the limit, only the two zeroes α⋆(d) and α⋆(d) correspond to possible values of αn,d/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For the asymmetric case, where no perturbation is necessary, the connection between Gd and the proportion of frozen variables has been studied in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' While we cannot derive a picture as detailed as in [9], we can show that ht(αn,t/n) is no less than ht evaluated at one of the zeroes, which provides a sufficient substitute for the exact asymptotic characterisation of αn,t/n: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14 (Lower bound on the rank increase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any d > 0, ht � αn,t/n � ≥ ht (α⋆(t)) + ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19) Recall that the principal aim of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15) was to establish a connection to the rank formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3), which comes in terms of an optimization problem over [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The function Rd attains its minimum on [0, 1] either for α ∈ {0, 1} or for α ∈ (0, 1) such that R′ d(α) = d2φd (α) (α + φd (1 − φd (α)) − 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='20) The little calculation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='20) shows that R′ d(α) = 0 if and only if Gd(α) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Indeed, α⋆(d) and α⋆(d) are the two only minimizers of Rd on [0, 1]: Rd (α⋆(d)) = Rd (α⋆(d)) = min α∈[0,1] Rd(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='21) Thus doubtlessly, the lower bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19) establishes a connection to the minimizers of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Given Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14, it is now a matter of analysis to prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2), which we complete next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9 1 d α⋆(d) α0(d) α⋆(d) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9 1 d minα∈[0,1] Rd(α) Figure 2: Left: Plot of α⋆(d), α0(d) and α⋆(d), which are distinct for d > e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Right: Plot of the function d �→ minα∈[0,1] Rd(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 7The existence and uniqueness of α0(d) are straightforward to check, see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 11 The rank of sparse symmetric matrices over arbitrary fields 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8 Lower bound on the expected rank: Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 subject to Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14 An application of Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14 now gives the following lower bound for 1 nE � rkF � Tn,d/n[θ] �� : 1 nE � rkF � Tn,d/n[θ] �� ≥ 1 n n−1 � m=εn � E � rkF � Tm+1,(dm/n)/m[θ] �� − E � rkF � Tm,(dm/n)/m[θ] ��� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='22) = 1 n n−1 � m=εn E � hdm/n � αm,d/n �� + on,P (1) ≥ 1 n n−1 � m=εn hdm/n (α⋆(dm/n)) + on,P (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The sum 1 n �n−1 m=εn hmd/n (α⋆(md/n)) can be treated as a Riemann sum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=', 1 nE � rkF � Tn,d/n[θ] �� ≥ � 1 ε hds (α∗ (ds)) ds + on,P (1) = 1 d � d εd ht (α⋆(t)) dt + on,P (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Taking the appropriate limits on both sides gives lim inf P →∞ lim inf n→∞ inf N≥n, JN∈SymN(F∗) 1 nE � rkF � Tn,d/n[θ] �� ≥ 1 d � d εd ht (α⋆(t)) dt and since we can choose ε arbitrarily small, we conclude that lim inf P →∞ lim inf n→∞ inf N≥n, JN∈SymN(F∗) 1 nE � rkF � Tn,d/n[θ] �� ≥ 1 d � d 0 ht (α⋆(t)) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then indeed, as we prove in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3, the derived integral expression coincides with the desired rank formula: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15 (Integral evaluation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any d ≥ 0, � d 0 ht (α⋆(t)) dt = d · Rd (α⋆(d)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Now, the combination of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='21) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15 yields that 1 d � d 0 ht (α⋆(t)) dt = Rd (α⋆(d)) = min α∈[0,1] Rd(α) and therefore by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8) that lim inf n→∞ inf N≥n, JN∈SymN(F∗) 1 nE � rkF � Tn,d/n �� ≥ lim inf P →∞ lim inf n→∞ inf N≥n, JN∈SymN(F∗) 1 nE � rkF � Tn,d/n[θ] �� ≥ min α∈[0,1] Rd(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='23) By definition of Tn,p = T (N) n,p , we have rkF � T (N) N,p � = rkF (AN,p) and consequently inf Jn∈Symn(F∗) 1 nE � rkF � An,d/n �� = inf N=n, JN∈SymN(F∗) 1 nE � rkF � T (N) n,d/n �� ≥ inf N≥n, JN∈SymN(F∗) 1 nE � rkF � T (N) n,d/n �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='24) Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 now follows from the combination of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='23) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9 Discussion The understanding of the ensemble of adjacency matrices of Erd˝os-Rényi random graphs has seen major advances during the last two decades, in particular with respect to its real rank and spectral properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Prominently, ln(n)/n is a threshold for the singularity of these matrices [5, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' More generally, in the regime where pn ∈ [c ln(n)/n, 1/2] for c > 0 and for more general real matrix entries as considered in the current article, Costello and Vu [16] show that with high probability, the nullity of An,pn is exactly equal to the number of isolated vertices in the underlying Erd˝os-Rényi random graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In the same spirit, DeMichele, Moreira and Glasgow [18] show that for pn = ω(1/n), with high probability, the nullity of An,pn coincides with the number of isolated vertices in the graph that arises from Gn,d/n after an application of the Karp-Sipser algorithm described in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In an associated random matrix process where edges are revealed one after the other, Addario-Berry and Eslava [1] derive a hitting time theorem in the 12 The rank of sparse symmetric matrices over arbitrary fields sense that with high probability, the matrix becomes singular at the exact moment when there are no zero rows and columns left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In the challenging sparse regime where pn = d/n for fixed d > 0, much less is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Notably, there is the asymptotic rational rank formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3) for An,d/n by Bordenave, Lelarge and Salez [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Recently, building on the machinery of [8], Ferber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' [23] have shown that the k-core for k ≥ 3 is non-singular with high probability, thereby resolving an open conjecture of Vu from 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This work has been inspired by recent advances on the rank of random matrices in the context of random constraint satisfaction problems, in particular work on the k-XORSAT problem [4, 12, 19, 20, 28] and a model inspired by random code ensembles [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In this context, it is natural to consider the matrices not only over the reals, but as binary matrices or more generally, matrices over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Correspondingly, this article crucially builds on the methodology developed in [4, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' However, because of the symmetry of our model, virtually all core ides have to be developed differently in comparison to [4, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' First of all, we modify the perturbation according to Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' While the basic idea of a perturbation as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8 in the context of random graphical models goes back to information theory [27], it has since been successfully applied to the study of random inference problems and random factor graphs [11, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The basis for an application to asymmetric sparse random matrices, in combination with the conceptualisation of linear relations, has been laid out in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In comparison to this previous application, the perturbation in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8 is of a slightly different flavour, since it cannot be straightforwardly interpreted as the addition of unary factor nodes in the underlying graphical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In [10] and the earlier version [4], as well as in results on random factor graphs, the perturbation has proven to be particularly useful when combined with the Aizenman-Sims-Starr scheme from mathematical physics, which brings us to our next modification: Instead of combining the pinning operation with the Aizenman-Sims-Starr scheme, we apply the telescoping argument (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4) and therefore compare the matrices Tm+1,d/n and Tm,d/n rather than Tn+1,d/(n+1) and Tn,d/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This is due to the fact that an application of the Aizenman-Sims-Starr scheme as in [10] would require knowledge about the event (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='16) that we do not have, and comes at the price of pursuing a different route to characterise αn,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We therefore introduce frailly frozen variables, which are probably the most essential difference between this article and the previous work on asymmetric matrices [4, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Finally, we believe that the methods developed in this article will generalise to broader symmetric matrix structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' It would also be interesting to see whether the fraction of frozen variables in the unperturbed matrix An,d/n satisfies an anti-concentration result as its asymmetric counterpart [9], or whether the two models behave differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In hindsight, the rank formula Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 gives us some information about the perturbed matrix and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14 shows that there are essentially two cases: In the first case, the proportion of frailly frozen variables xn,t/n is approximately zero and the proportion of frozen variables αn,t/n is approximately α⋆(d) or α⋆(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In the second case, the proportion of frailly frozen variables xn,t/n is approximately α⋆(d) − α⋆(d) and the proportion of frozen variables αn,t/n is approximately α⋆(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' From simulations, it seems likely that only the first case corresponds to the actual asymptotic behaviour of the perturbed matrices under consideration, but we cannot exclude the second case at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 3 Matrix perturbations In this short section, we prove the two most important properties of the matrix perturbation introduced in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8: In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, we construct the coupling from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6, which ensures that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=', for any two large square matrices that differ by one in their size, their canonical perturbation is based on the same row- and column-perturbation matrices (compare (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2, we then prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10 on the joint deletion of short proper relations in both the perturbed A and its transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 Coupling of perturbation matrices: Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6 To couple the matrices Θr[θr, n1|n2], we couple the locations of their non-zero entries row by row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For a given row k, the basic idea is to construct a coupling (jk,n1)n1≥1 of uniformly distributed random variables jk,n1 ∼ Unif([n1]) on increasing integer intervals, such that for any two random variables, P (jk,n0 ̸= jk,n1) = dTV (Unif([n0]), Unif([n1])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For the overall coupling, we then take the product distribution over the rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' More precisely, let (uk,ℓ)k,ℓ≥1 be an array of independent random variables such that for all k, ℓ ≥ 1, uk,ℓ is uniformly distributed on [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any n1 ∈ N, set jk,n1 = max{ℓ ∈ [n1] : uk,ℓ = ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1) Since uk,1 = 1, the set above is nonempty, and it is straightforward to verify that jk,n1 ∼ Unif([n1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any θr, n2 ∈ N, n1 ∈ [n2], let Θr[θr, n1|n2] ∈ Fθr×n2 be the matrix where row k ∈ [θr] has its unique non-zero entry in column jk,n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since the definition of jk,n1 only depends on n1, but not on θr or n2, this coupling satisfies properties (i) and (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 13 The rank of sparse symmetric matrices over arbitrary fields Consider now n0 ≤ n1 ≤ n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then Θr[θr, n0|n2] = Θr[θr, n1|n2] if and only if jk,n0 = jk,n1 for all k ∈ [θr], or equivalently uk,n0+1 < n0 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , uk,n1 < n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, P (Θr[θr, n0|n2] = Θr[θr, n1|n2]) = n1 � k=n0+1 �k − 1 k �θr = �n0 n1 �θr , so that the coupling satisfies (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The coupling of {Θc[m1|m2, θc] : θc, m2 ≥ 1, m1 ∈ [m2]} can be constructed along the same lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 Perturbation eliminates most short proper relations: Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10 Recall the definition of the canonical perturbation from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In this section, we prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10, which ensures that for any δ > 0 and ℓ ∈ N≥2, the canonical perturbation of any (almost) square matrix A, as well as its transpose, are (δ, ℓ)-free with probability arbitrarily close to one, provided that the matrix dimension and the perturbation parameter P are chosen large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The main ingredient in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10 is the following lemma: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 ([10, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let δ > 0 and ℓ ∈ N≥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then there exists P ′ = P ′(δ, ℓ) ∈ N such that for any P ≥ P ′ the following holds: For any matrix A ∈ Fm×n P �� A Θr[θr, n|n] � is (δ, ℓ)-free � ≥ 1 − δ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2) provided that θr ∼ Unif([P]) and is independent of the coupling {Θr[θr, n1|n2] : θr, n2 ≥ 1, n1 ∈ [n2]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 is a minor adaptation of [10, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' While the exact wording is for P = P ′(δ, ℓ) rather than all P ≥ P ′(δ, ℓ), its proof shows that all choices of P > 4ℓ3/δ4 imply (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ■ Before we prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10, we observe the following simple consequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let δ > 0 and L ∈ N≥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then there exists P ′ = P ′(δ, L) ∈ N such that for any P ≥ P ′ the following holds: For any matrix A ∈ Fm×n and n1 ∈ [n] P �� A Θr[θr, n1|n] � is (δ, ℓ)-free for 2 ≤ ℓ ≤ L � ≥ �n1 n �P − δ, provided that θr ∼ Unif([P]) and is independent of the coupling {Θr[θr, n1|n2] : θr, n2 ≥ 1, n1 ∈ [n2]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Fix δ > 0 and L ∈ N≥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any ℓ ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , L}, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 guarantees the existence of Pℓ = Pℓ(δ/L, ℓ) ∈ N such that for any P ≥ Pℓ and θr ∼ Unif([P]), P �� A Θr[θr, n|n] � is (δ, ℓ)-free � ≥ P �� A Θr[θr, n|n] � is (δ/L, ℓ)-free � ≥ 1 − δ/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3) Let P ′ = max2≤ℓ≤L Pℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then for any P ≥ P ′ and θr ∼ Unif([P]), by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3) and a union bound, P �� A Θr[θr, n|n] � is (δ, ℓ)-free for 2 ≤ ℓ ≤ L � ≥ 1 − L � ℓ=2 P �� A Θr[θr, n|n] � is not (δ, ℓ)-free � ≥ 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4) By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6 (iii), P (Θr[θr, n|n] = Θr[θr, n1|n]) = E � (n1/n)θr� ≥ (n1/n)P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, P �� A Θr[θr, n|n] � (δ, ℓ)-free for 2 ≤ ℓ ≤ L � ≤ P �� A Θr[θr, n1|n] � (δ, ℓ)-free for 2 ≤ ℓ ≤ L � + 1 − �n1 n �P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5) Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5) yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Fix δ > 0, L ∈ N≥2 and s ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For n, P ∈ N, let θ = (θr, θc) ∼ Unif([P]2) and A ∈ F(n+s)×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' With the coupling from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6 and A′ := (A Θc[n + s|n + s, θc]), A[θ] = � A Θc[n + s|n + s, θc] Θr[θr, n|n] 0θr×θc � = � A′ Θr[θr, n|n + θc] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6) Conditionally on A′ and θc, because of independence of the row and column perturbations, Θr[θr, n|n + θc] is distributed as the perturbation in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 with the ensuing choice of n1 and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thus, for any a > 1, if P(δ/a, L) is chosen large enough, conditioning on A′ and θc in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6) yields that for P ≥ P(δ/a, L), P (A[θ] is (δ, ℓ)-free for 2 ≤ ℓ ≤ L) ≥ P (A[θ] is (δ/a, ℓ)-free for 2 ≤ ℓ ≤ L) ≥ � n n + P �P − δ/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7) 14 The rank of sparse symmetric matrices over arbitrary fields By an analogous argument, also for P ≥ P(δ/a, L), P � A[θ]T is (δ, ℓ)-free for 2 ≤ ℓ ≤ L � ≥ � n + s n + s + P �P − δ/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since P(B1 ∩ B2) ≥ P(B1) + P(B2) − 1 for any two events B1, B2, we conclude that P � Both A[θ] and A[θ]T are (δ, ℓ)-free for 2 ≤ ℓ ≤ L � ≥ � n + s n + s + P �P + � n n + P �P − 2δ/a − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In particular, lim sup P →∞ lim sup n→∞ sup A∈F(n+s)×n P � A[θ] or A[θ]T is not (δ, ℓ)-free for some 2 ≤ ℓ ≤ L � ≤ 2δ/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since this is upper bound holds for any a > 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' It is natural to wonder whether there is a possibility to perturb a symmetric matrix A such that the perturbed matrix A[θ] is symmetric as well and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' However, simply choosing Θc[n|n, θc] = Θr[θr, n|n]T does not have the desired effect: For (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7) to hold, it is crucial that both the number of rows as well as the columns of the non-zero indices of Θr[θr, n|n + θc] are chosen uniformly given A′ = (A Θc[n|n, θc]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thus, the above perturbation technique necessarily destroys the matrix symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ■ 4 Frozen variables: General properties & stability The principle aim of this section is to derive general properties of the various types of frozen variables as well as to prove stability of the proportions of types in the transition from Tn,t/n[θ] to Tn+1,t/n[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In this sense, our main result of this section, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11, asserts that the proportions of the various types remain nearly unchanged when we grow the matrix from n to n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 How the type of a variable encodes rank change under row- and column removal We first present basic deterministic implications of the type of a variable that are used throughout the article, and that indicate the significance of the types of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' More specifically, we are ultimately interested in the rank decrease upon simultaneous removal of row i and column i from a given matrix A ∈ Fm×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In this section, we prove that the type of i according to Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13 completely determines the ensuing rank change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The starting point is the following lemma on frozen variables: Living up to their name, in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4, frozen variables were characterised as coordinates that take the value zero in any kernel vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The following lemma shows how the rank of any given matrix changes, if a column that corresponds to a frozen variable is removed from it: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 ([18, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let A ∈ Fm×n and i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then i ∈ F(A) ⇐⇒ rk (A) − rk (A ⟨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Recall that we denote the ith standard unit vector in F1×n by en(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' While the linear dependencies of column i of A with the other columns of A may be intricate, attaching en(i) at the bottom of A surely renders column i linearly independent of all the other columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thus rk � A en(i) � = 1 + rk (A ⟨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, by Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3, i is frozen in A if and only if en(i) is in the row span of A, so rk � A en(i) � = rk (A) + 1 {i /∈ F(A)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The next lemma demonstrates that, generally, column removal and row addition cannot “unfreeze” variables: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let A ∈ Fm×n, b ∈ Fm×1, c ∈ F1×n and i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then (i) i ∈ F ((A b)) =⇒ i ∈ F(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (ii) i ∈ F(A) =⇒ i ∈ F �� A c �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Both statements immediately follow from the characterisation of frozen variables from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1: variable i is frozen in A if and only if column i does not lie in the linear span of the other columns of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 15 The rank of sparse symmetric matrices over arbitrary fields While Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 shows that addition of rows can only enlarge the set of frozen variables, the next lemma studies the consequences of row removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Indeed, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 illustrates that the removal of a row has the same effect as addition of a unit vector (which effectively forbids to use the corresponding row in representations): Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any matrix A ∈ Fm×n, i ∈ [n] and j ∈ [m], i ∈ F(A ⟨j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩) ⇐⇒ i ∈ F �� A em(j)T �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Throughout the proof, we abbreviate B = � A em(j)T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Assume that i ∈ F (A ⟨j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since B ⟨j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ only has a zero-column appended at the right in comparison to A ⟨j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩, i is also frozen in B ⟨j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 (ii), adding a row cannot unfreeze variables, so i ∈ F (B) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Next, assume that i ∈ F(B) and let y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , ym) be a representation of {i} in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since no row of B apart from j has a non-zero entry in column n + 1, yj = (yB)n+1 = 0, which implies that y ⟨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' j⟩ is a representation of {i} in A ⟨j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We next take a closer look at the frailly frozen variables, which were characterised as those variables that unfreeze under removal of the identically indexed row (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12 (i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since on the other hand, variables can never freeze under row removal, we obtain the following corollary of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 (ii), which expresses that the frailly frozen variables are exactly those variables that are classified differently in the matrix with one appropriately chosen row less than in the original matrix: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any A ∈ Fm×n and i ∈ [m ∧ n], i is frailly frozen in A ⇐⇒ i ∈ F(A)∆F (A ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13, we have claimed that for any matrix A ∈ Fm×n, the set [m ∧ n] can be partitioned into five types of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The next proposition shows that this claim is justified, since any variable in [m ∧ n] is either frailly frozen, firmly frozen or unfrozen in A, and if it is frailly frozen in A, then it must also be frailly frozen in AT : Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let A ∈ Fm×n and i ∈ [m ∧ n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then i is frailly frozen in A ⇐⇒ i is frailly frozen in AT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5 by means of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 and the following observation: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any matrix A ∈ Fm×n, vectors b ∈ Fm×1, c ∈ F1×n and f ∈ F, rk � A c � − rk(A) = 0 and rk � A b c f � − rk (A b) = 1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1) if and only if rk (A b) − rk(A) = 0 and rk � A b c f � − rk � A c � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2) Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Denote rk � A b c f � − rk(A) by h and assume that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then h = rk (A b) − rk(A) + rk � A b c f � − rk (A b) ≥ rk � A b c f � − rk (A b) = 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3) as well as h = rk � A b c f � − rk � A c � + rk � A c � − rk(A) ≤ 1 + rk � A c � − rk(A) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4) Therefore, h = 1, and we must have equality throughout (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2) then follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The converse implication can be shown to be true analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The assertion is an immediate consequence of the characterisation of frozen variables in terms of rank decrease upon column removal from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 in combination with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6 applied to A ⟨i, i⟩ , A ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ , A ⟨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩ and A, since the rank of a matrix is identical to that of its transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The final result of this section, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7, connects the five variable categories X(A), Y(A), Z(A), U(A), V(A) from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13 to the following rank changes under symmetric row and column removal: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any A ∈ Fm×n and i ∈ [m ∧ n], (i) i ∈ Y(A) ⇐⇒ rk(A) − rk(A ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩) = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 16 The rank of sparse symmetric matrices over arbitrary fields (ii) i ∈ X(A) ∪ U(A) ∪ V(A) ⇐⇒ rk(A) − rk(A ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (iii) i ∈ Z(A) ⇐⇒ rk(A) − rk(A ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thus, rk(A) − rk(A ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩) = 1{i ∈ X(A)} + 2 · 1{i ∈ Y(A)} + 1{i ∈ U(A)} + 1{i ∈ V(A)} = 1 + 1{i ∈ Y(A)} − 1{i ∈ Z(A)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let i ∈ [m ∧ n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 yields the representation rk(A) − rk(A ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩) = rk(A) − rk(A ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩) + rk(A ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩) − rk(A ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩) = 1 � i ∈ F(AT ) � + 1 {i ∈ F(A ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6) Identities (i)-(iii) now follow from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6) by an application of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 Appending a row to a (δ, ℓ)-free matrix In the present section, we discuss how the rank of a (δ, ℓ)-free matrix A changes upon the attachment of a single row b with exactly ℓ non-zero entries, which are chosen uniformly from a subset of the columns of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Recall that in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6), we had observed that for a general vector b to be in the row span of A, it is sufficient that supp (b) ⊆ F(A) and necessary that either supp (b) ⊆ F(A) or supp (b) forms a proper relation in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' These considerations show that in the complete absence of “short” proper relations in A, rank stagnation upon attachment of a vector b with ℓ non-zero entries can be equivalently described by the event that all variables of supp (b) are frozen in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8, which revisits an argument from the proof of [10, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4], shows how to transfer the above reasoning to matrices with few short proper relations, where the dominant reason for a rank stagnation upon attachment of a vector should still be the event that all variables in its support are frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For convenience of the reader, we revisit the main step of the argument in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For this, let A ∈ Fm×n and PRℓ(A) = {I ⊆ [n] : I is a proper relation of A with |I| = ℓ} and PR(A) = ∪∞ ℓ=2 PRℓ(A) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7) be the set of proper relations of A of size ℓ ≥ 2 as well as the set of all proper relations of A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8 ([10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Fix δ > 0, ℓ ∈ N≥2 and s ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any sequence ((bn−s,n, bn−s+1,n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , bn,n))n∈N such that for all n and n1 ∈ [n] \\ [n − s − 1], bn1,n ∈ F1×n and supp(bn1,n) is uniformly distributed over all ℓ-subsets of [n1], sup m∈{n−s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=',n+s} n1∈{n−s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=',n} sup A∈Fm×n: A is (δ,ℓ)−free P (supp (bn1,n) ∈ PRℓ(A)) ≤ δℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' + on(1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8) and sup m∈{n−s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=',n+s} n1∈{n−s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=',n} sup A∈Fm×n: A is (δ,ℓ)−free �����E � rkF � A bn1,n �� − rkF(A) − � 1 − �|F(A) ∩ [n1]| n1 �ℓ������ ≤ δℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' + on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Observe that E � rkF � A bn1,n �� − rkF(A) = 1 − P (bn1,n is in the span of the rows of A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10) As discussed in the beginning of the subsection, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6) gives P (supp (bn1,n) ⊆ F(A)) ≤P (bn1,n is in the span of the rows of A) ≤P (supp (bn1,n) ∈ PRℓ(A)) + P (supp (bn1,n) ⊆ F(A)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11) For any (δ, ℓ)-free matrix A ∈ Fm×n, |PRℓ(A)| ≤ δnℓ, and therefore P (supp (bn1,n) ∈ PRℓ(A)) ≤ |PRℓ(A)| �n1 ℓ � ≤ δℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' � n n − s − ℓ �ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12) Taking the supremum over all (δ, ℓ)-free matrices A ∈ Fm×n, then m ∈ {n − s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , n + s} and n1 ∈ {n − s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , n} yields (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' To estimate P (supp (bn1,n) ⊆ F(A)), let α(A) = |F(A) ∩ [n1]| /n1 be the proportion of frozen variables of A among [n1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then ��P (supp (bn1,n) ⊆ F(A)) − α(A)ℓ�� = ���� �n1α(A) ℓ ���n1 ℓ � − α(A)ℓ ���� = O(1/n) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13) uniformly in m, n1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10) - (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13) yields (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 17 The rank of sparse symmetric matrices over arbitrary fields 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 Stability of types As outlined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5, a central ingredient in our proof strategy is to show that the proportion of frozen variables in Tn,t/n[θ] is close to that in Tn+1,t/n[θ], which is the core theme of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Specifically, we look at the extended variable types X, Y, Z, U and V from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For each of these types, we define the share it has among the variables of a matrix with perturbation, where the artificial row-perturbation columns are not taken into account: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9 (Proportions of types).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (i) For A ∈ Fn×n and W ∈ {X, Y, Z, U, V}, we use the non-calligraphic lowercase letter w to denote the proportion of variables i ∈ [n] of the corresponding type: w(A[θ]) = |W(A[θ]) ∩ [n]| n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (ii) For A ∈ Fn×n, we denote the vector of all proportions by ζ(A[θ]) = (x(A[θ]), y(A[θ]), z(A[θ]), u(A[θ]), v(A[θ])) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (iii) For A = Tn,t/n and wn,t/n ∈ {xn,t/n, yn,t/n, zn,t/n, un,t/n, vn,t/n, ζn,t/n}, we simply write wn,t/n = w � Tn,t/n[θ] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ♦ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10 (Summation of proportions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By definition, for any matrix A ∈ Fn×n, ∥ζ(A[θ])∥1 = x(A[θ]) + y(A[θ]) + z(A[θ]) + u(A[θ]) + v(A[θ]) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14) Moreover, recall that αn,p and αT n,p denote the proportions of frozen variables among [n] in Tn,p[θ] and Tn,p[θ]T , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' With the above definitions, αn,t/n = xn,t/n + yn,t/n + vn,t/n and αT n,t/n = xn,t/n + yn,t/n + un,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15) ■ With the notation of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9, the main result of the remainder of Section 4 is the following proposition: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 (Stability of types).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any d > 0, and w ∈ {x, y, z, u, v, ζ}, E ��wn,t/n − wn+1,t/n �� 1 = on,P (1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In light of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15), it is tempting to conjecture that the proportions αn,d/n of frozen variables of Tn,d/n[θ] converge in a suitable sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Unfortunately, this conjecture turns out to be incorrect, and one of the implications of our present proof is that αn,d/n does not converge for d > e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Despite this complication, the strictly weaker statement of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 is sufficient for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The rest of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 is organized as follows: In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, we study the impact of symmetric row- and column relabelling on the proper relations and variable types of a given matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2, building on Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, we prove that any fixed variable is unlikely to change from frozen to unfrozen, or the other way round, under one-step matrix growth of Tn,d/n[θ] and a related matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Finally, we present the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 Row- and column exchangeability In the following proofs, exchangeability arguments play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We prepare these arguments in the current section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Throughout this section, for k ∈ N, let Sk denote the symmetric group of [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let A ∈ Fn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For a permutation π ∈ Sn, define the matrix Aπ by setting Aπ(i, j) = A(π−1(i), π−1(j)), for i, j ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='16) ♦ Aπ is the matrix that arises from A through joint relabelling of the rows and columns according to i �→ π−1(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any π ∈ Sn and p ∈ [0, 1], T π n,p[θ] d= Tn,p[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Recall the definition of Tn,p in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3), according to which Tn,p(i, j) = AN,p(τ(i), τ(j)) for i, j ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Hence, T π n,p(i, j) = Tn,p(π−1(i), π−1(j)) = AN,p(τ ◦ π−1(i), τ ◦ π−1(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 18 The rank of sparse symmetric matrices over arbitrary fields Since τ is a uniform permutation of [N], also τ ◦π−1 is a uniform permutation of [N], where we view π as a permutation of [N] that leaves {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , N} fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thus, T π n,p d= Tn,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Independence of Tn,p, τ and the row- and column-perturbation matrices now implies that T π n,p[θ] d= Tn,p[θ], as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let π ∈ Sn and I = {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , ik} ⊂ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Setting Iπ = {π(i1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , π(ik)}, P(I is a proper relation of Tn,p[θ]) = P(Iπ is a proper relation of Tn,p[θ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Note that I is a proper relation of Tn,p[θ] ⇐⇒ Iπ is a proper relation of T π n,p[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The desired result now follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any A ∈ Fn×n, π ∈ Sn, and W ∈ {X, Y, Z, U, V}, i ∈ W(A[θ]) ⇐⇒ π(i) ∈ W(Aπ[θ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' As a consequence, ζ(A[θ]) = ζ(Aπ[θ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By the determinantal rank characterisation and the Leibniz determinant formula, the rank of A stays unchanged under the permutation π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, i ∈ F(A) ⇐⇒ rk (A) − rk (A ⟨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩) = 1 ⇐⇒ rk (Aπ) − rk (Aπ ⟨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' π(i)⟩) = 1 ⇐⇒ π(i) ∈ F(Aπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17) Analogously, i ∈ F(A ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩) ⇐⇒ π(i) ∈ F(Aπ ⟨π(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='18) The desired results now follow from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In particular, we will make frequent use of the following corollary: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any i ∈ [n], p ∈ [0, 1] and W ∈ {X, Y, Z, U, V}, P (i ∈ W (Tn,p[θ])) = E [wn,p] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let π ∈ Sn be the transposition of n + 1 and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13 together imply that P (n + 1 ∈ W (Tn,p[θ])) = P � π(n + 1) ∈ W � T π n,p[θ] �� = P (i ∈ W (Tn,p[θ])) and therefore P (n + 1 ∈ W (Tn,p[θ])) = E � 1 n + 1 n+1 � i=1 1 {i ∈ W (Tn,p[θ])} � = E [wn,p] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 Freezing and unfreezing under row- and column removal Building upon the symmetry arguments of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, we now prove the two main lemmas that are needed to attack Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11, which makes a statement about the expected differences of the various proportions of variable types in Tn,t/n[θ] and Tn+1,t/n[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since the type of variable i ∈ [n] with respect to the matrix A ∈ Fn×n is defined solely in terms of the membership of i in each of the sets F(A), F(AT ), F(A ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩) and F(AT ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩) (see Definitions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13) and the matrices Tn,t/n[θ] and Tn+1,t/n[θ] are reasonably alike, it seems like a viable strategy to show that any given variable is unlikely to change its membership in each of the aforementioned sets of frozen variables in the transition from Tn,t/n[θ] to Tn+1,t/n[θ], which is precisely what we show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In this sense, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17 shows that any fixed variable is unlikely to be frozen in exactly one of the matrices Tn,t/n[θ] or Tn+1,t/n[θ]: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17 (One-step matrix growth, original matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Fix d, δ > 0 and L ∈ N≥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then for any i ∈ [n], P � i ∈ F � Tn,t/n[θ] � ∆F � Tn+1,t/n[θ] �� ≤ 2(L + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='δ + P (Po(d) ≥ L) + on,P (1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Conveniently, the pair (Tn,t/n[θ], Tn+1,t/n[θ]) is identically distributed to (Tn,t/n[θ]T , Tn+1,t/n[θ]T ), so it is enough to work with non-transposed matrices in the above considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In the same spirit, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='18 shows that any fixed variable is unlikely to be frozen in exactly one of the matrices Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ or Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩: 19 The rank of sparse symmetric matrices over arbitrary fields Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='18 (One-step matrix growth, row-deleted matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Fix d, δ > 0 and L ∈ N≥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then for any i ∈ [n], P � i ∈ F � Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ � ∆F � Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ �� ≤ 2(L + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='δ + (L + 2)P (Po (d) ≥ L) + on,P (1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' While Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='18 is structurally similar to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17, its proof proceeds differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This is due to the fact that the removal of row i makes this index special, and the exchangeability arguments that are used in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17 do not apply directly to the modified setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We use Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8 to overcome this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Finally, we also prove a third lemma which shows that a small deterministic increase in θ is unlikely to change whether variable i is frozen or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19 is not used in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 and will only become relevant in Section 5, but since it is similar in spirit to the previous two lemmas, we include it here: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19 (Deterministic perturbation shift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let µ = (µr, µc) ∈ N2, A ∈ Fm×n and i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then P (i ∈ F (A[θ]) ∆F (A[θ + µ])) ≤ µr + µc P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' While the proofs of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='18 heavily depend on the structure of Tn,t/n, the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19 only uses properties of the perturbation, and thus the result is true for arbitrary matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Two good events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Before we turn to the proofs of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19, we define two good events that will be used here and later throughout the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For p ∈ [0, 1], let Rn,p = � both Tn,p[θ] and Tn,p[θ]T are (δ, ℓ)-free for 2 ≤ ℓ ≤ L � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19) and Pn = {Θr[θr, n|n] = Θr[θr, n + 1|n + 1] ⟨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' n + 1⟩ , Θc[n|n, θc] = Θc[n + 1|n + 1, θc] ⟨n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='20) Rn,p ensures that the rank increase upon attaching rows and columns can be controlled as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8, while the benefit of Pn is that when growing the matrix from n to n + 1, the perturbation stays unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6, P (Rn,p) ≥ 1 + on,P (1) and P (Pn) = 1 + on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='21) The bound P � Rc n,p � ≤ on,P (1) holds uniformly in p ∈ [0, 1], since it is based on Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In the following, we frequently work on the intersection of Rn,t/n and Pn, which is a sufficiently likely event by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We now prove Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19 in their order of appearance: Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since i can either freeze or unfreeze when the matrix is grown, P � i ∈ F � Tn,t/n[θ] � ∆F � Tn+1,t/n[θ] �� = P � i ∈ F � Tn,t/n[θ] � \\F � Tn+1,t/n[θ] �� + P � i ∈ F � Tn+1,t/n[θ] � \\F � Tn,t/n[θ] �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We bound both cases separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (i) Unfreezing: To bound the probability that i is frozen in Tn,t/n[θ], but not in Tn+1,t/n[θ], we first show that on Pn, i ∈ F � Tn,t/n[θ] � \\F � Tn+1,t/n[θ] � =⇒ {i, n + 1} is a proper relation in Tn+1,t/n[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='22) Assume that Pn holds and i ∈ F � Tn,t/n[θ] � \\F(Tn+1,t/n[θ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then Tn+1,t/n[θ] arises from Tn,t/n[θ] through the symmetric attachment of a row and a column, which we may break into two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2, attaching a row cannot unfreeze i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=', i ∈ F � Tn,t/n[θ] � =⇒ i ∈ F � Tn+1,t/n[θ] ⟨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' n + 1⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In particular, there exists a representation y of {i} in Tn+1,t/n[θ] ⟨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' n + 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Attaching column n + 1 and using the representation y on the resulting matrix Tn+1,t/n[θ] yields {i} ⊆ supp � yTn+1,t/n[θ] � ⊆ {i, n + 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since i ̸∈ F � Tn+1,t/n[θ] � by assumption, we conclude that supp � yTn+1,t/n[θ] � = {i, n + 1} , which implies that {i, n + 1} is a proper relation in Tn+1,t/n[θ], since the existence of the representation y ensures that n + 1 cannot be frozen in Tn+1,t/n[θ] without i being frozen in Tn+1,t/n[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This proves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 20 The rank of sparse symmetric matrices over arbitrary fields The next step is to show that, on the good event Rn+1,t/n, the probability that {i, n + 1} forms a proper relation in Tn+1,t/n[θ] is small: This is an immediate consequence of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14, which asserts that the probability to be a proper relation is the same for any pair {i1, i2} for 1 ≤ i1 < i2 ≤ n + 1 and the observation that on Rn+1,t/n, there are at most δ(n + 1 + P)2 proper relations of length two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, P � {i, n + 1} is a proper relation in Tn+1,t/n[θ], Rn+1,t/n � ≤ 2δ + on,P (1), uniformly in t ∈ [0, d] and thus also P � i ∈ F � Tn,t/n[θ] � \\ F � Tn+1,t/n[θ] � , Pn, Rn+1,t/n � ≤ 2δ + on,P (1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='23) (ii) Freezing: To bound the probability that i is frozen in Tn+1,t/n[θ], but not in Tn,t/n[θ], we show that on Pn, i ∈ F � Tn+1,t/n[θ] � \\F � Tn,t/n[θ] � , i /∈ supp � Tn+1,t/n(n + 1, ) � =⇒ {i} ∪ supp � Tn+1,t/n[θ](n + 1, ) � is a proper relation in Tn,t/n[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='24) Assume that Pn holds, i ∈ F(Tn+1,t/n[θ])\\F � Tn,t/n[θ] � and i /∈ supp � Tn+1,t/n(n + 1, ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then the matrix Tn,t/n[θ] arises from Tn+1,t/n[θ] through symmetric removal of a column and a row, which we may break into two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2, removing a column cannot unfreeze i: i ∈ F � Tn+1,t/n[θ] � =⇒ i ∈ F � Tn+1,t/n[θ] ⟨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' n + 1⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This implies in particular that there exists a representation y of {i} in Tn+1,t/n[θ] ⟨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' n + 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' If there was a representation y of {i} in Tn+1,t/n[θ] ⟨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' n + 1⟩ with yn+1 = 0, then shortening y to y ⟨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' n + 1⟩ would be a representation of {i} in Tn,t/n[θ], in contrast to our assumption that i is not frozen in Tn,t/n[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thus, all representations y of {i} in Tn+1,t/n[θ] ⟨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' n + 1⟩ have their (n + 1)st coordinate different from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since we assume that i /∈ supp � Tn+1,t/n(n + 1, ) � , we conclude that supp � y ⟨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' n + 1⟩ Tn,t/n[θ] � = {i} ∪ supp � Tn+1,t/n[θ](n + 1, ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This implies that {i} ∪ supp � Tn+1,t/n[θ](n + 1, ) � is a proper relation of Tn,t/n[θ], since it contains the non-frozen variable i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thus, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='24) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The next step is to show that on the good event Rn,t/n, the probability that {i} ∪ supp � Tn+1,t/n[θ](n + 1, ) � forms a proper relation in Tn,t/n[θ] is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We first upper-bound the probability that row n + 1 has too many non-zero entries, which is due to the sparsity of the matrix Tn+1,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By [25, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10], we can upper bound P ���supp � Tn+1,t/n[θ](n + 1, ) ��� ≥ L � ≤P (Bin (n, t/n) ≥ L) + P (Pc n) ≤P (Po (t) ≥ L) + P (Pc n) + on(1) ≤ P (Po (d) ≥ L) + on(1) uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14, the probability to be a proper relation in Tn,t/n[θ] is the same for any subset of [n] of cardinality ��{i} ∪ supp � Tn+1,t/n[θ](n + 1, ) ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' If row n + 1 has at most L − 1 non-zero entries and Rn,t/n holds, then Tn,t/n[θ] is (δ, ��{i} ∪ supp � Tn+1,t/n[θ](n + 1, ) ���)-free, and P � {i} ∪ supp � Tn+1,t/n[θ](n + 1, ) � ∈ PR � Tn,t/n[θ] � , ��{i} ∪ supp � Tn+1,t/n[θ](n + 1, ) ��� ≤ L, Rn,t/n � ≤ L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='δ + on,P (1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Finally, since P � i ∈ supp � Tn+1,t/n(n + 1, ) �� = P � Tn+1,t/n(n + 1, i) = 1 � = t/n ≤ d/n = on(1) uniformly in t ∈ [0, d], we conclude from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='24) that P � i ∈ F � Tn+1,t/n[θ] � \\ F � Tn,t/n[θ] � , Pn, Rn+1,t/n � ≤ (L + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='δ + P (Po(d) ≥ L) + on,P (1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='25) uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='21), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='23) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='25) finishes the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Again, we relate a status change of i to the existence of a proper relation: On a sufficiently likely event Sn, i ∈ F � Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ � ∆F � Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ � =⇒ supp � Tn,t/n(, i) � is a proper relation in Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T or in Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='26) 21 The rank of sparse symmetric matrices over arbitrary fields Definition of Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The event that we work on is composed of three parts: First, we define Sn,1 = Pn ∩ � Tn+1,t/n[θ](n + 1, i) = 0, Θr[θr, n|n](, i) = 0θr×1, Θc[n|n, θc](i, k) = 01×θc � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On Sn,1, the non-zero entries of column (or equivalently row) i in all involved matrices are contained in [n], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=', supp � Tn,t/n(, i) � = supp � Tn,t/n[θ](, i) � = supp � Tn+1,t/n[θ](, i) � = supp � Tn+1,t/n(, i) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' From the construction of the perturbation Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6 and the definition of Tn+1,t/n, it is immediate that P � Sc n,1 � ≤ 2P n + 1 + d n + 2P n = on(1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Next, let Sn,2 = � for all j ∈ supp(Tn,t/n[θ](, i)) : j /∈ F � Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T � ∆F � Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T �� be the event that no element of the support of column i has a different status in Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T than in Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since (Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T , Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T ) conditionally on Sn,1 and � Tn−1,t/n[θ], Tn,t/n[θ] � conditionally on Pn−1 have the same law, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14, P(Sc n,2) ≤ LP � 1 ∈ F � Tn−1,t/n[θ] � ∆F � Tn,t/n[θ] �� + P(|supp(Tn,t/n[θ](, i))| > L) + P(Sc n,1) + P � Pc n−1 � ≤ (2 + (L + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' )Lδ + (L + 1)P (Po(d) ≥ L) + on,P (1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Finally, let Sn,3 = � Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T and Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T are (δ, ℓ)-free for 2 ≤ ℓ ≤ L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since (Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T , Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T ) conditionally on Sn,1 and � Tn−1,t/n[θ], Tn,t/n[θ] � conditionally on Pn−1 have the same law, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10 implies that P(Sc n,3) = on,P (1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We set Sn = Sn,1 ∩ Sn,2 ∩ Sn,3, so that P(Sc n) ≤ (2 + (L + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' )Lδ + (L + 1)P (Po(d) ≥ L) + on,P (1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='27) Proof of implication (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Now suppose that Sn holds and that supp(Tn,t/n(, i)) is neither a proper relation in Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T nor in Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since the support may contain frozen variables, there are four cases: Case 1: supp(Tn,t/n(, i)) neither has a representation in Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T nor in Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The non-existence of a representation of supp(Tn,t/n ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ (, i)) in Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T in particular implies that column i of Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ is not in the linear span of the other columns of Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thus, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, i is frozen in Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The same reasoning implies that i is frozen in Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Case 2: supp(Tn,t/n(, i)) has a representation both in Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T and in Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since we assume that supp(Tn,t/n(, i)) is neither a proper relation in Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T nor in Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T , all variables in ∅ ̸= supp(Tn,t/n(, i)) must be frozen both in Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T and in Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In this case, the existence of the respective representations ensures that column i of Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ is contained in the linear span of the other columns of Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ and that column i of Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ is contained in the linear span of the other columns of Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thus, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, i is neither frozen in Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ nor in Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Case 3: supp(Tn,t/n(, i)) has a representation in Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T , but none in Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Again, all variables in ∅ ̸= supp(Tn,t/n(, i)) must be frozen in Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T , but there must exist a variable that is not frozen in Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This possibility is excluded by Sn,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Case 4: supp(Tn,t/n(, i)) has a representation in Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T , but none in Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By the same reasoning as in case 3, this cannot happen on Sn,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 22 The rank of sparse symmetric matrices over arbitrary fields Cases 1 to 4 imply (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='26), which gives P(i ∈ F � Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ � ∆F � Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ � ) ≤P � Sn, supp(Tn,t/n(, i)) is a proper relation in Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T or in Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T � + P (Sc n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Finally, since Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T and Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T are (δ, ℓ)-free for 2 ≤ ℓ ≤ L on Sn, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8, P(Sn, supp(Tn,t/n(, i)) is a proper relation in Tn,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T or in Tn+1,t/n[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i⟩T ) ≤2L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='δ + P (Po(d) ≥ L) + on(1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='28) Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='27) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='28) yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The matrix A[θ + µ] arises from A[θ] through the attachment of µr independent unit rows and µc independent unit columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We split this row- and column-attachment into two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since A is non-random, dTV (F(A[θ]), F(A[θ + (0, µc)])) ≤ dTV (θ, θ + (0, µc)) ≤ µc P , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='29) and dTV (F(A[θ + (0, µc)]), F(A[θ + µ])) ≤ µr P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='30) By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2, increasing the number of columns can only diminish the number of frozen variables among the first n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, for i ∈ [n], using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='29), P (i ∈ F (A[θ]) ∆F (A[θ + (0, µc)])) = P (i ∈ F (A[θ])) − P (i ∈ F (A[θ + (0, µc)])) ≤ µc P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='31) Similarly, also by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2, increasing the number of rows can only enlarge the number of frozen variables among the first n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='30), P (i ∈ F (A[θ + (0, µc)]) ∆F (A[θ + µ])) = P (i ∈ F (A[θ + µ)])) − P (i ∈ F (A[θ + (0, µc)])) ≤ µr P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='32) The claim follows by combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='31) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' As the final result of this subsection, we note an immediate consequence of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19, that will be used in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='20 shows that for any i ∈ [n], removal of a bounded number of uniformly chosen rows is unlikely to unfreeze i, even if row i is forbidden to be among the removed rows: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='20 (Random row-removal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any A ∈ Fm×n, k ∈ [m], i ∈ [n] and a uniformly chosen k-subset J ⊆ [m]\\ {i}, P (i ∈ F (A[θ]) ∆F (A[θ] ⟨J ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩)) ≤ k P + � 1 − 1 m �k k(k − 1) 2m + k m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='33) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The proof is based on Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19: First, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 shows that row removal has the same effect on whether i is frozen or not as addition of a unit column vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The latter operation then can be treated as a slight change in the column perturbation, and therefore falls under the scope of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We first replace J by a set obtained from sampling with replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let j′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , j′ k ∈ [m] be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' uniform indices and J ′ = ∪k s=1 {j′ s} such that P(J ̸= J ′) = dTV(J , J ′) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=', we take an optimal coupling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then P (i ∈ F (A[θ]) ∆F (A[θ] ⟨J ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩)) ≤ P � i ∈ F (A[θ]) ∆F � A[θ] � J ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ��� + dTV(J , J ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='34) Furthermore, an application of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='34) gives P (i ∈ F (A[θ]) ∆F (A[θ] ⟨J ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩)) ≤ P(i ∈ F (A[θ]) ∆F (A[θ + (0, k)])) + dTV(J , J ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='35) Since dTV(J , J ′) ≤ (1−1/m)kk(k−1)/(2m)+k/m (see [24] in combination with the observation that J ′ samples from [m] rather than [m] \\ {i}, for example), the claim now follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='35) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 23 The rank of sparse symmetric matrices over arbitrary fields 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 Stability of types: Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 With Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='18, we are now in the position to prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11: Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Observe that for any W ∈ {X, Y, Z, U, V}, the sequence (1 � i ∈ W � Tn,t/n[θ] �� − 1 � i ∈ W � Tn+1,t/n[θ] �� )i∈[n] consists of identically distributed random variables: This is a consequence of the fact that Tn,t/n is a submatrix of Tn+1,t/n, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, E ��wn,t/n − wn+1,t/n �� = 1 nE ����� n � i=1 � 1 � i ∈ W � Tn,t/n[θ] �� − 1 � i ∈ W � Tn+1,t/n[θ] ��� ����� + on(1) ≤ E ��1 � 1 ∈ W � Tn,t/n[θ] �� − 1 � 1 ∈ W � Tn+1,t/n[θ] ���� + on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='36) We next bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='36) for the different types separately: Case 1: wn,t/n = xn,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4 yields the identity 1 � 1 ∈ X � Tn,t/n[θ] �� = 1 � 1 ∈ F � Tn,t/n[θ] � ∆F � Tn,t/n[θ] ⟨1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='37) Using (B1∆B2)∆(B3∆B4) ⊆ (B1∆B3) ∪ (B2∆B4) for any sets B1, B2, B3, B4 and plugging (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='37) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='36) yields the upper bound P � 1 ∈ F � Tn,t/n[θ] � ∆F � Tn+1,t/n[θ] �� + P � 1 ∈ F � Tn,t/n[θ] ⟨1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ � ∆F � Tn+1,t/n[θ] ⟨1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ �� + on(1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='38) for E|xn,t/n − xn+1,t/n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='18 now imply that E|xn,t/n − xn+1,t/n| ≤ 4(L + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='δ + (L + 3)P (Po (d) ≥ L) + on,P (1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='39) In particular, lim sup P →∞ lim sup n→∞ sup N≥n,JN∈SymN(F∗) sup t∈[0,d] E|xn,t/n − xn+1,t/n| ≤ 4(L + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='δ + 3LP (Po (d) ≥ L) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='40) Since the left hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='40) does not depend on L and δ, we can send δ ↓ 0 followed by L → ∞ to conclude that lim sup P →∞ lim sup n→∞ sup N≥n,JN∈SymN(F∗) sup t∈[0,d] E|xn,t/n − xn+1,t/n| = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='41) or equivalently, E|xn,t/n − xn+1,t/n| = on,P (1) uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Case 2: wn,t/n = yn,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12 of completely frozen variables and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 (ii) on row addition yield the identity 1 � 1 ∈ Y � Tn,t/n[θ] �� = 1 � 1 ∈ F � Tn,t/n[θ] ⟨1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ � ∩ F � Tn,t/n[θ]T ⟨1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='42) Using (B1 ∩ B2)∆(B3 ∩ B4) ⊆ (B1∆B3) ∪ (B2∆B4) for any sets B1, B2, B3, B4 and plugging (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='42) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='36) yields the upper bound P � 1 ∈ F � Tn,t/n[θ] ⟨1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ � ∆F � Tn+1,t/n[θ] ⟨1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ �� + P � 1 ∈ F(Tn,t/n[θ]T ⟨1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩)∆F(Tn+1,t/n[θ]T ⟨1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩) � + on(1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='43) for E|yn,t/n − yn+1,t/n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='18 then yields E|yn,t/n − yn+1,t/n| ≤ 4(L + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='δ + 2(L + 2)P (Po (d) ≥ L) + on,P (1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Now the same limiting argument as in Case 1 yields E|yn,t/n − yn+1,t/n| = on,P (1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Case 3: wn,t/n = zn,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 of frozen variables yields the identity 1 � 1 ∈ Z � Tn,t/n[θ] �� = 1 � 1 /∈ F � Tn,t/n[θ] � ∪ F � Tn,t/n[θ]T �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='44) Using (B1 ∪ B2)∆(B3 ∪ B4) ⊆ (B1∆B3) ∪ (B2∆B4) for any sets B1, B2, B3, B4 and plugging (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='44) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='36) yields the upper bound P � 1 ∈ F � Tn,t/n[θ] � ∆F � Tn+1,t/n[θ] �� + P � 1 ∈ F � Tn,t/n[θ]T � ∆F � Tn+1,t/n[θ]T �� + on(1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='45) for E|zn,t/n − zn+1,t/n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17 then yields E|zn,t/n − zn+1,t/n| ≤ 4(L + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='δ + 2P (Po(d) ≥ L) + on,P (1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 24 The rank of sparse symmetric matrices over arbitrary fields Now the same limiting argument as in Case 1 yields E|zn,t/n − zn+1,t/n| = on,P (1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Case 4: wn,t/n = un,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 of frozen variables and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 (ii) on row addition yield the identity 1 � 1 ∈ U � Tn,t/n[θ] �� = 1 � 1 ∈ F � Tn,t/n[θ]T ⟨1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ � \\ F � Tn,t/n[θ] �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='46) Using (B1 \\ B2)∆(B3 \\ B4) ⊆ (B1∆B3) ∪ (B2∆B4) for any sets B1, B2, B3, B4 and plugging (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='46) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='36) yields the upper bound P � 1 ∈ F � Tn,t/n[θ] � ∆F � Tn+1,t/n[θ] �� + P � 1 ∈ F(Tn,t/n[θ]T ⟨1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩)∆F(Tn+1,t/n[θ]T ⟨1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩) � + on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='47) for E ��un,t/n − un+1,t/n ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='18 then give E|un,t/n − un+1,t/n| ≤ 4(L + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='δ + (L + 3)P (Po (d) ≥ L) + on,P (1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Now the same limiting argument as in Case 1 yields E|un,t/n − un+1,t/n| = on,P,(1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Case 5: wn,t/n = vn,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This is completely analogous to Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Case 6: wn,t/n = ζn,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This is an immediate consequence of Cases 1-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 5 Type fixed point equations As laid out in detail in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5, for our lower bound on the rank to be tight, we need further means to restrict the potential values of the proportion αn,t/n of frozen variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In this section, with the help of the stability properties of the types that we derived in Section 4, we derive asymptotic fixed point equations for the proportions of finer types yn,t/n, un,t/n and vt,t/n in Tn,t/n[θ], as well as a lower bound for zn,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Correspondingly, the single main result of this section is Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The proof of the characterisations in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 is based on a detailed analysis of the connection of the type of variable n + 1 in the larger matrix Tn+1,t/n[θ] to the types of the non-zero entries of row n + 1 in the smaller matrix Tn,t/n[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In this way, we can relate the proportions of types to certain functions of other proportions, that simply correspond to the choices of the non-zero entries of row n + 1 and therefore comparatively easy to evaluate, see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Of course, the details are considerably more involved, but indeed, this proof scheme is quite similar to the deduction of the heuristic fixed point equation in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5, where we relate the type of the new coordinate to the types of its neighbours by the combination of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 Section overview The main goal of this section is to derive the following fixed point equations for the types from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 (Type fixed point equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any n ≥ 0 and d > 0, yn,t/n = 1 − φt � xn,t/n + yn,t/n + un,t/n � − φt � xn,t/n + yn,t/n + vn,t/n � + φt � xn,t/n + yn,t/n � + ¯oP(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1) un,t/n = φt � xn,t/n + yn,t/n + un,t/n � − φt � xn,t/n + yn,t/n � + ¯oP(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2) vn,t/n = φt � xn,t/n + yn,t/n + vn,t/n � − φt � xn,t/n + yn,t/n � + ¯oP(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3) zn,t/n ≥ φt � yn,t/n � + ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4) For the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, in whose course we also work with a more general matrix model, we give names to the functions on the right hand sides of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1) to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We use the following suggestive notation: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 (Type functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let G denote the set of non-decreasing functions g : [0, 1] → [0, 1] and ∆4 be the four-dimensional standard simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We then define the following three functions Y, U, V : ∆4 × G → [0, 1] by setting: (i) Y (ζ, g) = 1 − g(x + y + u) − g(x + y + v) + g(x + y) for (ζ, g) ∈ ∆4 × G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (ii) U (ζ, g) = g(x + y + u) − g(x + y) for (ζ, g) ∈ ∆4 × G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (iii) V (ζ, g) = g(x + y + v) − g(x + y) for (ζ, g) ∈ ∆4 × G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ♦ 25 The rank of sparse symmetric matrices over arbitrary fields The proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 is split into two main parts: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' First, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 reduces the approximation of the types through the type functions to the separate approximation of conditional type probabilities in a larger matrix through the type functions and approximation of ζn+1,t/n through ζn,t/n: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let t ∈ [0, d] with d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any W ∈ {Y, U, V } and K ∈ Z≥2, E ��wn,t/n − W � ζn,t/n, φt ��� ≤E ��P � n + 1 ∈ W � Tn+1,t/n[θ] � ��ζn,t/n � − W � ζn,t/n, φt ��� (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5) + (5 + 8K2)E∥ζn,t/n − ζn+1,t/n∥∞ + 4 − 4(1 − 1/K)5 + 10 (1 + 2d) /K, and E �� zn,t/n − φt � yn,t/n ��−� ≤E �� P � n + 1 ∈ Z � Tn+1,t/n[θ] � ��ζn,t/n � − φt � yn,t/n ��−� (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6) + (5 + 8K2)E∥ζn,t/n − ζn+1,t/n∥∞ + 4 − 4(1 − 1/K)5 + 10 (1 + 2d) /K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The extensive proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 is deferred to Appendix C, since it is separate from our main proof strategy and rather technical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since thanks to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11, we have good control over the differences E∥ζn,t/n − ζn+1,t/n∥∞ already, it only remains to take care of the conditional probabilities in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This is exactly what the second Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4 does by illustrating the probabilistic interpretation of the type functions: It shows that the type functions (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' φt) are approximations (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' lower bounds) of the probabilities that column n + 1 in Tn+1,t/n[θ] has the type associated with the function (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' type Z), conditionally on ζn,t/n: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4 (Conditional probabilities and type functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any W ∈ {Y, U, V } and d > 0, P � n + 1 ∈ W � Tn+1,t/n[θ] � ��ζn,t/n � − W � ζn,t/n, φt � = ¯oP(1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7) while P � n + 1 ∈ Z � Tn+1,t/n[θ] � ��ζn,t/n � − φt � yn,t/n � ≥ ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8) The proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4 is presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' With Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4 in hand, we are ready to prove Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1: Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 subject to Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let W ∈ {Y, U, V } and K ∈ N≥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3, an application of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7) to the right hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5) together with the fact ∥ · ∥∞ ≤ ∥ · ∥1 gives E ��wn,t/n − W � ζn,t/n, φt ��� ≤ on,P (1) + 4 − 4(1 − 1/K)5 + 10(1 + 2d)/K uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9) Since (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9) holds true for any K ≥ 2, and the left hand side does not depend on K, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9) implies that lim sup P →∞ lim sup n→∞ sup N≥n,JN∈SymN(F∗) sup 0≤t≤d E ��wn,t/n − W � ζn,t/n, φt ��� = 0, which gives (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1) to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Analogously, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3, an application of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6) to the right hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8) together with the fact ∥ · ∥∞ ≤ ∥ · ∥1 gives E �� zn,t/n − φt � yn,t/n ��−� ≤ on,P (1) + 4 − 4(1 − 1/K)5 + 10(1 + 2d)/K uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10) As before , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10) implies that � zn,t/n − φt � yn,t/n ��− = ¯oP(1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=', zn,t/n − φt � yn,t/n � ≥ ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' It thus only remains to prove Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 Conditional probabilities and type functions: Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4 As outlined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, the only ingredient in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 that is still lacking a proof is Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This gap is closed in the current section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In addition, we prove a more general version of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4, which holds true for arbitrary square matrices and more general types of symmetric row- and column-attachment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We believe that this extension will prove useful in future applications of our strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' More precisely, for any positive integer n, let A ∈ Fn×n be an arbitrary square matrix and let h be an integer-valued random variable with probability generating function ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Given h, let h ∈ F1×n be a random vector whose non-zero entries are chosen uniformly at random from the set �[n] h � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Throughout this section, we write Ah = � A hT h 0 � to denote the matrix A after symmetric row- and column-attachment of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Finally, we omit the explicit dependence of the proportions on the underlying matrix and simply write x for x(A[θ]) throughout Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The quantities y, z, u, v and ζ are defined analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The main result of this section is the following generalised version of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4: 26 The rank of sparse symmetric matrices over arbitrary fields Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5 (Approximating the type probabilities for column n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any A ∈ Fn×n, L ∈ N≥2, δ > 0 and W ∈ {Y, U, V }, E ��P � n + 1 ∈ W � Ah[θ] � ��ζ � − W(ζ, ψ) �� ≤ 6L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='δ + 7P (h ≥ L) + on,P (1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11) and E �� P � n + 1 ∈ Z � Ah[θ] � ��ζ � − ψ (y) �−� ≤ 2P (h ≥ L) + on,P (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12) Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6 (Error terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We emphasize that the error terms in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5 and the rest of this section are uniform in A and ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In the current more general setting, it becomes evident that the distribution of the type of the new column n + 1 only depends on A through the proportions of the types in A[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ■ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4 is now a direct consequence of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5: Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4 subject to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In the set-up of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5, let A = Tn,t/n and h = Tn+1,t/n(n+ 1, ) ⟨;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' n + 1⟩, such that ψ becomes the probability generating function of a Bin(n, t/n)-variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We first look at (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For W ∈ {Y, U, V }, by the triangle inequality, E ��P � n + 1 ∈ W � Tn+1,t/n[θ] � ��ζn,t/n � − W � ζn,t/n, φt ��� ≤ E ��P � n + 1 ∈ W � Tn+1,t/n[θ] � ��ζn,t/n � − W � ζn,t/n, ψ ��� + E ��W � ζn,t/n, ψ � − W � ζn,t/n, φt ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13) By [25, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10], sup r∈[0,1] |ψ(r) − φt(r)| ≤ ∞ � k=0 |P (Bin(n, t/n) = k) − P (Po (t) = k)| = dTV (Bin (n, t/n) , Po (t)) ≤ d2/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14) Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7) then follows from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8) follows anagolously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In the remainder of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2, we carry out the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For this, we first introduce five type events in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 that establish a connection between the type of n + 1 and the types of supp (h) in the underlying matrix Ah[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since the non-zero coordinates of h are chosen uniformly given h, we can then estimate the probabilities of the type events in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2, and complete the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5 in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 Type events As announced, in this subsection, we introduce a number of “type” events that are solely defined in terms of supp (h) and A[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' These events capture the main causes for variable n + 1 to belong to a particular set W � Ah[θ] � in terms of whether all variables in supp (h) are frozen with respect to A[θ] or A[θ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In this sense, we show that the probability that n + 1 does not have a certain type on its matching type event is small in Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' As in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2, throughout this section, we will frequently work on two good events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The first event Pn will be the same as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='20), since it only involves the perturbation and ensures that the perturbations in A[θ] and Ah[θ] agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='21), P (Pn) = 1 + on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Secondly, and analogously to the definition of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19), we define an event R that is used to make our target matrix (δ, ℓ)-free, so that we can apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' More precisely, we denote by R = R(δ, L) the good event that both A[θ] and A[θ]T are (δ, ℓ)-free for 2 ≤ ℓ ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10 gives that P (R) ≥ 1 + on,P (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15) Before we introduce the actual type events in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9, we first define two basic events that are used to decide whether variable n + 1 is firmly frozen in Ah[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7 (Basic events).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Given A ∈ Fn×n and h as above, we define the following events: F = {supp (h) ⊆ F (A[θ])}, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='16) Ftr = {supp (h) ⊆ F � A[θ]T � }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17) ♦ The following preparatory lemma then shows that the main reason for the new variable n + 1 to be firmly frozen in Ah[θ] is that not all of the variables in supp (h) are frozen in A[θ]T , and thus the event Fc tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This observation is later used to characterise the other possible types of n + 1 in terms of the support of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any δ > 0, L ∈ N≥2 and A ∈ Fn×n, P � n + 1 is firmly frozen in Ah[θ], Ftr � = on(1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='18) and P � n + 1 is not firmly frozen in Ah[θ], Fc tr � ≤ L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='δ + P (h ≥ L) + on,P (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19) 27 The rank of sparse symmetric matrices over arbitrary fields Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (i) We first show (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By definition, n + 1 is firmly frozen in Ah[θ] if and only if it is frozen in Ah[θ] ⟨n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the good event Pn, removal of row n + 1 leaves us with the matrix A[θ] plus the additional column (h 01×θr)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, n + 1 is firmly frozen in Ah[θ], Pn =⇒ (h 01×θr)T cannot be linearly combined by the columns of A[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' =⇒ supp (h) ̸⊆ F � A[θ]T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, on Ftr, supp (h) ⊆ F � A[θ]T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, P � n + 1 is firmly frozen in Ah[θ], Ftr � = P � n + 1 is firmly frozen in Ah[θ], Ftr, Pn � + on(1) = on(1), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (ii) We next prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By definition, if n + 1 is not firmly frozen in Ah[θ], it is not frozen in Ah[θ] ⟨n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the good event Pn, removal of row n + 1 leaves us with the matrix A[θ] plus the additional column (h 01×θr)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since n + 1 is not frozen in this matrix, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 gives that n + 1 is not firmly frozen in Ah[θ], Pn =⇒ (h 01×θr)T can be lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' combined by the columns of A[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, on Fc tr, supp (h) ̸⊆ F � A[θ]T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This implies that both supp (h) and supp (h) \\F � A[θ]T � are non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' If additionally, (h 01×θr)T can be linearly combined by the columns of A[θ], supp (h) \\F � A[θ]T � is a relation of A[θ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Hence, by Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 (iii), n + 1 is not firmly frozen in Ah[θ], Fc tr, Pn =⇒ supp (h) is a proper relation of A[θ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15), P � n + 1 is not firmly frozen in Ah[θ], Fc tr, Pn � ≤ P � supp (h) is a proper relation in A[θ]T � ≤ L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='δ + P (h ≥ L) + on,P (1), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' With Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8, we are now in the position to characterise the type of variable n + 1 in terms of the role of the variables in supp (h) in A[θ] and A[θ]T through the following events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9 (Type events).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' With the notation of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7, let Y = Fc ∩ Fc tr, U = Fc ∩ Ftr, V = F ∩ Fc tr, XZ = F ∩ Ftr and Z◦ = {supp (h) ⊆ Y (A[θ])}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ♦ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By construction, the four events Y, U, V, XZ are pairwise disjoint, and their union Y ⊎ U ⊎ V ⊎ XZ gives the whole sample space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ■ In the following two Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12, we first show that for each choice of W ∈ {Y, U, V, XZ}, the probability that n + 1 does not have the type corresponding to W on W is small and then that the probability that n + 1 /∈ Z(Ah[θ]) on Z◦ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This offers an almost complete description of the type of n + 1 in terms of the events in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 deals with the simpler cases W ∈ {Y, U, V, XZ}, where the type events are intersections of basic events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any δ > 0, L ∈ N≥2, A ∈ Fn×n and W ∈ {Y, U, V }, P � n + 1 /∈ W � Ah[θ] � , W � ≤ 2L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='δ + 2P (h ≥ L) + on,P (1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='20) as well as P � n + 1 ̸∈ X � Ah[θ] � ∪ Z � Ah[θ] � , XZ � = on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='21) 28 The rank of sparse symmetric matrices over arbitrary fields Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We show the claim for each of the possible variable types separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Completely frozen variables - (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='20) for W = Y : By definition, if n + 1 is not completely frozen in Ah[θ], then it is not firmly frozen in Ah[θ] or not firmly frozen in Ah[θ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8 also applies to Ah[θ]T , a union bound gives P � n + 1 ̸∈ Y � Ah[θ] � , Y � ≤ P � n + 1 not firmly frozen in Ah[θ], Fc tr � + P � n + 1 not firmly frozen in Ah[θ]T , Fc� ≤ 2L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='δ + 2P (h ≥ L) + on,P (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' One-sided firmly frozen variables - (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='20) for W ∈ {U, V }: If n + 1 ̸∈ U � Ah[θ] � , then, by definition, either n + 1 is not firmly frozen in Ah[θ]T , or, if this is not the case, it is frozen in Ah[θ] and firmly frozen in Ah[θ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In the latter case, the symmetry of frailly frozen variables under transposition (see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5) implies that n + 1 is also firmly frozen in Ah[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We conclude that if n + 1 ̸∈ U � Ah[θ] � , then either n + 1 is not firmly frozen in Ah[θ]T or n + 1 is firmly frozen in Ah[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Again, by a union bound and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8, P � n + 1 ̸∈ U � Ah[θ] � , U � ≤ P � n + 1 not firmly frozen in Ah[θ]T , Fc� + P � n + 1 firmly frozen in Ah[θ], Ftr � ≤ L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='δ + P (h ≥ L) + on,P (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The claim for W = V follows analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Frailly frozen or two-sided non-frozen variables - (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='21): If n + 1 ̸∈ X � Ah[θ] � ∪ Z � Ah[θ] � , then by definition, n + 1 is firmly frozen in Ah[θ] or Ah[θ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By a union bound and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8, P � n + 1 ̸∈ X � Ah[θ] � ∪ Z � Ah[θ] � , XZ � ≤ P � n + 1 firmly frozen in Ah[θ], Ftr � + P � n + 1 firmly frozen in Ah[θ]T , F � = on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We have the following analogous lemma for the event Z◦: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any L ∈ N≥2 and A ∈ Fn×n, P � n + 1 ̸∈ Z � Ah[θ] � , Z◦ � ≤ P (h ≥ L) + on,P (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The first (and main) step is to prove that on the intersection of Z◦ with a sufficiently likely event, the (n + 1)st row in Ah[θ] can be linearly combined by the other rows of Ah[θ], from which it follows through Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 that n + 1 is not frozen in Ah[θ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On Z◦ ∩ Pn, Ah[θ](n + 1, ) = (h 01×(θc+1)) and all variables in supp (h) = supp(Ah[θ](n + 1, )) are firmly frozen in A[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Ideally, to derive the desired linear combination of Ah[θ](n + 1, ) by the other rows of Ah[θ], we would like to take one representation for each i ∈ supp (h), and then simply sum over the representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Alas, the matrix Ah[θ] has one more column than A[θ], and it is not clear that for the existing representations, also the entries of column n + 1 sum to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, we are looking for representations of i ∈ supp (h) that expressly do not use one of the rows in supp (h), if such representations exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In fact, on Z◦, since any i ∈ supp (h) is firmly frozen in A[θ], there exists a representation of i that does not use row i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' To take care of the other rows corresponding to elements of supp (h), we define the event C = {for all i ∈ supp (h), i /∈ F (A[θ] ⟨supp (h) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩) ∆F (A[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The event C is sufficiently likely for our purposes, as P (Cc) ≤P (h ≥ L) + � i∈[n] L−1 � k=2 L n P (i ∈ F (A[θ] ⟨supp (h) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩) ∆F (A[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩) |i ∈ supp (h) , h = k) P (i ∈ supp (h) h = k) ≤P (h ≥ L) + L2 P + on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Here, in the last step, we have used Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='20, which states that for any i ∈ [n] and k ≤ L, P (i ∈ F (A[θ] ⟨supp (h) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩) ∆F (A[θ] ⟨i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩) |i ∈ supp (h) , h = k) ≤ L P + on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By design, on the event C ∩ Z◦, any i ∈ supp (h) is frozen in A[θ] ⟨supp (h) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In particular, there exists a representation of {i} in A[θ] ⟨supp (h) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the good event Pn, each such representation can be extended to a representation b = (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , bn+θr) of {i} in Ah[θ] ⟨n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ such that bAh[θ] ⟨n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩ = en+θc(i) and bk = 0 for k ∈ supp (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='22) 29 The rank of sparse symmetric matrices over arbitrary fields Thus, on the event Z◦ ∩ C ∩ Pn, any i ∈ supp (h) is frozen in Ah[θ] ⟨n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We conclude that the (n + 1)st row in Ah[θ] can be linearly combined by the other rows of Ah[θ] (this is also true if supp (h) = ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, n + 1 is not frozen in Ah[θ]T , which only leaves the possibility n + 1 ∈ V � Ah[θ] � ∪ Z � Ah[θ] � on Z◦ ∩ C ∩ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, since Z◦ ⊆ Ftr, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='18), n + 1 cannot be firmly frozen in Ah[θ] on the event Z◦ ∩ C ∩ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, n + 1 ∈ Z � Ah[θ] � and we arrive at P � n + 1 ̸∈ Z � Ah[θ] � , Z◦, C � = on(1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=', P � n + 1 ̸∈ Z � Ah[θ] � , Z◦ � ≤ P (Cc) + on(1) ≤ P (h ≥ L) + L2 P + on(1) = P (h ≥ L) + on,P (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 Probabilities of type events In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, we have related the type of n + 1 in Ah[θ] to the occurrence of a bunch of type events, which are formulated in terms of supp (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We now approximate the conditional probabilities of the type events through the corresponding functions Y, U, V from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 and ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In this way, we build the connection between the event � n + 1 ∈ W � Ah[θ] �� and W(ζ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For the current section, recall that we use boldface letters w to abbreviate the proportions w(A[θ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We then show that conditionally on the vector ζ, for any W ∈ {Y, U, V }, the function W(ζ, ψ) is a good approximation of the probability of W, while ψ is a good approximation of Z◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since the type events are defined solely in terms of the membership of supp (h) in the sets W(A[θ]) and h is chosen independently of A[θ], this basically reduces to a comparison between drawing supp (h) with and without replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any L ∈ N≥2, W ∈ {Y, U, V }, |P (W|ζ) − W(ζ, ψ)| ≤ P (h ≥ L) + on,P (1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='23) and |P (Z◦|ζ) − ψ (y)| ≤ P (h ≥ L) + on,P (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='24) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (i) We first prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='23) for W = Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Recall from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9 that Y = Fc ∩ Fc tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By the inclusion-exclusion principle, P (Y) = P (Fc) + P (Fc tr) − P (Fc ∪ Fc tr) = 1 − P (F) − P (Ftr) + P (F ∩ Ftr) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='25) Moreover, by Definitions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13, (a) F coincides with the event that {supp (h) ⊆ X (A[θ]) ∪ Y (A[θ]) ∪ V (A[θ])}, (b) Ftr coincides with the event that {supp (h) ⊆ X (A[θ]) ∪ Y (A[θ]) ∪ U (A[θ])} and (c) F ∩ Ftr coincides with the event that {supp (h) ⊆ X (A[θ]) ∪ Y (A[θ])}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Given the number of non-zero entries h of h, the positions of these non-zero entries are chosen uniformly at random from all h-subsets of [n], and independently of A[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Moreover, by [24], for any k ≥ 0, ����� �(x+y+v)n n � �n k � − (x + y + v)k + �(x+y+u)n n � �n k � − (x + y + u)k − �(x+y)n n � �n k � + (x + y)k ����� ≤ 3k(k − 1) 2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='26) Thus |P (Y|ζ) − Y (ζ, ψ)| ≤ ∞ � k=0 P (h = k) ����� �(x+y+v)n n � �n k � − (x + y + v)k + �(x+y+u)n n � �n k � − (x + y + u)k − �(x+y)n n � �n k � + (x + y)k ����� ≤ P (h ≥ L) + 3L(L − 1) 2n = P (h ≥ L) + on,P (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (ii) We next prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='23) for W = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Recall from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9 that U = Fc ∩ Ftr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Moreover, (a) Fc coincides with the event that {supp (h) ∩ (U (A[θ]) ∪ Z (A[θ])) ̸= ∅} and (b) Ftr coincides with the event that {supp (h) ⊆ X (A[θ]) ∪ Y (A[θ]) ∪ U (A[θ])}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 30 The rank of sparse symmetric matrices over arbitrary fields In other words, U coincides with the event that supp (h) is a subset of X (A[θ]) ∪ Y (A[θ]) ∪ U (A[θ]), but not of X (A[θ]) ∪ Y (A[θ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' As before, using the total variation estimate between sampling with and without replacement of [24], for any k ≥ 0, ����� �(x+y+u)n n � �n k � − �(x+y)n n � �n k � − � (x + y + u)k − (x + y)k� ����� ≤ 2k(k − 1) 2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='27) Thus, |P (U|ζ) − U(ζ, ψ)| = ∞ � k=0 P (h = k) ����� �(x+y+u)n n � �n k � − �(x+y)n n � �n k � − � (x + y + u)k − (x + y)k� ����� ≤ P (h ≥ L) + L(L − 1) n = P (h ≥ L) + on,P (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (iii) By symmetry, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='23) for W = V follows as in (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (iv) We finally prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Given the number of non-zero entries h of h, the positions of these non-zero entries are chosen uniformly at random from all h-subsets of [n], and independently of A[θ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thus, conditionally on h and the proportions of types ζ in A[θ], the event Z◦ holds if and only if all of these h positions are chosen from the set Y(A[θ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, P (Z◦|ζ, h) = �ny h � �n h � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='28) On the other hand, by [24], for any fixed k ≥ 0, ����� �ny k � �n k � − yk ����� ≤ k(k − 1) 2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='29) Therefore, for any L ∈ N≥2, |P (Z◦|ζ) − ψ(y)| ≤ ∞ � k=0 P (h = k) ����� �ny k � �n k � − yk ����� ≤ P (h ≥ L) + sup 0≤k≤L ����� �ny k � �n k � − yk ����� ≤ P (h ≥ L) + L(L − 1) 2n = P (h ≥ L) + on,P (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 Approximating the type probabilities for column n + 1: Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5 With the results of the previous two subsections, we are now in the position to prove Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' At least one-sided firmly frozen variables - proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11): For W ∈ {Y, U, V }, by the triangle inequality, E ��P � n + 1 ∈ W � Ah[θ] � ��ζ � − W(ζ, ψ) �� ≤ E ��P � n + 1 ∈ W � Ah[θ] � ��ζ � − P (W|ζ) �� + E |P (W|ζ) − W(ζ, ψ)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='30) We bound both summands on the right hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='30) separately, beginning with the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By conditional Jensen’s inequality and the tower property, E ���P � n + 1 ∈ W � Ah[θ] � ���ζ � − P (W|ζ) ��� ≤ E ��1 � n + 1 ∈ W � Ah[θ] �� − 1W �� (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='31) ≤ P � n + 1 ∈ W � Ah[θ] � , Wc� + P � n + 1 ̸∈ W � Ah[θ] � , W � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Now, let IW = {Y, U, V, XZ} \\ {W}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since the type events apart from Z◦ are pairwise disjoint (see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10), Wc = � W′∈IW W′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='32) Thus, with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='32) and using the abbreviation XZ � Ah[θ] � to denote the union X � Ah[θ] � ∪ Z � Ah[θ] � , we obtain P � n + 1 ∈ W � Ah[θ] � , Wc� ≤ � W ′∈IW P � n + 1 ∈ W � Ah[θ] � , W′� ≤ � W ′∈IW P � n + 1 /∈ W′ � Ah[θ] � , W′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='33) 31 The rank of sparse symmetric matrices over arbitrary fields Plugging (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='33) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='31) and using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 on all four summands yields E ���P � n + 1 ∈ W � Ah[θ] � ���ζ � − P (W|ζ) ��� ≤ 6L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='δ + 6P(h ≥ L) + on,P (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='34) Finally, the upper bound on the second summand E |P (W|ζ) − W(ζ, ψ)| on the right hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='30) follows immediately from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='23) in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Plugging the two bounds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='23) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='34) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='30) gives (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Nowhere frozen variables - proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12): Since (a + b)− ≤ a− + b− and a− ≤ |a|, E �� P � n + 1 ∈ Z � Ah[θ] � ��ζ � − ψ (y) �−� ≤E �� P � n + 1 ∈ Z � Ah[θ] � ��ζ � − P (Z◦|ζ) �−� + E |P (Z◦|ζ) − ψ (y)| ≤P � n + 1 ̸∈ Z � Ah[θ] � , Z◦ � + E |P (Z◦|ζ) − ψ (y)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12) now follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='24) in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 6 Analysis of the rank-difference In Section 2, we have reduced the lower bound of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 to Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This section is devoted to the proof of those three results, of which Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14 requires the most efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Our starting points here are Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, the fixed point equations for the proportions of frozen types, and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14): yn,t/n = 1 − φt(xn,t/n + yn,t/n + un,t/n) − φt(xn,t/n + yn,t/n + vn,t/n) + φt(xn,t/n + yn,t/n) + ¯oP(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1) un,t/n = φt(xn,t/n + yn,t/n + un,t/n) − φt(xn,t/n + yn,t/n) + ¯oP(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2) vn,t/n = φt(xn,t/n + yn,t/n + vn,t/n) − φt(xn,t/n + yn,t/n) + ¯oP(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3) zn,t/n ≥ φt(yn,t/n) + ¯oP(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4) xn,t/n + yn,t/n + zn,t/n + un,t/n + vn,t/n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5) The combination of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5) gives that xn,t/n + zn,t/n = 1 − yn,t/n − un,t/n − vn,t/n = φt � xn,t/n + yn,t/n � + ¯oP(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6) Equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1) to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6), as well as Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11, are the main results from the previous sections and the proofs in this section highly depend on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 The rank increase: Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 Recall the function ht : [0, 1] → R, ht (α) = α + 1 − φt (α) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13) as well as Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 (The rank increase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any d > 0, E � rkF � Tn+1,t/n[θ] � − rkF � Tn,t/n[θ] �� = E � ht � αn,t/n �� + on,P (1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Recall the good event Pn from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On Pn, the matrix Tn,t/n[θ] arises from the matrix Tn+1,t/n[θ] through removal of the (n + 1)st row and column, and therefore, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7 gives the following representation of their rank difference in terms of the type of n + 1: rkF � Tn+1,t/n[θ] � − rkF � Tn,t/n[θ] � =1 � n + 1 ∈ X � Tn+1,t/n[θ] �� + 2 · 1 � n + 1 ∈ Y � Tn+1,t/n[θ] �� + 1 � n + 1 ∈ U � Tn+1,t/n[θ] �� + 1 � n + 1 ∈ V � Tn+1,t/n[θ] �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, in any case, ��rkF � Tn+1,t/n[θ] � − rkF � Tn,t/n[θ] ��� ≤ 2P + 2, since both matrices can be obtained from Tn,t/n by adding at most P + 1 rows and at most P + 1 columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='21), the above equation holds with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Hence, E � rkF � Tn+1,t/n[θ] � − rkF � Tn,t/n[θ] �� = P � n + 1 ∈ X � Tn+1,t/n[θ] �� + 2 · P � n + 1 ∈ Y � Tn+1,t/n[θ] �� + P � n + 1 ∈ U � Tn+1,t/n[θ] �� + P � n + 1 ∈ V � Tn+1,t/n[θ] �� + on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7) On the other hand, by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='16, for any W ∈ {X, Y, Z, U, V} and any i ∈ [n + 1], P(i ∈ W � Tn+1,t/n[θ] � ) = E[wn+1,t/n], and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11 shows that E � wn+1,t/n � = E � wn,t/n � + on,P (1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 32 The rank of sparse symmetric matrices over arbitrary fields Therefore, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7) reduces to E � rkF � Tn+1,t/n[θ] � − rkF � Tn,t/n[θ] �� = E � xn,t/n + 2yn,t/n + un,t/n + vn,t/n � + on,P (1), uniformly in t ∈ [0, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since αn,t/n = xn,t/n + yn,t/n + vn,t/n as observed in in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15), the combination of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2) gives that E � rkF � Tn+1,t/n[θ] � − rkF � Tn,t/n[θ] �� =E �� xn,t/n + yn,t/n + vn,t/n � + � yn,t/n + un,t/n �� + on,P (1) =E � αn,t/n + 1 − φt � αn,t/n �� + on,P (1) = E � ht � αn,t/n �� + on,P (1), uniformly in t ∈ [0, d], as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 Lower bound on the rank increase: Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14 In this section, we prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14 (Lower bound on the rank increase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any d > 0, ht � αn,t/n � ≥ ht (α⋆(t)) + ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19) The proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14 heavily depends on the properties of the function Gt defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17) and its zeroes: Recall that Gt : [0, 1] → R, Gt(α) = α + φt (1 − φt (α)) − 1 and α⋆(t) and α⋆(t) were defined as the smallest and the largest zeroes of Gt in [0, 1], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Moreover, α0(t) denotes the unique zero of the increasing function Ξt : [0, 1] → R, Ξt(α) = α + φt(α) − 1, which is also always a zero of Gt (see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' With this terminology, we note the following properties of Gt and its zeroes: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 (Useful properties of Gt and its zeroes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' see [9, Section 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For t ∈ [0, e], Gt is strictly increasing and has a unique zero: α⋆(t) = α0(t) = α⋆(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For t ∈ (e, ∞), Gt has exactly three distinct zeroes α⋆(t) < α0(t) < α⋆(t), and α0(t) ≥ 1 − ln t/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For all t ≥ 0, α⋆(t) = 1 − φt (α⋆(t)) and α⋆(t) = 1 − φt (α⋆(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For t ∈ (e, ∞), Gt is positive on (α⋆(t), α0(t)) ∪ (α⋆(t), 1] and negative on [0, α⋆(t)) ∪ (α0(t), α⋆(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Moreover, Gt is strictly increasing on [α⋆(t), 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For t ̸= e, Gt and G′ t have no common zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For t = e, their unique common zero is given by α0(e) = 1−1/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For all t > 0 and α ∈ [0, 1] \\ {α⋆(t), α⋆(t)}, Rt(α⋆(t)) = Rt(α⋆(t)) < Rt(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The functions t �→ α⋆(t), t �→ α0(t) and t �→ α⋆(t) are differentiable on [0, ∞) with continuous derivatives on (0, e) ∪ (e, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let (bn,P,N,JN,t)n,P,N∈Z+,JN∈SymN(F∗),t∈[0,d] ⊆ [0, 1] be an arbitrary family of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' If Gt(bn,P,N,JN,t) = ¯oP(1), then also min {|bn,P,N,JN,t − α⋆(t)| , |bn,P,N,JN,t − α0(t)| , |bn,P,N,JN,t − α⋆(t)|} = ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We emphasize that a large part of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 is covered by the results of [9, Section 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, several of the specific properties that we need only arise in proofs, and are correspondingly difficult to cite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For the sake of completeness and easy reference, we therefore give a proof of all properties that we need in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' As a last preparation for the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14, we prove two short lemmas on the evaluation of ht at specific points: The first lemma shows that α⋆(t) and α⋆(t) minimize ht among the zeroes of Gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' It is a direct consequence of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any t ≥ 0, ht (α⋆(t)) = ht (α⋆(t)) ≤ ht (α0(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By items 1 and 2 in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, α⋆(t) = α0(t) = α⋆(t) for t ≤ e and α⋆(t) > α0(t) ≥ 1 − ln t/t for t > e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Taking the derivative of ht w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' α, we have h′ t(α) = 1 − tφt(α) = 1 − tet(α−1), so α �→ ht(α) is a strictly increasing function on [0, 1 − ln t/t] and a strictly decreasing function on [1 − ln t/t, 1] (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9) and thus ht (α⋆(t)) ≤ ht (α0(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' It thus only remains to show that ht (α⋆(t)) = ht (α⋆(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By item 3 in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, 1 − α⋆(t) = φt(α⋆(t)) and 1 − α⋆(t) = φt(α⋆(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' It now directly follows that ht (α⋆(t)) = α⋆(t) + 1 − φt (α⋆(t)) = α⋆(t) + α⋆(t) = α⋆(t) + 1 − φt (α⋆(t)) = ht (α⋆(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 33 The rank of sparse symmetric matrices over arbitrary fields The second lemma is a consequence of the type fixed point equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1) to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6): Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any d > 0, ht � xn,t/n + yn,t/n + un,t/n � = ht � xn,t/n + yn,t/n + vn,t/n � + ¯oP(1) = ht � xn,t/n + yn,t/n � + ¯oP(1) ≤ ht � yn,t/n � + ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The first and second equalities in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10) follow directly from eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, the combination of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6) gives ht � xn,t/n + yn,t/n � = xn,t/n + yn,t/n + 1 − φt � xn,t/n + yn,t/n � = yn,t/n + 1 − zn,t/n + ¯oP(1) ≤ yn,t/n + 1 − φt � yn,t/n � + ¯oP(1) = ht � yn,t/n � + ¯oP(1), and thus the last inequality in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' With Lemmas 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 in hand, we are finally in the position to prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Define ¯τn = 1 � h′ t(xn,t/n + yn,t/n) ≥ 0 � and ¯ηn = 1 − ¯τn = 1 � h′ t(xn,t/n + yn,t/n) < 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11) Since ¯τn + ¯ηn = 1, we divide equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19) into two parts as follows: ¯τn � ht (α⋆(t)) − ht � αn,t/n �� ≤ ¯oP(1), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12) and ¯ηn � ht (α⋆(t)) − ht � αn,t/n �� ≤ ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13) In the absence of the error terms ¯oP(1) and under the assumption that ¯τn ≡ 1 (or ¯ηn ≡ 1), the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14 would amount to an analytic treatment of the properties of ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Unfortunately, we have to deal with the error terms and both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In the ensuing argument, we therefore fall back upon Taylor’s Theorem with Lagrange Remainder, item 8 in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 and the following two facts: (i) For any function g and υ ∈ {0, 1}, if υa = υb, then υg(a) = υg(b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (ii) For a family of differentiable functions (gt)t∈[0,d], if there exists a uniform bound b such that supt∈[0,d],s∈[0,1] |g′ t(s)| ≤ b, then gt(a′ t) = gt(at) + ¯oP(1) for any random variables at, a′ t ∈ [0, 1], a′ t = at + ¯oP(1) ∈ [0, 1] and t ∈ [0, d], since |gt(a′ t) − gt(at)| ≤ b |a′ t − at|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By definition of ¯τn, ¯τnh′ t � xn,t/n + yn,t/n � ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Fix ε ∈ (0, 1/d) and let b = inft∈[εd,d] � t2e−t� > 0, such that supα∈[0,1] h′′ t (α) ≤ −b for t ∈ [εd, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then by Taylor’s Theorem with Lagrange remainder, for t ∈ [εd, d], ¯τnht(yn,t/n) ≤ ¯τn � ht � xn,t/n + yn,t/n � − h′ t � xn,t/n + yn,t/n � xn,t/n − b 2x2 n,t/n � ≤ ¯τnht � xn,t/n + yn,t/n � − ¯τn1{t ∈ [εd, d]} b 2x2 n,t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Analogously, let c = 1 − εd > 0 such that infα∈[0,1] h′ t(α) ≥ c for t ∈ [0, εd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then by Taylor’s Theorem with Lagrange remainder, for t ∈ [0, εd), ¯τnht(yn,t/n) ≤¯τn � ht � xn,t/n + yn,t/n � − cxn,t/n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10) shows that for all t ∈ [0, d], ¯τnht(yn,t/n) ≥ ¯τnht � xn,t/n + yn,t/n � + ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14) Since xn,t/n ∈ [0, 1], (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14) implies that min � b 2, c � ¯τnx2 n,t/n ≤ ¯τn1{t ∈ [εd, d]} b 2x2 n,t/n + ¯τn1{t ∈ [0, εd)}cxn,t/n ≤ ¯oP(1), and we conclude that that ¯τnx2 n,t/n = ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The Cauchy-Schwarz inequality E[¯τnxn,t/n] ≤ E[¯τnx2 n,t/n]1/2 then yields ¯τnxn,t/n = ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15) 34 The rank of sparse symmetric matrices over arbitrary fields Since αn,t/n = xn,t/n + yn,t/n + vn,t/n and αT n,t/n = xn,t/n + yn,t/n + un,t/n, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15) in combination with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3) implies that ¯τnαn,t/n = ¯τn � yn,t/n + vn,t/n + ¯oP(1) � = ¯τn � 1 − φt � xn,t/n + yn,t/n + un,t/n � + ¯oP(1) � = ¯τn(1 − φt(αT n,t/n) + ¯oP(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Analogously, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15) in combination with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2) implies that ¯τnαT n,t/n = ¯τn � 1 − φt � αn,t/n � + ¯oP(1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Hence, ¯τnαn,t/n = ¯τn(1 − φt(αT n,t/n) + ¯oP(1)) = ¯τn(1 − φt � 1 − φt � αn,t/n �� + ¯oP(1)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=', ¯τnGt � αn,t/n � = ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='16) Let βn,t/n = ¯τnαn,t/n + ¯ηnα⋆(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since Gt(α⋆(t)) = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='16) implies that Gt(βn,t/n) = ¯τnGt � αn,t/n � + ¯ηnGt (α⋆(t)) = ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17) Hence, item 8 in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 implies that min ���βn,t/n − α⋆(t) �� , ��βn,t/n − α0(t) �� , ��βn,t/n − α⋆(t) ��� = ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2, ht (α⋆(t)) = ht (α⋆(t)) ≤ ht (α0(t)), so ht (α⋆(t)) ≤ ht � βn,t/n � + ¯oP(1) = ¯τnht � αn,t/n � + ¯ηnht (α⋆(t)) + ¯oP(1), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12) follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By definition of ¯ηn, ¯ηnh′ t � xn,t/n + yn,t/n � < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since the function ht is strictly increasing on [0, 1 − ln(t)/t], this implies that ¯ηn � xn,t/n + yn,t/n � ≥ ¯ηn (1 − ln t/t), so that ¯ηnαn,t/n ≥ ¯ηn � xn,t/n + yn,t/n � ≥ ¯ηn (1 − ln t/t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='18) Another application of Taylor’s Theorem with Lagrange remainder to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3) as in the argument leading to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15) yields that ¯ηnun,t/n = ¯oP(1) and ¯ηnvn,t/n = ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Hence, by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1), ¯ηnyn,t/n = ¯ηn(1 − φt(xn,t/n + yn,t/n) + ¯oP(1)) and ¯ηnαn,t/n = ¯ηn(xn,t/n + yn,t/n + ¯oP(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19) Let β′ n,t/n = ¯ηn1 � αn,t/n > α⋆(t) � αn,t/n + � 1 − ¯ηn1 � αn,t/n > α⋆(t) �� α⋆(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then β′ n,t/n ≥ α⋆(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since Gt(α⋆(t)) = 0, by eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19), Gt(β′ n,t/n) =¯ηn1 � αn,t/n > α⋆(t) � � αn,t/n + φt � 1 − φt � αn,t/n �� − 1 � ≤¯ηn1 � αn,t/n > α⋆(t) � � xn,t/n + yn,t/n + zn,t/n − 1 + ¯oP(1) � ≤ ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, by item 4 in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, Gt is strictly increasing on [α⋆(t), 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Hence Gt(β′ n,t/n) ≥ Gt (α⋆(t)) = 0, so that Gt(β′ n,t/n) = ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then the combination of item 8 in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 and β′ n,t/n ≥ α⋆(t) yields that β′ n,t/n = α⋆(t) + ¯oP(1), which leads to ¯ηnαn,t/n ≤ ¯ηnβ′ n,t/n = ¯ηn (α⋆(t) + ¯oP(1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='20) Hence, by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='18) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='20), ¯ηnht � αn,t/n � ≥ ¯ηnht � β′ n,t/n � = ¯ηn (ht (α⋆(t)) + ¯oP(1)) , and thus (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The combination of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13) gives (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 35 The rank of sparse symmetric matrices over arbitrary fields 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 Integral evaluation: Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15 In this section, we prove Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15 from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15 (Integral evaluation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any d ≥ 0, � d 0 ht (α⋆(t)) dt = d · Rd (α⋆(d)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Throughout the proof, we use the abbreviations q(d) = � d 0 ht(α⋆(t)) dt = � d 0 (α⋆(t) − φt (α⋆(t)) + 1) dt and r(d) = d · Rd(α⋆(d)) = 2d − dφd (1 − φd(α⋆(d))) − dφd (α⋆(d)) − d2φd (α⋆(d)) (1 − α⋆(d)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We have q(0) = r(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Moreover, by item 7 in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, the function t �→ α⋆(t) is continuous on [0, ∞), which then transfers to the functions d �→ q(d) and d �→ r(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In order to prove that q(d) = r(d) for all d ≥ 0, it is thus sufficient to certify that q′(d) = r′(d) for all d ∈ (0, e) ∪ (e, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Recall that the derivative of d �→ α⋆(d) is continuous on (0, e) ∪ (e, ∞) by item 7 in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' To derive an expression for r′(d), we compute the partial derivatives of the function (d, α) �→ φd (1 − φd(α)) at (d, α⋆(d)) for d ̸= e, where we use that α⋆(d) is a zero of Gd, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=', φd (1 − φd (α⋆(d))) = 1 − α⋆(d): ∂ ∂dφd (1 − φd(α)) ��� α=α⋆(d) = − φd (α⋆(d)) φd (1 − φd (α⋆(d))) (1 − d (1 − α⋆(d))) = − φd (α⋆(d)) (1 − α⋆(d)) (1 − d (1 − α⋆(d))) as well as ∂ ∂αφd (1 − φd(α)) ��� α=α⋆(d) = −d2φd (α⋆(d)) φd (1 − φd (α⋆(d))) = −d2φd (α⋆(d)) (1 − α⋆(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Hence, r′(d) =2 − φd (1 − φd(α⋆(d))) − d ∂ ∂dφd (1 − φd(α)) ��� α=α⋆(d) − d ∂ ∂αφd (1 − φd(α)) ��� α=α⋆(d) dα⋆(d) dd (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='21) − φd (α⋆(d)) − dφd (α⋆(d)) � α⋆(d) − 1 + ddα⋆(d) dd � − 2dφd (α⋆(d)) (1 − α⋆(d)) − d2φd (α⋆(d)) � α⋆(d) − 1 + ddα⋆(d) dd � (1 − α⋆(d)) + d2φd (α⋆(d)) dα⋆(d) dd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Substituting the two partial derivatives of (d, α) �→ φd (1 − φd(α)) into (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='21), we see that the sum of the terms with dα⋆(d)/dd on the right hand side vanishes and r′(d) = 2 − φd (1 − φd(α⋆(d))) − φd (α⋆(d)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, q′(d) = α⋆(d) − φd (α⋆(d)) + 1 = r′(d) since φd (1 − φd (α⋆(d))) = 1 − α⋆(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thus r(d) = q(d) for all d ≥ 0, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This research was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 945045, and by the NWO Gravitation project NETWORKS under grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='002.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Characteristic vectors of bordered matrices with infinite dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Annals of Mathematics, 62(3), 1955.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 37 The rank of sparse symmetric matrices over arbitrary fields Appendices A Useful properties of the functions Gt and Rt For t ≥ 0, recall the rank function Rt : [0, 1] → R, Rt(α) = 2 − φt (1 − φt(α)) − (1 + t(1 − α))φt(α), defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2), as well as Gt : [0, 1] → R, Gt(α) = α + φt (1 − φt(α)) − 1, defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The following lemma shows that for all t ≥ 0, Gt has at least one zero α0(t), that additionally satisfies the equation α0(t) = 1 − φt(α0(t)): Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 (See [9, Section 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any t ≥ 0, the function Ξt : [0, 1] → R, Ξt(α) = α + φt(α) − 1, has a unique zero α0(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Moreover, Gt(α0(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We have Ξ′ t(α) = 1 + tφt(α) > 0 as well as Ξt(0) = e−t − 1 ≤ 0 and Ξt(1) = 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thus, Ξt has a unique zero α0 ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Moreover, Gt(α0(t)) = α0(t) − 1 + φt(1 − φt(α0(t))) = −φt(α0(t)) + φt(α0(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Recall that α⋆(t) and α⋆(t) denote the smallest and largest zero of Gt, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We now prove Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 (Useful properties of Gt and its zeroes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' see [9, Section 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For t ∈ [0, e], Gt is strictly increasing and has a unique zero: α⋆(t) = α0(t) = α⋆(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For t ∈ (e, ∞), Gt has exactly three distinct zeroes α⋆(t) < α0(t) < α⋆(t), and α0(t) ≥ 1 − ln t/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For all t ≥ 0, α⋆(t) = 1 − φt (α⋆(t)) and α⋆(t) = 1 − φt (α⋆(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For t ∈ (e, ∞), Gt is positive on (α⋆(t), α0(t)) ∪ (α⋆(t), 1] and negative on [0, α⋆(t)) ∪ (α0(t), α⋆(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Moreover, Gt is strictly increasing on [α⋆(t), 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For t ̸= e, Gt and G′ t have no common zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For t = e, their unique common zero is given by α0(e) = 1−1/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For all t > 0 and α ∈ [0, 1] \\ {α⋆(t), α⋆(t)}, Rt(α⋆(t)) = Rt(α⋆(t)) < Rt(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The functions t �→ α⋆(t), t �→ α0(t) and t �→ α⋆(t) are differentiable on [0, ∞) with continuous derivatives on (0, e) ∪ (e, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let (bn,P,N,JN,t)n,P,N∈Z+,JN∈SymN(F∗),t∈[0,d] ⊆ [0, 1] be an arbitrary family of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' If Gt(bn,P,N,JN,t) = ¯oP(1), then also min {|bn,P,N,JN,t − α⋆(t)| , |bn,P,N,JN,t − α0(t)| , |bn,P,N,JN,t − α⋆(t)|} = ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' First observe that G0(α) = α, so G0 is strictly increasing with a unique zero in α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Taking the first and second derivative of α �→ Gt, we have G′ t(α) = 1 − t2φt(α)φt (1 − φt(α)) and G′′ t (α) = t3φt(α)φt (1 − φt (α)) (tφt(α) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since t > 0, G′′ t (α) < 0 precisely when α < 1 − ln t/t, and G′′ t (α) > 0 precisely when α > 1 − ln t/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thus, for any t > 0, the first derivative α �→ G′ t(α) is strictly decreasing on [0, 1 − ln t/t) and strictly increasing on (1 − ln t/t, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' When t ∈ (0, 1], 1 − ln t/t ≥ 1 and for all α ∈ [0, 1], G′ t(α) ≥ G′ t(1) = 1 − t2e−t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' When t ∈ (1, e], 1 − ln t/t ∈ (0, 1) and for all α ∈ [0, 1], G′ t(α) ≥ G′ t(1 − ln t/t) = 1 − t/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2) We conclude that for all t ∈ [0, e) and α ∈ [0, 1], G′ t(α) > 0 and that Gt is strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In this case, Gt has at most one zero, which is given by α0(t) from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For t = e, G′ e has exactly one zero in α = 1 − 1/e and is positive otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thus Ge is strictly increasing and has a unique zero, which is given by α0(e) = 1 − 1/e from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' & 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Suppose that t ∈ (e, ∞), so that 1 − ln t/t ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We first show that Gt has at most three zeroes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The proof of item 1 shows that G′ t is strictly decreasing on [0, 1 − ln t/t) and strictly increasing on (1 − ln t/t, 1] with G′ t(1 − ln t/t) = 1 − t/e < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Moreover, G′ t(0) = 1 − t2e−te−te−t ≥ 1 − t2e−t = G′ t(1) > 0, 38 The rank of sparse symmetric matrices over arbitrary fields where we have used that et/2 = eet/2−1 > e (1 + t/2 − 1) > t for t > e in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The intermediate value theorem now implies that G′ t has exactly two zeroes in [0, 1], which we denote by α1(t) < α2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By the above, α1(t) < 1 − ln t/t < α2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This implies for Gt that Gt is strictly increasing on [0, α1(t)) ∪ (α2(t), 1] and strictly decreasing on (α1(t), α2(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3) By the intermediate value theorem and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3), Gt has at most three zeroes in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We next argue that for t > e, Gt has exactly three zeroes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For this, observe that Gt(0) = e−te−t − 1 < 0 and Gt(1) = e−t > 0, Gt(1 − ln t/t) = e−1 − ln t/t = e−1 � 1 − ln t/eln t−1� > e−1 (1 − ln t/ (1 + ln t − 1)) = 0, and Gt(1 − 1/t) = −1/t + e−te−1 = −1/t + 1/ � eete−1−1� < −1/t + 1/ � e � 1 + te−1 − 1 �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By the intermediate value theorem, Gt has at least one zero in each of the intervals (0, 1 − ln t/t), (1 − ln t/t, 1 − 1/t) and (1 − 1/t, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' It follows that Gt has exactly three zeroes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We next show that for t > e, α0(t) is neither the largest nor the smallest zero, such that the zeroes α⋆(t), α⋆(t) and α0(t) are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' If ˙α is any zero of Gt, then ˙α = 1 − φt(1 − φt( ˙α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This implies that Gt(1 − φt( ˙α)) = 1 − φt( ˙α) + φt(1 − φt(1 − φt( ˙α))) − 1 = 1 − φt( ˙α) + φt( ˙α) − 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, 1 − φt( ˙α) is also a zero of Gt, for any t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For t > e, let αM(t) be the zero of Gt that is contained in the non-empty interval (1 − ln t/t, 1 − 1/t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since Gt has exactly the three zeros α⋆(t) < αM(t) < α⋆(t), by the above, 1 − φt(α⋆(t)) < 1 − φt(αM(t)) < 1 − φt(α⋆(t)) are also three distinct zeroes of Gt(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thus we must have that α⋆(t) = 1 − φt(α⋆(t)), αM(t) = 1 − φt(αM(t)) and α⋆(t) = 1 − φt(α⋆(t)), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4) which implies that αM(t) = α0(t) (see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1), as well as α0(t) ∈ (1 − ln t/t, 1 − 1/t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In particular, the proof of item 2 is now complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Moreover, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4) proves item 3 for t > e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For t ∈ [0, e] on the other hand, item 3 follows from the fact that α⋆(t) = α0(t) = α⋆(t) and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' As shown in the proof of item 2, for t > e, α⋆(t) ∈ (0, 1 − ln t/t), α0(t) ∈ (1 − ln t/t, 1 − 1/t) and α⋆(t) ∈ (1 − 1/t, 1) with Gt(0) < 0, Gt(1 − ln t/t) > 0, Gt(1 − 1/t) < 0 and Gt(1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This implies the first part of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Moreover, recall the two zeroes α1(t) < α2(t) of G′ t as well as observation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since there can be at most one zero in each of the intervals [0, α1(t)], [α1(t), α2(t)] and [α2(t), 1] and Gt has exactly three zeroes, we must have that α⋆(t) ∈ [0, α1(t)], α0(t) ∈ [α1(t), α2(t)] and α⋆(t) ∈ [α2(t), 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The second part of item 4 now follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Assume that ¯α(t) ∈ [0, 1] is a common zero of Gt and G′ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then ¯α(t) < 1, since Gt(1) = e−t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let b = φt(¯α(t))/(1 − ¯α(t)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We distinguish the following two cases: i) If b = 1, then ¯α(t) = 1−φt(¯α(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since ¯α(t) is a zero of G′ t, 0 = G′ t(¯α(t)) = 1−t2φ2 t(¯α(t)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' t > 0 and φt(¯α(t)) = 1/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By definition of φt, φt(¯α(t)) = et(¯α(t)−1), so ¯α(t) = 1 − ln t/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, ¯α(t) = 1 − φt(¯α(t)) = 1 − 1/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This is only possible for t = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In this case, ¯α(e) = 1 − 1/e = α0(e), and indeed, Ge(α0(e)) = G′ e(α0(e)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ii) If b ̸= 1, then φt(¯α(t))/b = 1 − ¯α(t) = φt(1 − φt(¯α(t))) = e−tφt(¯α(t)) = � et(¯α(t)−1)�b = φt(¯α(t))b > 0, where we have used Gt(¯α(t)) = 0 in the second step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, φt(¯α(t)) = b−1/(b−1) and 1 − ¯α(t) = b−1φt(¯α(t)) = b−b/(b−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5) Hence, by definition of φt and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5), t = (ln φt(¯α(t))) /(¯α(t) − 1) = bb/(b−1) ln b/(b − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6) Since also G′ t(¯α(t)) = 0, we have that 0 = 1 − t2φt(¯α(t))φt (1 − φt(¯α(t))) Gt(¯α(t))=0 = 1 − t2φt(¯α(t))(1 − ¯α(t)) = 1 − b (ln b)2 /(b − 1)2, 39 The rank of sparse symmetric matrices over arbitrary fields where we have used (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6) in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thus (b − 1)2 − b (ln b)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Hence (b − 1)b−1/2 − ln b = 0 since b − 1 and ln b have the same sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let l(c) = (c − 1)c−1/2 − ln c for c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then l(b) = 0 and l(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Taking the derivative of l, we have l′(c) = c−1/2/2 + c−3/2/2 − 1/c ≥ 0, with equality only if c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, l is strictly increasing on (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since b ̸= 1, we conclude that l(b) ̸= 0, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By item 3, Rt(α⋆(t)) = 2 − φt (1 − φt(α⋆(t))) − (1 + t(1 − α⋆(t)))φt(α⋆(t)) = 2 − φt (α⋆(t)) − (1 + tφt (α⋆(t)))φt(α⋆(t)) = 2 − φt (α⋆(t)) − φt (α⋆(t)) − tφt (α⋆(t)) φt (α⋆(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Analogously, Rt(α⋆(t)) = 2 − φt (α⋆(t)) − φt (α⋆(t)) − tφt (α⋆(t)) φt (α⋆(t)), so Rt(α⋆(t)) = Rt(α⋆(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' One the other hand, R′ t(α) = t2φt (α) Gt(α), so for t > 0, the sign of R′ t(α) is equal to the sign of Gt(α) for α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' i) For t ≤ e, as shown under item 1, Gt is strictly increasing with a unique zero in α⋆(t) = α⋆(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, Rt obtains its unique minimum in α = α⋆(t) = α⋆(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ii) For t > e, item 4 shows that Rt is strictly decreasing on (0, α⋆(t))∪(α0(t), α⋆(t)) and strictly increasing on (α⋆(t), α0(t)) ∪ (α⋆(t), 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, Rt attains its minimum either in α⋆(t) or in α⋆(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We conclude that for all α /∈ {α⋆(t), α⋆(t)}, Rt(α) > Rt(α⋆(t)) = Rt(α⋆(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We apply the implicit function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Consider the two-variable function ˜G : R≥0 × [0, 1] → R, ˜G(t, α) = Gt(α) = α − 1 + e−tet(α−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' ˜G has continuous partial derivatives and thus is differentiable on R>0 × (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By Items 1 and 2, for any t0 ≥ 0, ˜G(t0, α⋆(t0)) = 0, ˜G(t0, α0(t0)) = 0 and ˜G(t0, α⋆(t0)) = 0, and, by item 5, for t0 ̸= e, the partial derivative ∂α ˜G does not vanish in the respective zeroes: ∂α ˜G(t0, α⋆(t0)) ̸= 0, ∂α ˜G(t0, α0(t0)) ̸= 0 and ∂α ˜G(t0, α⋆(t0)) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For t0 ∈ (0, e) ∪ (e, ∞), the implicit function theorem provides the existence of continuously differentiable functions t �→ β⋆(t), t �→ β0(t) and t �→ β⋆(t) defined on an open set t0 ∈ T ⊂ [0, ∞) such that β⋆(t0) = α⋆(t0), β0(t0) = α0(t0), β⋆(t0) = α⋆(t0), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7) and ˜G(t, β⋆(t)) = ˜G(t, β0(t)) = ˜G(t, β⋆(t)) = 0 for all t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8) Assume now that t0 ∈ (0, e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In this case, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8) and item 1 imply that on T ∩(0, e), β⋆, β0 and β⋆ are identical to α0, since for any t ∈ (0, e), Gt has exactly one zero α0(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, for any t0 ∈ (0, e), the function t �→ α0(t) is continuously differentiable in t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let now t0 ∈ (e, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since in this case, the three zeroes of Gt are distinct by Item 2, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7) implies that β⋆(t0) < β0(t0) < β⋆(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since β⋆(t), β0(t) and β⋆(t) are continuous functions, we can further restrict T such that T ⊂ (e, ∞) and β⋆(t) < β0(t) < β⋆(t) for all t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then the combination of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8) and item 2 gives that on T, α⋆(t) = β⋆(t), α0(t) = β0(t) and α⋆(t) = β⋆(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, for any t0 ∈ (e, ∞), the functions t �→ α⋆(t), t �→ α0(t) and t �→ α⋆(t) are continuously differentiable in t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We finally consider continuity of t �→ α⋆(t) in the point t = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let a := lim supt→e α⋆(t) ∈ [0, 1] and suppose that a ̸= α⋆(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since α⋆(e) is the only zero of Ge, ˜G(e, a) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' As ˜G is a continuous function, there exists δ > 0 such that for all (t, α) ∈ Uδ := [e−δ, e+δ]×[a+δ, a−δ], | ˜G(t, α)− ˜G(e, a)| ≤ | ˜G(e, a)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In particular, ˜G(t, α) ̸= 0 for all (t, α) ∈ Uδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' On the other hand, by definition of a, there exists tδ ∈ [e − δ, e + δ] \\ {e} with α⋆(tδ) > a − δ, such that (tδ, α⋆(tδ)) ∈ Uδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' But ˜G(tδ, α⋆(tδ)) = 0, which gives the desired contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We conclude that lim supt→e α⋆(t) = α⋆(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Analogously, it can be shown that lim inft→e α⋆(t) = α⋆(e) and therefore, t �→ α⋆(t) is continuous in t = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Similarly, one can show that t �→ α⋆(t) is continuous in t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The results for α0(t) and α⋆(t) follow along the same lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 40 The rank of sparse symmetric matrices over arbitrary fields 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' As in the proof of item 7, we use ˜G to denote the two-variable function (t, α) �→ Gt(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For ε ≥ 0, let Uε = {(t, α) ∈ [0, d] × [0, 1] : | ˜G(t, α)| ≤ ε} ⊆ R2, such that U0 = {(t, α) ∈ [0, d] × [0, 1] : α ∈ {α⋆(t), α0(t), α⋆(t)}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For b ∈ R2 and A ⊂ R2, let d (b, A) := infa∈A ∥a − b∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We next argue that lim ε→0 sup x∈Uε d (x, U0) =: lim ε→0 ∆ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9) Indeed, suppose that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then there exist δ > 0, εn ↓ 0 and xn ∈ Uεn such that for all n ≥ 1, d (xn, U0) ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' As a uniformly bounded sequence, (xn)n≥1 has a convergent subsequence (xnk)k≥1 with limit x∗, and d (x∗, U0) = lim k→∞ d (xnk, U0) ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' However, since ˜G is continuous, | ˜G(x∗)| = limk→∞ | ˜G (xnk) | ≤ limk→∞ εnk = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=', d (x∗, U0) = 0, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Recall that we assume that Gt(bn,P,N,JN,t) = ¯oP(1) for a family of random variables (bn,P,N,JN,t)n,P,N∈Z+,JN∈SymN(F∗),t∈[0,d] ⊆ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This is equivalent to lim inf P →∞ lim inf n→∞ inf N≥n,JN∈SymN(F∗) inf t∈[0,d] P (|Gt(bn,P,N,JN,t)| ≤ ε) = 1 for all ε > 0, since for any ε > 0 and B := supn,P,N∈N,JN∈SymN(F∗),t∈[0,d] |Gt(bn,P,N,JN,t)| < ∞, εP (|Gt(bn,P,N,JN,t)| > ε) ≤ E |Gt(bn,P,N,JN,t)| ≤ ε + BP (|Gt(bn,P,N,JN,t)| > ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We conclude that for any ε > 0, lim inf P →∞ lim inf n→∞ inf N≥n,JN∈SymN(F∗) inf t∈[0,d] P � (t, bn,P,N,SymN(F∗),t) ∈ Uε � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The definition of ∆ε then implies that for all ε > 0, lim inf P →∞ lim inf n→∞ inf N≥n,JN∈SymN(F∗) inf t∈[0,d] P � d � (t, bn,P,N,SymN(F∗),t), U0 � ≤ ∆ε � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10) On the event {d ((t, bn,P,N,JN,t), U0) ≤ ∆ε}, there exists (˜t, ˜α) ∈ U0 with |t − ˜t| ≤ ∆ε and |bn,P,N,JN,t − ˜α| ≤ ∆ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We next argue that for ε chosen small enough and thus ˜t close to t, also the value of ˜α ∈ {α⋆(˜t), α0(˜t), α⋆(˜t)} is close to one of α⋆(t), α0(t) or α⋆(t): Observe that since the functions s �→ α⋆(s), s �→ α0(s) and s �→ α⋆(s) are uniformly continuous on [0, d] by item 7, for any υ > 0, there exists ω > 0 such that for any t1, t2 ∈ [0, d] with |t1 − t2| ≤ ω, max {|α0(t1) − α0(t2)| , |α⋆(t1) − α⋆(t2)| , |α⋆(t1) − α⋆(t2)|} ≤ υ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11) As limε↓0 ∆ε = 0, we can choose ε such that ∆ε < min {υ/2, ω}, such that in particular ��t − ˜t �� ≤ ω and |bn,P,N,JN,t − ˜α| ≤ υ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12) Then by equations (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12), max ���α0(t) − α0(˜t) �� , ��α⋆(t) − α⋆(˜t) �� , ��α⋆(t) − α⋆(˜t) ��� ≤ υ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Combining the above inequality with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12), we conclude that Mn,P,N,JN,t := min {|bn,P,N,JN,t − α0(t)| , |bn,P,N,JN,t − α⋆(t)| , |bn,P,N,JN,t − α⋆(t)|} ≤ υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Hence, for any υ > 0 and ε sufficiently small, by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10), lim inf P →∞ lim inf n→∞ inf N≥n, JN∈SymN(F∗) inf t∈[0,d] P (Mn,P,N,JN,t ≤ υ) ≥ lim inf P →∞ lim inf n→∞ inf N≥n, JN∈SymN(F∗) inf t∈[0,d] P (d ((t, bn,P,N,JN,t), U0) ≤ ∆ε) = 1, which, by the above equivalent characterisation of ¯oP(1)-convergence, gives the claim: Mn,P,N,JN,t = ¯oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 41 The rank of sparse symmetric matrices over arbitrary fields B Upper bound via leaf-removal: Derivation of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 In this section, we briefly explain how to derive Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 from known results about the Karp-Sipser core of sparse Erd˝os-Rényi random graphs [3, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' To obtain the Karp-Sipser core of a graph G, we iteratively remove vertices of degree one and their unique neighbors from G, until only isolated vertices and a subgraph of minimum degree at least two remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We call the number of isolated vertices in the reduced graph IKS(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The derivation of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 rests on the following result: Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 ([3, 26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For any d > 0, IKS(Gn,d/n) n P −→ γ⋆ + γ⋆ + γ⋆γ⋆ d − 1, n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Here, γ⋆ is the smallest root of the equation x = d exp(−d exp(−x)) and γ⋆ = d exp(−γ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 is of importance in our setting since crucially, removal of degree-one vertices and their neighbours does not change the nullity of the corresponding adjacency matrix (see [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, irrespective of the field or the matrix entries, rkF � An,d/n � n = 1 − nulF(An,d/n) n ≤ 1 − IKS(Gn,d/n) n a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' and Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 thus implies that for any d, ε > 0 and any field F, lim n→∞ P � sup Jn∈Symn(F∗) rkF � An,d/n � n ≤ 2 − γ⋆ + γ⋆ + γ⋆γ⋆ d + ε � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Thus, it remains to relate the limit from Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 to the rank function Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For this, observe that any root x⋆ of the equation x = d exp(−d exp(−x)) satisfies x⋆ ∈ (0, d) as well as Gd(1 − x⋆/d) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Items 1 and 2 in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 thus imply that γ⋆ = d(1 − α⋆) and γ⋆ = d(1 − α⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Moreover, item 3 in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 as well as Gd(α⋆) = 0 give that 2 − γ⋆ + γ⋆ + γ⋆γ⋆ d = 2 − d(1 − α⋆) + d(1 − α⋆) + d2(1 − α⋆)(1 − α⋆) d = 2 − φd(1 − φd(α⋆)) − φd(α⋆) − dφd(α⋆)(1 − α⋆) = Rd(α⋆) = min α∈[0,1] Rd(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Here, in the last step, we have used Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, part 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This concludes the derivation of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 from Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' C Difference approximation via conditional expectations: proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 For a differentiable function f : Rk �→ R, let ∇f be the gradient of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We prove the following more general version of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3: Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Fix a dimension k ∈ N and K > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Let Z1, Z2 and X be defined on the same probability space with convex codomains RZ1, RZ2 ⊂ Rk and RX ⊂ R respectively, such that RX is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then for any differentiable functions f, g : Rk → R, E |f(Z2) − g(Z2)| ≤ � sup ζ∈RZ1 |f(ζ)| + sup x∈RX |x| � � 4K2E∥Z1 − Z2∥∞ + 2 − 2(1 − 1/K)k� + k sup ζ∈RZ2 ∥∇f(ζ)∥∞ E∥Z1 − Z2∥∞ + E |f(Z1) − E [X|Z1]| + E |E [X|Z2] − g(Z2)| + 2k K � sup ζ∈RZ1 ∥∇f(ζ)∥∞ + sup ζ∈RZ2 ∥∇g(ζ)∥∞ � (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1) and E � (f(Z2) − g(Z2))−� ≤ � sup ζ∈RZ1 |f(ζ)| + sup x∈RX |x| � � 4K2E∥Z1 − Z2∥∞ + 2 − 2(1 − 1/K)k� + k sup ζ∈RZ2 ∥∇f(ζ)∥∞ E∥Z1 − Z2∥∞ + E � (f(Z1) − E [X|Z1])−� + E � (E [X|Z2] − g(Z2))−� + 2k K � sup ζ∈RZ1 ∥∇f(ζ)∥∞ + sup ζ∈RZ2 ∥∇g(ζ)∥∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2) 42 The rank of sparse symmetric matrices over arbitrary fields The philosophy behind Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 is that for sufficiently nice functions f and g, good control over E∥Z1 − Z2∥∞, E |E [X|Z1] − f(Z1)| and E |E [X|Z2] − g(Z2)| allows to bound the difference of f(Z2) and g(Z2) in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Indeed, Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1 is designed to deal with situations where Z1 and Z2 are close, and it might be helpful to keep this in mind during the following proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' To prove Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, we partition Rk into small hypercubes: For K > 0, let Zk/K = � q ∈ Rk : Kq ∈ Zk� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We define the k-dimensional, half-open hypercube with side-length r > 0 and center s = (s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , sk) ∈ Rk as Ds(r) = {(t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , tk) : ti − si ∈ [−r/2, r/2), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' , k} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The following lemma shows that if Z1 and Z2 are close, they are likely to be found within the same box, after the application of a random uniform translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' This random translation ensures that the rather arbitrary random variables Z1, Z2 do not always take values in the boundary of the partitioning hypercubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Fix a dimension k ∈ N and hypercube edge length 1/K > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then for any two random vectors Z1, Z2 ∈ Rk that are defined on the same probability space, an independently and uniformly chosen “shift” vector ξK ∈ (0, 1/K]k and 0 < ε < 1/K, � q∈Zk/K E [|1{Z1 − ξK ∈ Dq (1/K)} − 1{Z2 − ξK ∈ Dq (1/K)}|] ≤ 4 ε E∥Z1 − Z2∥∞ + 2 − 2(1 − Kε)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For the sake of brevity, we omit the range of summation from � q∈Zk/K throughout this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Fix any hypercube Dq (1/K) and let j ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' For Zj − ξK to fall into Dq (1/K) and Z3−j − ξK to fall into a distinct box, one of the following two cases must happen: (a) Zj − ξK is in the “inner part” Dq (1/K − ε) of the box, but Z3−j − ξK /∈ Dq (1/K) (“separation”), or (b) Zj − ξK is in the “ε-boundary” Dq (1/K) \\ Dq (1/K − ε) of the box (“boundary”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We call the separation event S(j) q , and the boundary event B(j) q , which yields the almost sure upper bound |1{Z1 − ξK ∈ Dq (1/K)} − 1{Z2 − ξK ∈ Dq (1/K)}| ≤ 1S(1) q + 1B(1) q + 1S(2) q + 1B(2) q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4) It thus remains to upper bound the right hand side of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4) in expectation and then sum over q ∈ Zk/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Separation: Deterministically, for j ∈ {1, 2}, 1S(j) q ≤ 1{Zj − ξK ∈ Dq (1/K − ε) , ∥Z2 − Z1∥∞ ≥ ε/2} ≤ 1{Zj − ξK ∈ Dq (1/K − ε)}2 ε ∥Z2 − Z1∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5) Summing over q ∈ Zk/K in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5) and taking expectation gives E �� 1S(j) q � ≤ 2 εE [∥Z2 − Z1∥∞] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6) Boundary: This is the case where the benefit of the random translation ξK becomes apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Again, let j ∈ {1, 2} and fix q ∈ Zk/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Conditionally on Zj, the random variable Zj − ξK − q is uniformly distributed over the box �k i=1[(Zj)i − qi − 1/K, (Zj)i − qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Therefore, P � B(j) q ��Zj � = Kkλ � k � i=1 [(Zj)i − qi − 1/K, (Zj)i − qi) ∩ (D0 (1/K) \\ D0 (1/K − ε)) � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7) where λ denotes the k-dimensional Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Now, since also the boxes �k i=1[(Zj)i − qi − 1/K, (Zj)i − qi), q ∈ Zk/K, partition Rk, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='7) further yields that E �� 1B(j) q � = E �� P � B(j) q ��Zj �� = Kkλ (D0 (1/K) \\ D0 (1/K − ε)) = 1 − (1 − Kε)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8) The claim now follows from summing (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='4) over q ∈ Zk/K, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We next turn to the proof of Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof of Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Again, for brevity, we omit the range of summation from � q∈Zk/K throughout the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' As in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2, let ξK ∈ (0, 1/K]k be a uniformly chosen “shift” vector that is independent of (Z1, Z2, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We first distinguish the possible hypercube-locations for Z2 − ξK and apply the tower property to get E |f(Z2) − g(Z2)| = � E [E [|f(Z2) − g(Z2)| 1{Z2 − ξK ∈ Dq (1/K)}|ξK]] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9) 43 The rank of sparse symmetric matrices over arbitrary fields Given ξK, on the event {Z2 − ξK ∈ Dq (1/K)}, Z2 is located in the hypercube Dq+ξK (1/K) of sidelength 1/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Since the hypercubes are small and f, g are continuous, the values of f and g should not fluctuate too much on Dq+ξK (1/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' More precisely, let t ∈ Dq+ξK (1/K) ∩ RZ2 be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' If Dq+ξK (1/K) ∩ RZ2 = ∅, let t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then by the mean value theorem, E [|f(Z2) − f(t)| 1{Z2 − ξK ∈ Dq (1/K)}|ξK] ≤ k K sup ζ∈RZ2 ∥∇f(ζ)∥∞ P (Z2 − ξK ∈ Dq (1/K) |ξK) , and E [|g(Z2) − g(t)| 1{Z2 − ξK ∈ Dq (1/K)}|ξK] ≤ k K sup ζ∈RZ2 ∥∇g(ζ)∥∞ P (Z2 − ξK ∈ Dq (1/K) |ξK) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' In the last two displays, both sides are zero if Dq+ξK (1/K) ∩ RZ2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' By the triangle inequality, we get E [|f(Z2) − g(Z2)| 1{Z2 − ξK ∈ Dq (1/K)}|ξK] (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10) ≤ � |f(t) − g(t)| + k K sup ζ∈RZ2 (∥∇f(ζ)∥∞ + ∥∇g(ζ)∥∞) � P (Z2 − ξK ∈ Dq (1/K) |ξK) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='11) ≤ |E [(f(Z2) − g(Z2)) 1{Z2 − ξK ∈ Dq (1/K)}|ξK]| (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12) + 2k K sup ζ∈RZ2 (∥∇f(ζ)∥∞ + ∥∇g(ζ)∥∞) P (Z2 − ξK ∈ Dq (1/K) |ξK) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='13) where now the modulus is outside of the expectation in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12) in comparison to (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Summing (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12) over q ∈ Zk/K and applying the triangle inequality together yield that � |E [(f(Z2) − g(Z2)) 1{Z2 − ξK ∈ Dq (1/K)}|ξK]| (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14) ≤ � E [|f(Z2) − f(Z1)| 1{Z2 − ξK ∈ Dq (1/K)}|ξK] (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15) + � E [|f(Z1)| |1{Z2 − ξK ∈ Dq (1/K)} − 1{Z1 − ξK ∈ Dq (1/K)}| |ξK] (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='16) + � |E [f(Z1)1{Z1 − ξK ∈ Dq (1/K)} − g(Z2)1{Z2 − ξK ∈ Dq (1/K)}|ξK]| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17) For (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15), since � 1{Z2 − ξK ∈ Dq (1/K)} = 1, again the mean value theorem implies that � E [|f(Z2) − f(Z1)| 1{Z2 − ξK ∈ Dq (1/K)}|ξK] ≤ E |f(Z2) − f(Z1)| ≤ k sup ζ∈RZ2 ∥∇f(ζ)∥∞ E∥Z1 − Z2∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='18) Taking expectation in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='16), then an application of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2 gives that � E [|f(Z1)| |1{Z2 − ξK ∈ Dq (1/K)} − 1{Z1 − ξK ∈ Dq (1/K)}|] ≤ sup ζ∈RZ1 |f(ζ)| �4 εE∥Z1 − Z2∥∞ + 2 − 2(1 − Kε)k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='19) Finally, using the triangle inequality once more, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17) can again be divided into three sub-parts as follows: � |E [f(Z1)1{Z1 − ξK ∈ Dq (1/K)} − g(Z2)1{Z2 − ξK ∈ Dq (1/K)}|ξK]| (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='20) ≤ � E [|f(Z1) − E [X|Z1]| 1{Z1 − ξK ∈ Dq (1/K)}|ξK] (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='21) + � E [|E [X|Z2] − g(Z2)| 1{Z2 − ξK ∈ Dq (1/K)}|ξK] (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='22) + � |E [E [X|Z1] 1{Z1 − ξK ∈ Dq (1/K)}|ξK] − E [E [X|Z2] 1{Z2 − ξK ∈ Dq (1/K)}|ξK]| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='23) Since � 1{Z2 − ξK ∈ Dq (1/K)} = 1 and ξK and (Z1, Z2, X) are independent, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='21) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='22) reduce to � E [|f(Z1) − E [X|Z1]| 1{Z1 − ξK ∈ Dq (1/K)}|ξK] = E [|f(Z1) − E [X|Z1]|] , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='24) and � E [|E [X|Z2] − g(Z2)| 1{Z2 − ξK ∈ Dq (1/K)}|ξK] = E [|E [X|Z2] − g(Z2)|] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='25) 44 The rank of sparse symmetric matrices over arbitrary fields Let now i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Again, since ξK and (Z1, Z2, X) are independent, each expectation in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='23) can be simplified as E [E [X|Zi] 1{Zi − ξK ∈ Dq (1/K)}|ξK] = E [E [X|Zi, ξK] 1{Zi − ξK ∈ Dq (1/K)}|ξK] = E [X · 1{Zi − ξK ∈ Dq (1/K)}|ξK] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='26) Plugging identity (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='26) into (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='23) and the triangle inequality yield � |E [E [X|Z1] 1{Z1 − ξK ∈ Dq (1/K)}|ξK] − E [E [X|Z2] 1{Z2 − ξK ∈ Dq (1/K)}|ξK]| ≤ � E [|X| |1{Z1 − ξK ∈ Dq (1/K)} − 1{Z2 − ξK ∈ Dq (1/K)}| |ξK] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='27) Now again, by Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2, � E [E [|X| |1{Z1 − ξK ∈ Dq (1/K)} − 1{Z2 − ξK ∈ Dq (1/K)}| |ξK]] ≤ sup x∈RX |x| �4 εE∥Z1 − Z2∥∞ + 2 − 2(1 − Kε)k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='28) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1) now follows by combining the bounds (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9) – (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='28) and the choice ε = 1/K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' The proof of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2) follows along the same lines, since the triangle inequality (a + b)− ≤ a− + b− and Jensen’s inequality (E [a])− ≤ E[(a)−] hold for the negative part, as well as a− ≤ |a|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Indeed, the only difference between the proofs is that we replace all absolute values |·| in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='9) to (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='12), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='14), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='17), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='20), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='22) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='25) by the corresponding negative parts, while we keep the absolute values in all other bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='3 is an immediate consequence of Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1: In the notation of Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1, let k = 5 and fix any K > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' We choose Z1 = ζn+1,t/n, Z2 = ζn,t/n and X = 1 � n + 1 ∈ W � Tn+1,t/n[θ] �� for W ∈ {Y, U, V} with codomains RZ1 = RZ2 = [0, 1]5 and RX = [0, 1], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Next, let f : Rk → R be the projection onto the coordinate of ζ corresponding to w ∈ {y, u, v}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' f(ζ) = f((x, y, z, u, v)) = w, and g : Rk → R, g(ζ) = W (ζ, φt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5) follows from (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='1) by checking that (i) E |f(Z2) − g(Z2)| = E ��wn,t/n − W(ζn,t/n, φt) ��;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (ii) E [X|Z1] = wn+1,t/n = f(Z1) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (iii) supζ∈[0,1]5 |f(ζ)| = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (iv) supx∈[0,1] |x| = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (v) supζ∈[0,1]5 ∥∇f(ζ)∥∞ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' (vi) supζ∈[0,1]5 ∥∇g(ζ)∥∞ ≤ 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' Analogously, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='6) follows from (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='2) by choosing f (ζ) = z, g (ζ) = φt (y) and X = 1 � n + 1 ∈ Z � Tn+1,t/n[θ] �� , while the other parameters are as in the derivation of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} +page_content=' 45' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFOT4oBgHgl3EQf_jRR/content/2301.12978v1.pdf'} diff --git a/atFLT4oBgHgl3EQfXS8c/vector_store/index.faiss b/atFLT4oBgHgl3EQfXS8c/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..222aa00787415c33941598cd0ea31a02c89126fd --- /dev/null +++ b/atFLT4oBgHgl3EQfXS8c/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f7dfb2d25e9e8a46596d1731a0c9f34b60c7acb7b81fc77b66ac74a7b405785f +size 3735597 diff --git a/b9E3T4oBgHgl3EQfdwpP/content/tmp_files/2301.04537v1.pdf.txt b/b9E3T4oBgHgl3EQfdwpP/content/tmp_files/2301.04537v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fe212b14509168f0c8fdc03cb63f1a77d8d5c5ce --- /dev/null +++ b/b9E3T4oBgHgl3EQfdwpP/content/tmp_files/2301.04537v1.pdf.txt @@ -0,0 +1,964 @@ +Formally Exact Simulations of Mesoscale +Exciton Diffusion in a Photosynthetic +Aggregate +Leonel Varvelo,†,¶ Jacob K. Lynd,†,¶ Brian Citty,† Oliver Kühn,‡ and Doran I. G. +B. Raccah∗,† +†Department of Chemistry, Southern Methodist University, PO Box 750314, Dallas, TX, USA +‡Institute of Physics, University of Rostock, Albert-Einstein-Str. 23-24, 18059 Rostock, Germany +¶Co-first author: These authors contributed equally and authorship order was determined by a +coin-toss. All authors agree that these authors may list themselves in either order for their +CV/Resume and other purposes. +E-mail: doranb@smu.edu +Abstract +The photosynthetic apparatus of plants and bacte- +ria combine atomically precise pigment-protein com- +plexes with dynamic membrane architectures to con- +trol energy transfer on the 10-100 nm length scales. +Recently, synthetic materials have integrated photo- +synthetic antennae proteins to enhance exciton trans- +port, though the influence of artificial packing on the +excited-state dynamics in these biohybrid materials re- +mains unclear. Here, we use the adaptive Hierarchy +of Pure States (adHOPS) to perform a formally exact +simulation of excitation energy transfer within artifi- +cial aggregates of light harvesting complex 2 (LH2) +with a range of packing densities. We find that LH2 ag- +gregates support a remarkable exciton diffusion length +ranging from 100 nm at a biological packing density +to 300 nm at the densest packing previously suggested +in an artificial aggregate. The unprecedented scale of +these calculations also underscores the efficiency with +which adHOPS simulates excited-state processes in re- +alistic molecular materials. +Graphical TOC Entry +1 +arXiv:2301.04537v1 [physics.chem-ph] 11 Jan 2023 + +ho / y(k)(t; z* +R +Ekiyi.Ln l yr +Artificia) +L. = 300 nm +n,j +(t) = E +Formally Exact +La = 200 nm +Simulations +g2(t) = +Z +n +()u +Ld = 100 nm +n.0 +Biological +L2 +d +RMain Text +The ability to control energy transfer on the 10 - 100 +nm length scale (i.e., the ‘mesoscale’) is essential to +designing new materials with applications in opto- +electronics, photocatalysis, and light harvesting. The +photosynthetic apparatus of plants and bacteria com- +bines atomically precise structures of individual pig- +ment protein complexes with a dynamic membrane +architecture that has both inspired new light harvest- +ing materials and stimulated advances in theoretical1–6 +and experimental7–11 characterization of excitation en- +ergy transfer and charge separation. More recently, +biohybrid materials that integrate photosynthetic pro- +teins have performed remarkably well.12–18 +Controlling excited-state processes in artificial as- +semblies of pigment-protein complexes requires un- +derstanding the mechanism of excitation transport on +the mesoscale. +Light harvesting complex 2 (LH2) +from purple bacteria is a widely studied biological an- +tenna protein19–22 that has been previously incorpo- +rated into artificial materials.12,13 Early mechanistic +studies of excitation energy transfer between LH2 pro- +teins in biological assemblies suggest an incoherent +mechanism of transport (i.e., excitations ‘hopping’) +between the dipole-allowed bright states.21,23 Recent +simulations of two-dimensional LH2 aggregates in ar- +tificial assemblies have suggested the possibility of co- +herent transport on a relatively-long 500 fs timescale +(using the single-D1 Davydov ansatz)20 or dark-state +mediated transport arising between close-packed LH2 +pairs (using a Lindblad master equation).19 On the +other hand, a generalized master equation simula- +tion suggests a dominantly incoherent mechanism of +transport even for very close-packed LH2 rings.24 +While generalized master equations, such as gener- +alized Förster theory, are accurate at biological inter- +ring distances,21 changes in aggregate packing can in- +fluence the appropriateness of different approximation +schemes. Thus, assigning the mechanism of excita- +tion transport in artificial LH2 aggregates, without im- +posing a priori constraints, requires a formally exact +method that can simulate excitation energy transport +across a large number of bacteriochlorophyll. +While there exist many formally exact methods for +calculating exciton dynamics, such as the multi-D1 +Davydov ansatz,25 Hierarchical Equations of Motion +(HEOM),26 Hierarchy of Pure States (HOPS),27 and +quasi-adiabatic path integrals (QUAPI),28 they are +limited to small aggregates by the rapid scaling of their +computational expense with the number of simulated +molecules. Recent advances in reduced scaling tech- +niques - ranging from modular path integrals29,30 to +tensor-contracted methods31–36 and adaptive basis set +techniques37 - raise the possibility of simulating ex- +citation transport in mesoscale photosynthetic aggre- +gates using formally exact methods, but such calcula- +tions have not been reported to date. +In this paper, we perform the first formally exact +simulations of mesoscale excitation transport in LH2 +aggregates using the adaptive Hierarchy of Pure States +(adHOPS) method.37 Our calculations suggest LH2 +aggregates can support a remarkable excitation diffu- +sion length of up to 300 nm. +We model excitation energy transport using an open +quantum system Hamiltonian38 +ˆH = ˆHS +∑ +n,q +κqn ˆLn( ˆa† +qn + ˆaqn)+∑ +n,q +ℏωqn( ˆa† +qn ˆaqn +1/2) +(1) +where ωqn is the frequency of a harmonic oscillator +with corresponding creation and annihilation opera- +tors ˆa† +qn and ˆaqn, and ˆLn is an operator (ˆLn = |n⟩⟨n|) +that couples the nth pigment to its independent envi- +ronment described by bath modes {qn}. The influence +of the thermal environment on the electronic (i.e., sys- +tem) energy levels is described by a correlation func- +tion +Cn(t) = ℏ +π +� ∞ +0 dωJn(ω) +� +coth(ℏβω/2)cos(ωt)−isin(ωt) +� +(2) +which we decompose into a sum of exponentials called +‘correlation function modes’ indexed by jn +Cn(t) = ∑ +jn +g jne−γjnt/ℏ. +(3) +where β = +1 +kBT is the inverse temperature and Jn(ω) is +the spectral density. +We simulate the exciton dynamics using the Hierar- +chy of Pure States (HOPS),27 a formally exact solution +to the open quantum system Hamiltonian (Eq. (1)). +In HOPS, the full state of the system and bath is ex- +pressed as a collection of wave functions indexed by +a vector⃗k, where |ψ(⃗0) +t +⟩ is the physical wave function +and the remainder are referred to as ‘auxiliary wave +functions.’ The reduced density matrix for the system +is given by an ensemble average (E[·]) over Ntra j wave +2 + +function trajectories +ρS = E +� +|ψ(⃗0)(t;zt)⟩⟨ψ(⃗0)(t;zt)| +� +(4) +subject to a complex, stochastic noise zt where com- +ponents associated with individual thermal environ- +ments zn,t are defined by E[zn,t] = 0, E[zn,tzn,s] = 0, +and E[z∗ +n,tzn,s] = Cn(t − s). For notational simplicity, +we will refer to |ψ(t,zt)⟩ as |ψt⟩. The time-evolution +of the HOPS wave functions is then given by +ℏ∂t|ψ(⃗k) +t +⟩ = +� +−i ˆHS −⃗k ·⃗γ −Γt +∑ +n +ˆLn(z∗ +n,t +ξn,t) +� +|ψ(⃗k) +t +⟩ ++∑ +n, j +kjnγ jn ˆLn|ψ(⃗k−⃗ejn) +t +⟩ +−∑ +n, j +(gjn +γjn +)(ˆL† +n −⟨ˆL† +n⟩t)|ψ(⃗k+⃗ejn) +t +⟩ +(5) +where +ξn,t = 1 +ℏ +� t +0 dsC∗ +n(t −s)⟨L† +n⟩s +(6) +is a memory term that causes a drift in the effective +noise, +⟨ˆL† +n⟩t = ⟨ψ(⃗0) +t +|ˆL† +n|ψ(⃗0) +t +⟩, +(7) +and +Γt =∑ +n +⟨ˆLn⟩t Re[z∗ +n,t +ξn,t] +−∑ +n, j +Re +��g jn +γjn +� +⟨ψ(⃗0) +t +|ˆL† +n|ψ(⃗ejn) +t +⟩ +� ++∑ +n, j +⟨ˆL† +n⟩t Re +��g jn +γ jn +� +⟨ψ(⃗0) +t +|ψ(⃗ejn) +t +⟩ +� +(8) +ensures normalization of the physical wave function. +In these equations, kjn is the jnth element of the index +vector ⃗k, and ⃗γ is a vector of the exponential factors +of the correlation function modes. Here we employ +the "triangular truncation" scheme, which restricts the +hierarchy of auxiliary wave functions to a finite depth +kmax, such that {⃗k ∈ A : ∑i ki ≤ kmax}, though other +static filtering approaches have been proposed.39 +The computational expense of HOPS scales poorly +with system size because the number of auxiliary wave +functions in the hierarchy increases rapidly with the +total number of exponential modes defining the corre- +lation functions. However, the physical wave function +of a HOPS trajectory is localized in the basis of sys- +tem states (S) by the interaction with the thermal bath +(‘dynamic localization’), which induces a correspond- +ing localization in the basis of auxiliary wave func- +tions (A).37 As a result, in extended aggregates, the +vast majority of auxiliary wave functions are unoccu- +pied at any single point in time. If unoccupied system +states and auxiliary wave functions could be efficiently +eliminated from the calculation, the computational ex- +pense of HOPS would not increase with system size +(i.e., O(1) scaling) for sufficiently large aggregates.37 +The adaptive hierarchy of pure states (adHOPS) ap- +proach constructs a time-evolving reduced basis to +provide a numerically tractable formulation of HOPS +trajectories for large molecular aggregates.37 Every ut +(‘update time’) femtoseconds, adHOPS constructs a +reduced basis as a direct sum of reduced system and +auxiliary bases (St +�At). The algorithm determines +the error introduced by neglecting each basis element +as an upper-bound on the Euclidean norm of the result- +ing displacement in the full derivative vector (Eq. (5)). +It then truncates basis elements in order of increasing +error until user-defined error bounds for the hierarchy +(δA) and then the state (δS) bases are saturated. Previ- +ously, adHOPS has replicated the dynamics of HOPS +in a linear chain with a dramatically reduced basis +that demonstrated size-invariant (i.e., O(1)) scaling for +sufficiently large aggregates.37 +The version of adHOPS used for the calculations re- +ported here is available in the MesoHOPS package ver- +sion 1.1.40 Additional details about both the algorithm +and the calculation parameters are available in the Sup- +porting Information. We check the convergence of ad- +HOPS results by running a series of adHOPS ensem- +bles with increasingly strict parameters until the char- +acteristic observables are within 2% of the most exact +calculations (see section S4 of the Supporting Infor- +mation for details). +The LH2 monomer (Fig. 1a-b) contains two rings +of bacteriochlorophyll (BChl) that absorb at 800 nm +(B800 ring, blue) and 850 nm (B850 ring, green), re- +spectively. The B800 ring is comprised of 9 widely- +spaced and weakly-coupled BChls and funnels excita- +tion to the B850 ring, which is responsible for trans- +port between LH2 monomers. The tight packing of +the 18 BChls (organized into αβ pairs) in the B850 +ring gives rise to strong electronic couplings and delo- +calized eigenstates. Since inter-LH2 transport occurs +predominately between B850 rings,41 we neglect the +B800 pigments found in the 1NKZ pdb structure.42 +3 + +Figure 1: LH2 monomer. (a) Side view: the B850 +ring (18 BChl) and the B800 ring (9 BChl) are shown +in green and blue, respectively. (b) Top view. (c) The +distance from equilibrium as a function of time (black) +compared to a bi-exponential fit (green). (d) Boltz- +mann distribution (green line) compared to adHOPS +equilibrium populations (grey squares) of eigenstates +ordered by absolute index (|ν|). σ = 160 cm−1 for +site energy static disorder. Convergence parameters +are given in Table S6 of the Supporting Information. +We model the electronic energy levels of the B850 +ring using previously established parameters.43 In the +electronic (‘system’) Hamiltonian, the vertical excita- +tion energies (or ‘site energies’) of the α and β chloro- +phylls are 12690 cm−1 and 12070 cm−1, respectively, +and the electronic couplings (Vn,m) within and between +αβ pairs are 307 cm−1 and 237 cm−1, respectively +(see section S1 of the Supporting Information for full +B850 Hamiltonian). In keeping with previous spectro- +scopic assignments, we also include static disorder on +the site energy of each chlorophyll as Gaussian fluctu- +ations with a standard deviation of σ = 160 cm−1.22,43 +Finally, we model the thermal environment of each +pigment with a Drude-Lorentz spectral density charac- +terized by a reorganization timescale (γ0n = 53 cm−1) +and reorganization energy (λn = 65 cm−1)43 +Jn(ω) = 2λnγ0n +ω +ω2 +γ2 +0n +. +(9) +The corresponding correlation function is composed of +one high temperature mode, kMats Matsubara modes, +and an additional mode to ensure Im[α(0)] = 0 (see +section S2 of the Supporting Information for details). +We characterize the relaxation of the exciton popula- +tion towards equilibrium using the 1-norm of a differ- +ence vector +||P(t)−Peq||1 +(10) +where P(t) and Peq are the eigenstate population vec- +tor of the B850 ring at time t and its equilibrium value, +respectively. Using these parameters, the degenerate +optically bright (ν = ±1) states in the B850 ring relax +to equilibrium on two timescales (τ1 = 25 fs and τ2 = +150 fs, Fig. 1c), which agree with those found by +global kinetic fits of B850 intraband relaxation in 2D +electronic spectroscopy (50 fs and 150 fs).44 The equi- +librium eigenstate populations (Fig. 1d) form a Boltz- +mann distribution, slightly perturbed by the electron- +vibrational coupling. +We construct a hexagonally-packed supercomplex +containing 37 B850 rings (666 BChl) organized as +three concentric shells around a donor ring (Fig. 2a) +with center-to-center ring distances (R) of 6.5 nm. This +inter-ring distance represents the closest packing of +LH2 proteins proposed for artificial aggregates.19,45 +The inter-ring couplings ˜Vn,m are calculated using the +ideal dipole approximation +˜Vn,m = C +⃗dn · ⃗dm −3(⃗dn ·⃗rn,m)(⃗dm ·⃗rn,m) +R3n,m +(11) +where the coupling constant C = 348,000 ˚A3 · cm−1, +⃗dn is a unit vector along the direction of the transi- +tion dipole moment of pigment n, and ⃗rn,m (Rn,m) is +the unit vector (distance) between Mg atoms in pig- +ments n and m. In addition to the site energy static dis- +order included in each ring, we add an angular static +disorder represented by randomly orienting the B850 +rings in each trajectory. We find that static disorder +in the site energies suppresses transport while angular +disorder has negligible impact (see section S7 of the +Supporting Information for details). +Exciton transport in LH2 aggregates exhibits two +distinct regimes: a superdiffusive transport at short +times and ongoing diffusive transport at times longer +than 200 fs. +To quantify transport in our adHOPS +simulations, we calculate the mean-squared deviation +4 + +(a) LH2 side +(b) LH2 top +B850 +B800 +(c) Distance from Eq. +(d) Eig. Distribution +0.3 +1.5 +adHOPS +adHOPS +fit +Boltzmann +Dist. to eq. +Population +0.2 +1.0 +0.5 - +0.1 +0.0 +0.0 +0 +0.25 +0.5 +0 +3 +6 +9 +Time (ps) +Eigenstate Index +(DFigure 2: Exciton transport in a mesoscale aggregate of LH2. (a) Schematic of a hexagonally-packed B850 com- +plex with three concentric shells. (b) Mean squared deviation (MSD) of the excitation from adHOPS dynamics +(thick line), power-law fits (thin lines), and kinetic model (dashed line). (c) adHOPS (solid lines) and kinetic +model (dashed lines) population dynamics for the donor and three concentric shells of B850 rings (shown in +color-coded inset). σ = 160 cm−1 for site energy static disorder, and angular static disorder is given by randomly +orienting all B850 rings in each trajectory. Convergence parameters are given in Table S6 of the Supporting +Information. +(MSD, grey line, Fig. 2b) of the exciton distribution +MSD(t) = ∑ +n +Pn(t)R2 +0n +(12) +where Pn is the population of the nth B850 ring and +n = 0 signifies the donor. To characterize the mech- +anism of transport, we fit the MSD to a power law +(MSD(t) = Dtα) in different time windows (black +lines, Fig. 2b). From 0 to 50 fs, the MSD exhibits su- +perdiffusive behavior (α = 1.74 ± 0.11). Between 50 +and 200 fs, however, the MSD behavior shifts rapidly +from a convex to a linear form, consistent with coher- +ent transport giving way to diffusive transport on the +timescale of vibrational reorganization (ℏ/γ0n ≈ 100 +fs). From 200 fs onward, the MSD remains diffusive +(α = 0.95±0.04) until the exciton reaches the bound- +aries of the complex. +The 37-mer studied here is the minimal aggregate +sufficient for simulating the turn over between coher- +ent and diffusive transport without edge effects. By the +onset of diffusive transport (200 fs), the exciton pop- +ulation summed over B850 rings forming the second +concentric shell (inset, Fig. 2c) is nearly 10% of the +total population (medium green line, Fig. 2c), which +would suppress further transport in a smaller aggre- +gate. +The later-time diffusive transport of the 37-mer is re- +produced by a kinetic model characterized by a single +rate of transport. We fit the population dynamics of +a B850 dimer (Fig. 3a) with center-to-center distance +R = 6.5 nm to a single rate, κdimer = 0.44±0.04 ps−1 +(see section S6 of the Supporting Information for de- +tails). We time-evolve the vector of B850 ring popula- +tions (P(t)) according to +˙P(t) = K P(t), +(13) +where the rate matrix (K) connects nearest-neighbor +rings via symmetric transport rates equal to κdimer. +Starting from 200 fs, the population dynamics of the +kinetic model (dashed lines) reproduce the results of +adHOPS calculations (solid lines, Fig. 2c). This en- +sures that the kinetic model also reproduces the MSD +dynamics of diffusive exciton transport in the LH2 ag- +gregate (dashed line, Fig. 2b). We have also explored +the possibility of the dark-state shelving mechanism +recently proposed for close-packed LH2,19 but found +that inter-ring transport is dominated by coupling be- +tween bright states (see section S8 of the Supporting +Information for details). +The excitation diffusion length (Ld) in LH2 aggre- +gates predicted by our kinetic model greatly exceeds +that of prototypical organic semiconductors, even for +the relatively loose packing associated with biologi- +cal membranes. In Fig. 3b, we show the dimer rates +of transport as a function of the packing distance (R) +5 + +(a) 37-mer +(b) Mean-Squared Deviation +(c) Population Dynamics +Dt2 +Dt1 +Dt2 +Dt1 +100 - +1.0 - +80 +0.75 +(zwu) +lation +60 +Popul: +0.5 +% +40 +I adHOPS +0.25 +20 - +fit + rate model +-0 +0 +0.2 +0.4 +0.6 +0.8 +0 +0.2 +0.4 +0.6 +0.8 +NB850 = 37 +NBChl = 666 +Time (ps) +Time (ps)Table 1: LH2 exciton diffusion lengths in a hexagonally-packed lattice +R (nm) +method +D (nm2/psα) +α +Ld (nm) +6.5 +rate model +110±10. +1 +330±15 +adHOPS +100±10. +0.95±0.044 +270±54 +7.5 +rate model +43±2.5 +1 +210±6 +8.5 +rate model +19±1.0 +1 +140±4 +The uncertainty represents the 95% confidence interval. +Figure 3: LH2 dimer rates. (a) Schematic of a B850 +dimer with a center-to-center separation distance of R. +(b) Transport rates for an LH2 dimer with an inter-ring +separation of R. In all cases, σ = 160 cm−1 for site en- +ergy static disorder, and angular static disorder is given +by randomly orienting both B850 rings in each trajec- +tory. Convergence parameters are given in Table S6 of +the Supporting Information. +from 6.5 nm suggested in some artificial materials to +8.5 nm found in biological membranes. Table 1 reports +the excitation diffusion length in the kinetic model of +an infinite aggregate (see section S9 of the Supporting +Information) determined by +L2 +d = 6R2κdimerτ ≡ Dτ, +(14) +where R is the packing distance, κdimer is the LH2 +dimer rate, and τ is the lifetime of the exciton (as- +sumed to be 1 ns).45 At a biologically relevant inter- +ring distance of R = 8.5 nm, our excitation diffusion +length (Ld = 140 nm) is consistent with a previous +order of magnitude estimate46 and greatly exceeds +that expected for a prototypical organic semiconduc- +tor (< 30 nm).47,48 Moreover, our rate of transport at +R = 8.5 nm is smaller than some previous estimates +(e.g., 0.241 ps−1 in Ref. 24), suggesting other LH2 +Hamiltonians proposed in the literature could support +even longer excitation diffusion lengths. +We conclude that LH2 aggregates support exciton +transport on length scales exceeding those of simi- +lar artificial and natural materials. Excitation trans- +port in LH2 aggregates exhibits a brief ballistic period +(< 50 fs) followed by diffusive transport mediated by +bright-state coupling that supports long-range exciton +diffusion even at biological packing distances (R = 8.5 +nm, Ld = 140 nm). +We note that LH2 complexes, +despite belonging to the evolutionarily ‘early’ anoxy- +genic purple bacteria, support a surprisingly large Ld +compared to that previously calculated for the oxy- +genic photosystem II membrane (Ld = 50 nm),49,50 +suggesting that the need for regulation and not long- +range excitation energy transport has driven the design +of antenna complexes in higher plants. Moreover, in +artificially close-packed LH2 aggregates, our calcula- +tions suggest the exciton diffusion length can reach +up to 300 nm, a full order of magnitude larger than +a prototypical organic semiconductor, making LH2 a +promising antenna system for biohybrid materials. +In this paper, we have reported a formally exact +simulation of exciton dynamics in a mesoscale pho- +tosynthetic aggregate consisting of an unprecedented +37 LH2s with a total of 666 bacteriochlorophylls. +While some previous studies have simulated large LH2 +aggregates using approximate methods, the current +calculations provide an important benchmark result +where the accuracy of the simulation is limited only by +the parameterization of the model Hamiltonian. Our +results also demonstrate that adHOPS is suitable for +calculations of realistic mesoscale molecular materials +to address mechanistic questions about excited-state +processes. We expect the continued development of +adHOPS will enable a new generation of simulations +capable of probing larger and more complex materi- +als to reveal new strategies for controlling excited-state +processes. +Acknowledgement The authors thank Bailey Raber +6 + +(a) B850 Dimer +(b) Dimer Rates +100 +R +(ps-1 +Rate ( +10-2 +6.5 +7.5 +8.5 +R (nm)and Mohamed El Refaiy for assistance working with +the protein structure and the electronic Hamiltonian +for the extended aggregates, Julian Schmidt for help- +ful discussions and testing a preliminary version of +the adHOPS code, as well as Tarun Gera for editing +and review. +LV, BC, and DIGB acknowledge sup- +port from the Robert A. Welch Foundation (Grant N- +2026-20200401). JKL acknowledges support from a +Moody Fellowship. +OK acknowledges funding by +the Deutsche Forschungsgemeinschaft - SFB 1477 +"Light-Matter Interactions at Interfaces", project num- +ber 441234705”. +Supporting Information Available +Methodological details, all calculation parameters, +convergence studies, analysis of dark/bright state +mechanisms, and comparison of different types of +static disorder (PDF) +References +(1) Sumi, H. Theory on Rates of Excitation-Energy +Transfer between Molecular Aggregates through +Distributed Transition Dipoles with Application +to the Antenna System in Bacterial Photosynthe- +sis. J. Phys. Chem. B 1999, 103, 252–260. +(2) Scholes, G. D.; Jordanides, X. J.; Fleming, G. R. +Adapting the Förster Theory of Energy Transfer +for Modeling Dynamics in Aggregated Molec- +ular Assemblies. J. Phys. Chem. B 2001, 105, +1640–1651. +(3) Jang, S.; Newton, M. D.; Silbey, R. J. Multi- +chromophoric Förster Resonance Energy Trans- +fer. Phys. Rev. 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U.S.A. +2018, 115, E9523–E9531. +9 + diff --git a/b9E3T4oBgHgl3EQfdwpP/content/tmp_files/load_file.txt b/b9E3T4oBgHgl3EQfdwpP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0d24e74aa0f2615a1248db8381ba10ec59586250 --- /dev/null +++ b/b9E3T4oBgHgl3EQfdwpP/content/tmp_files/load_file.txt @@ -0,0 +1,930 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf,len=929 +page_content='Formally Exact Simulations of Mesoscale Exciton Diffusion in a Photosynthetic Aggregate Leonel Varvelo,†,¶ Jacob K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Lynd,†,¶ Brian Citty,† Oliver Kühn,‡ and Doran I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Raccah∗,† †Department of Chemistry, Southern Methodist University, PO Box 750314, Dallas, TX, USA ‡Institute of Physics, University of Rostock, Albert-Einstein-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' 23-24, 18059 Rostock, Germany ¶Co-first author: These authors contributed equally and authorship order was determined by a coin-toss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' All authors agree that these authors may list themselves in either order for their CV/Resume and other purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' E-mail: doranb@smu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='edu Abstract The photosynthetic apparatus of plants and bacte- ria combine atomically precise pigment-protein com- plexes with dynamic membrane architectures to con- trol energy transfer on the 10-100 nm length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Recently, synthetic materials have integrated photo- synthetic antennae proteins to enhance exciton trans- port, though the influence of artificial packing on the excited-state dynamics in these biohybrid materials re- mains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Here, we use the adaptive Hierarchy of Pure States (adHOPS) to perform a formally exact simulation of excitation energy transfer within artifi- cial aggregates of light harvesting complex 2 (LH2) with a range of packing densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' We find that LH2 ag- gregates support a remarkable exciton diffusion length ranging from 100 nm at a biological packing density to 300 nm at the densest packing previously suggested in an artificial aggregate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' The unprecedented scale of these calculations also underscores the efficiency with which adHOPS simulates excited-state processes in re- alistic molecular materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Graphical TOC Entry 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='04537v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='chem-ph] 11 Jan 2023 ho / y(k)(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' z* R Ekiyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='Ln l yr Artificia) L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' = 300 nm n,j (t) = E Formally Exact La = 200 nm Simulations g2(t) = Z n ()u Ld = 100 nm n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='0 Biological L2 d RMain Text The ability to control energy transfer on the 10 - 100 nm length scale (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=', the ‘mesoscale’) is essential to designing new materials with applications in opto- electronics, photocatalysis, and light harvesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' The photosynthetic apparatus of plants and bacteria com- bines atomically precise structures of individual pig- ment protein complexes with a dynamic membrane architecture that has both inspired new light harvest- ing materials and stimulated advances in theoretical1–6 and experimental7–11 characterization of excitation en- ergy transfer and charge separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' More recently, biohybrid materials that integrate photosynthetic pro- teins have performed remarkably well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='12–18 Controlling excited-state processes in artificial as- semblies of pigment-protein complexes requires un- derstanding the mechanism of excitation transport on the mesoscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Light harvesting complex 2 (LH2) from purple bacteria is a widely studied biological an- tenna protein19–22 that has been previously incorpo- rated into artificial materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='12,13 Early mechanistic studies of excitation energy transfer between LH2 pro- teins in biological assemblies suggest an incoherent mechanism of transport (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=', excitations ‘hopping’) between the dipole-allowed bright states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='21,23 Recent simulations of two-dimensional LH2 aggregates in ar- tificial assemblies have suggested the possibility of co- herent transport on a relatively-long 500 fs timescale (using the single-D1 Davydov ansatz)20 or dark-state mediated transport arising between close-packed LH2 pairs (using a Lindblad master equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='19 On the other hand, a generalized master equation simula- tion suggests a dominantly incoherent mechanism of transport even for very close-packed LH2 rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='24 While generalized master equations, such as gener- alized Förster theory, are accurate at biological inter- ring distances,21 changes in aggregate packing can in- fluence the appropriateness of different approximation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Thus, assigning the mechanism of excita- tion transport in artificial LH2 aggregates, without im- posing a priori constraints, requires a formally exact method that can simulate excitation energy transport across a large number of bacteriochlorophyll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' While there exist many formally exact methods for calculating exciton dynamics, such as the multi-D1 Davydov ansatz,25 Hierarchical Equations of Motion (HEOM),26 Hierarchy of Pure States (HOPS),27 and quasi-adiabatic path integrals (QUAPI),28 they are limited to small aggregates by the rapid scaling of their computational expense with the number of simulated molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Recent advances in reduced scaling tech- niques - ranging from modular path integrals29,30 to tensor-contracted methods31–36 and adaptive basis set techniques37 - raise the possibility of simulating ex- citation transport in mesoscale photosynthetic aggre- gates using formally exact methods, but such calcula- tions have not been reported to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' In this paper, we perform the first formally exact simulations of mesoscale excitation transport in LH2 aggregates using the adaptive Hierarchy of Pure States (adHOPS) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='37 Our calculations suggest LH2 aggregates can support a remarkable excitation diffu- sion length of up to 300 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' We model excitation energy transport using an open quantum system Hamiltonian38 ˆH = ˆHS +∑ n,q κqn ˆLn( ˆa† qn + ˆaqn)+∑ n,q ℏωqn( ˆa† qn ˆaqn +1/2) (1) where ωqn is the frequency of a harmonic oscillator with corresponding creation and annihilation opera- tors ˆa† qn and ˆaqn, and ˆLn is an operator (ˆLn = |n⟩⟨n|) that couples the nth pigment to its independent envi- ronment described by bath modes {qn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' The influence of the thermal environment on the electronic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=', sys- tem) energy levels is described by a correlation func- tion Cn(t) = ℏ π � ∞ 0 dωJn(ω) � coth(ℏβω/2)cos(ωt)−isin(ωt) � (2) which we decompose into a sum of exponentials called ‘correlation function modes’ indexed by jn Cn(t) = ∑ jn g jne−γjnt/ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' (3) where β = 1 kBT is the inverse temperature and Jn(ω) is the spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' We simulate the exciton dynamics using the Hierar- chy of Pure States (HOPS),27 a formally exact solution to the open quantum system Hamiltonian (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' In HOPS, the full state of the system and bath is ex- pressed as a collection of wave functions indexed by a vector⃗k, where |ψ(⃗0) t ⟩ is the physical wave function and the remainder are referred to as ‘auxiliary wave functions.’ The reduced density matrix for the system is given by an ensemble average (E[·]) over Ntra j wave 2 function trajectories ρS = E � |ψ(⃗0)(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='zt)⟩⟨ψ(⃗0)(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='zt)| � (4) subject to a complex, stochastic noise zt where com- ponents associated with individual thermal environ- ments zn,t are defined by E[zn,t] = 0, E[zn,tzn,s] = 0, and E[z∗ n,tzn,s] = Cn(t − s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' For notational simplicity, we will refer to |ψ(t,zt)⟩ as |ψt⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' The time-evolution of the HOPS wave functions is then given by ℏ∂t|ψ(⃗k) t ⟩ = � −i ˆHS −⃗k ·⃗γ −Γt +∑ n ˆLn(z∗ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='t +ξn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='t) � |ψ(⃗k) t ⟩ +∑ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' j kjnγ jn ˆLn|ψ(⃗k−⃗ejn) t ⟩ −∑ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' j (gjn γjn )(ˆL† n −⟨ˆL† n⟩t)|ψ(⃗k+⃗ejn) t ⟩ (5) where ξn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='t = 1 ℏ � t 0 dsC∗ n(t −s)⟨L† n⟩s (6) is a memory term that causes a drift in the effective noise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' ⟨ˆL† n⟩t = ⟨ψ(⃗0) t |ˆL† n|ψ(⃗0) t ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' (7) and Γt =∑ n ⟨ˆLn⟩t Re[z∗ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='t +ξn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='t] −∑ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' j Re ��g jn γjn � ⟨ψ(⃗0) t |ˆL† n|ψ(⃗ejn) t ⟩ � +∑ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' j ⟨ˆL† n⟩t Re ��g jn γ jn � ⟨ψ(⃗0) t |ψ(⃗ejn) t ⟩ � (8) ensures normalization of the physical wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' In these equations, kjn is the jnth element of the index vector ⃗k, and ⃗γ is a vector of the exponential factors of the correlation function modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Here we employ the "triangular truncation" scheme, which restricts the hierarchy of auxiliary wave functions to a finite depth kmax, such that {⃗k ∈ A : ∑i ki ≤ kmax}, though other static filtering approaches have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='39 The computational expense of HOPS scales poorly with system size because the number of auxiliary wave functions in the hierarchy increases rapidly with the total number of exponential modes defining the corre- lation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' However, the physical wave function of a HOPS trajectory is localized in the basis of sys- tem states (S) by the interaction with the thermal bath (‘dynamic localization’), which induces a correspond- ing localization in the basis of auxiliary wave func- tions (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='37 As a result, in extended aggregates, the vast majority of auxiliary wave functions are unoccu- pied at any single point in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' If unoccupied system states and auxiliary wave functions could be efficiently eliminated from the calculation, the computational ex- pense of HOPS would not increase with system size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=', O(1) scaling) for sufficiently large aggregates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='37 The adaptive hierarchy of pure states (adHOPS) ap- proach constructs a time-evolving reduced basis to provide a numerically tractable formulation of HOPS trajectories for large molecular aggregates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='37 Every ut (‘update time’) femtoseconds, adHOPS constructs a reduced basis as a direct sum of reduced system and auxiliary bases (St �At).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' The algorithm determines the error introduced by neglecting each basis element as an upper-bound on the Euclidean norm of the result- ing displacement in the full derivative vector (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' (5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' It then truncates basis elements in order of increasing error until user-defined error bounds for the hierarchy (δA) and then the state (δS) bases are saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Previ- ously, adHOPS has replicated the dynamics of HOPS in a linear chain with a dramatically reduced basis that demonstrated size-invariant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=', O(1)) scaling for sufficiently large aggregates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='37 The version of adHOPS used for the calculations re- ported here is available in the MesoHOPS package ver- sion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='40 Additional details about both the algorithm and the calculation parameters are available in the Sup- porting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' We check the convergence of ad- HOPS results by running a series of adHOPS ensem- bles with increasingly strict parameters until the char- acteristic observables are within 2% of the most exact calculations (see section S4 of the Supporting Infor- mation for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' The LH2 monomer (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' 1a-b) contains two rings of bacteriochlorophyll (BChl) that absorb at 800 nm (B800 ring, blue) and 850 nm (B850 ring, green), re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' The B800 ring is comprised of 9 widely- spaced and weakly-coupled BChls and funnels excita- tion to the B850 ring, which is responsible for trans- port between LH2 monomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' The tight packing of the 18 BChls (organized into αβ pairs) in the B850 ring gives rise to strong electronic couplings and delo- calized eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Since inter-LH2 transport occurs predominately between B850 rings,41 we neglect the B800 pigments found in the 1NKZ pdb structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='42 3 Figure 1: LH2 monomer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' (a) Side view: the B850 ring (18 BChl) and the B800 ring (9 BChl) are shown in green and blue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' (b) Top view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' (c) The distance from equilibrium as a function of time (black) compared to a bi-exponential fit (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' (d) Boltz- mann distribution (green line) compared to adHOPS equilibrium populations (grey squares) of eigenstates ordered by absolute index (|ν|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' σ = 160 cm−1 for site energy static disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Convergence parameters are given in Table S6 of the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' We model the electronic energy levels of the B850 ring using previously established parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='43 In the electronic (‘system’) Hamiltonian, the vertical excita- tion energies (or ‘site energies’) of the α and β chloro- phylls are 12690 cm−1 and 12070 cm−1, respectively, and the electronic couplings (Vn,m) within and between αβ pairs are 307 cm−1 and 237 cm−1, respectively (see section S1 of the Supporting Information for full B850 Hamiltonian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' In keeping with previous spectro- scopic assignments, we also include static disorder on the site energy of each chlorophyll as Gaussian fluctu- ations with a standard deviation of σ = 160 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='22,43 Finally, we model the thermal environment of each pigment with a Drude-Lorentz spectral density charac- terized by a reorganization timescale (γ0n = 53 cm−1) and reorganization energy (λn = 65 cm−1)43 Jn(ω) = 2λnγ0n ω ω2 +γ2 0n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' (9) The corresponding correlation function is composed of one high temperature mode, kMats Matsubara modes, and an additional mode to ensure Im[α(0)] = 0 (see section S2 of the Supporting Information for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' We characterize the relaxation of the exciton popula- tion towards equilibrium using the 1-norm of a differ- ence vector ||P(t)−Peq||1 (10) where P(t) and Peq are the eigenstate population vec- tor of the B850 ring at time t and its equilibrium value, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Using these parameters, the degenerate optically bright (ν = ±1) states in the B850 ring relax to equilibrium on two timescales (τ1 = 25 fs and τ2 = 150 fs, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' 1c), which agree with those found by global kinetic fits of B850 intraband relaxation in 2D electronic spectroscopy (50 fs and 150 fs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='44 The equi- librium eigenstate populations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' 1d) form a Boltz- mann distribution, slightly perturbed by the electron- vibrational coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' We construct a hexagonally-packed supercomplex containing 37 B850 rings (666 BChl) organized as three concentric shells around a donor ring (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' 2a) with center-to-center ring distances (R) of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' This inter-ring distance represents the closest packing of LH2 proteins proposed for artificial aggregates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='19,45 The inter-ring couplings ˜Vn,m are calculated using the ideal dipole approximation ˜Vn,m = C ⃗dn · ⃗dm −3(⃗dn ·⃗rn,m)(⃗dm ·⃗rn,m) R3n,m (11) where the coupling constant C = 348,000 ˚A3 · cm−1, ⃗dn is a unit vector along the direction of the transi- tion dipole moment of pigment n, and ⃗rn,m (Rn,m) is the unit vector (distance) between Mg atoms in pig- ments n and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' In addition to the site energy static dis- order included in each ring, we add an angular static disorder represented by randomly orienting the B850 rings in each trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' We find that static disorder in the site energies suppresses transport while angular disorder has negligible impact (see section S7 of the Supporting Information for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Exciton transport in LH2 aggregates exhibits two distinct regimes: a superdiffusive transport at short times and ongoing diffusive transport at times longer than 200 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' To quantify transport in our adHOPS simulations, we calculate the mean-squared deviation 4 (a) LH2 side (b) LH2 top B850 B800 (c) Distance from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' (d) Eig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Distribution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='5 adHOPS adHOPS fit Boltzmann Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Population 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='5 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='5 0 3 6 9 Time (ps) Eigenstate Index (DFigure 2: Exciton transport in a mesoscale aggregate of LH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' (a) Schematic of a hexagonally-packed B850 com- plex with three concentric shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' (b) Mean squared deviation (MSD) of the excitation from adHOPS dynamics (thick line), power-law fits (thin lines), and kinetic model (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' (c) adHOPS (solid lines) and kinetic model (dashed lines) population dynamics for the donor and three concentric shells of B850 rings (shown in color-coded inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' σ = 160 cm−1 for site energy static disorder, and angular static disorder is given by randomly orienting all B850 rings in each trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Convergence parameters are given in Table S6 of the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' (MSD, grey line, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' 2b) of the exciton distribution MSD(t) = ∑ n Pn(t)R2 0n (12) where Pn is the population of the nth B850 ring and n = 0 signifies the donor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' To characterize the mech- anism of transport, we fit the MSD to a power law (MSD(t) = Dtα) in different time windows (black lines, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' From 0 to 50 fs, the MSD exhibits su- perdiffusive behavior (α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Between 50 and 200 fs, however, the MSD behavior shifts rapidly from a convex to a linear form, consistent with coher- ent transport giving way to diffusive transport on the timescale of vibrational reorganization (ℏ/γ0n ≈ 100 fs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' From 200 fs onward, the MSD remains diffusive (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='95±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='04) until the exciton reaches the bound- aries of the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' The 37-mer studied here is the minimal aggregate sufficient for simulating the turn over between coher- ent and diffusive transport without edge effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' By the onset of diffusive transport (200 fs), the exciton pop- ulation summed over B850 rings forming the second concentric shell (inset, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' 2c) is nearly 10% of the total population (medium green line, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' 2c), which would suppress further transport in a smaller aggre- gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' The later-time diffusive transport of the 37-mer is re- produced by a kinetic model characterized by a single rate of transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' We fit the population dynamics of a B850 dimer (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' 3a) with center-to-center distance R = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='5 nm to a single rate, κdimer = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='44±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='04 ps−1 (see section S6 of the Supporting Information for de- tails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' We time-evolve the vector of B850 ring popula- tions (P(t)) according to ˙P(t) = K P(t), (13) where the rate matrix (K) connects nearest-neighbor rings via symmetric transport rates equal to κdimer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Starting from 200 fs, the population dynamics of the kinetic model (dashed lines) reproduce the results of adHOPS calculations (solid lines, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' This en- sures that the kinetic model also reproduces the MSD dynamics of diffusive exciton transport in the LH2 ag- gregate (dashed line, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' We have also explored the possibility of the dark-state shelving mechanism recently proposed for close-packed LH2,19 but found that inter-ring transport is dominated by coupling be- tween bright states (see section S8 of the Supporting Information for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' The excitation diffusion length (Ld) in LH2 aggre- gates predicted by our kinetic model greatly exceeds that of prototypical organic semiconductors, even for the relatively loose packing associated with biologi- cal membranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' 3b, we show the dimer rates of transport as a function of the packing distance (R) 5 (a) 37-mer (b) Mean-Squared Deviation (c) Population Dynamics Dt2 Dt1 Dt2 Dt1 100 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='0 - 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='75 (zwu) lation 60 Popul: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='5 % 40 I adHOPS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='25 20 - fit rate model 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='8 NB850 = 37 NBChl = 666 Time (ps) Time (ps)Table 1: LH2 exciton diffusion lengths in a hexagonally-packed lattice R (nm) method D (nm2/psα) α Ld (nm) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='5 rate model 110±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' 1 330±15 adHOPS 100±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='95±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='044 270±54 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='5 rate model 43±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='5 1 210±6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='5 rate model 19±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='0 1 140±4 The uncertainty represents the 95% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Figure 3: LH2 dimer rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' (a) Schematic of a B850 dimer with a center-to-center separation distance of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' (b) Transport rates for an LH2 dimer with an inter-ring separation of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' In all cases, σ = 160 cm−1 for site en- ergy static disorder, and angular static disorder is given by randomly orienting both B850 rings in each trajec- tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Convergence parameters are given in Table S6 of the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='5 nm suggested in some artificial materials to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='5 nm found in biological membranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Table 1 reports the excitation diffusion length in the kinetic model of an infinite aggregate (see section S9 of the Supporting Information) determined by L2 d = 6R2κdimerτ ≡ Dτ, (14) where R is the packing distance, κdimer is the LH2 dimer rate, and τ is the lifetime of the exciton (as- sumed to be 1 ns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='45 At a biologically relevant inter- ring distance of R = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='5 nm, our excitation diffusion length (Ld = 140 nm) is consistent with a previous order of magnitude estimate46 and greatly exceeds that expected for a prototypical organic semiconduc- tor (< 30 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='47,48 Moreover, our rate of transport at R = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='5 nm is smaller than some previous estimates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='241 ps−1 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' 24), suggesting other LH2 Hamiltonians proposed in the literature could support even longer excitation diffusion lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' We conclude that LH2 aggregates support exciton transport on length scales exceeding those of simi- lar artificial and natural materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Excitation trans- port in LH2 aggregates exhibits a brief ballistic period (< 50 fs) followed by diffusive transport mediated by bright-state coupling that supports long-range exciton diffusion even at biological packing distances (R = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='5 nm, Ld = 140 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' We note that LH2 complexes, despite belonging to the evolutionarily ‘early’ anoxy- genic purple bacteria, support a surprisingly large Ld compared to that previously calculated for the oxy- genic photosystem II membrane (Ld = 50 nm),49,50 suggesting that the need for regulation and not long- range excitation energy transport has driven the design of antenna complexes in higher plants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Moreover, in artificially close-packed LH2 aggregates, our calcula- tions suggest the exciton diffusion length can reach up to 300 nm, a full order of magnitude larger than a prototypical organic semiconductor, making LH2 a promising antenna system for biohybrid materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' In this paper, we have reported a formally exact simulation of exciton dynamics in a mesoscale pho- tosynthetic aggregate consisting of an unprecedented 37 LH2s with a total of 666 bacteriochlorophylls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' While some previous studies have simulated large LH2 aggregates using approximate methods, the current calculations provide an important benchmark result where the accuracy of the simulation is limited only by the parameterization of the model Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Our results also demonstrate that adHOPS is suitable for calculations of realistic mesoscale molecular materials to address mechanistic questions about excited-state processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' We expect the continued development of adHOPS will enable a new generation of simulations capable of probing larger and more complex materi- als to reveal new strategies for controlling excited-state processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Acknowledgement The authors thank Bailey Raber 6 (a) B850 Dimer (b) Dimer Rates 100 R (ps-1 Rate ( 10-2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='5 R (nm)and Mohamed El Refaiy for assistance working with the protein structure and the electronic Hamiltonian for the extended aggregates, Julian Schmidt for help- ful discussions and testing a preliminary version of the adHOPS code, as well as Tarun Gera for editing and review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' LV, BC, and DIGB acknowledge sup- port from the Robert A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Welch Foundation (Grant N- 2026-20200401).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' JKL acknowledges support from a Moody Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' OK acknowledges funding by the Deutsche Forschungsgemeinschaft - SFB 1477 "Light-Matter Interactions at Interfaces", project num- ber 441234705”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Supporting Information Available Methodological details, all calculation parameters, convergence studies, analysis of dark/bright state mechanisms, and comparison of different types of static disorder (PDF) References (1) Sumi, H.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' 2018, 115, E9523–E9531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} +page_content=' 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfdwpP/content/2301.04537v1.pdf'} diff --git a/bdE4T4oBgHgl3EQfog0k/content/tmp_files/2301.05184v1.pdf.txt b/bdE4T4oBgHgl3EQfog0k/content/tmp_files/2301.05184v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ab18c54513434b68d221b4395fb5ea4b512c105b --- /dev/null +++ b/bdE4T4oBgHgl3EQfog0k/content/tmp_files/2301.05184v1.pdf.txt @@ -0,0 +1,497 @@ +arXiv:2301.05184v1 [math.PR] 12 Jan 2023 +On some quasi-regenerative reliability system +Galina Zverkina∗† +December 2022 +1 +Introduction +We consider the reliability system consisting of two dependent restorable elements. Both restorable elements +have the same characteristics. However, the “speed” of work or repair depends on the full state of the system. +At first, the first (main) element works, and the serviceable second element does not work at full capacity - it +is in a warm reserve. +If the main element fails, then the reserve element starts working at full speed – it is now the main one. +Or if a working element has already been working for a long time, then the second element begins to warm up +in order to start working faster when the first element fails – to immediately switch to the active mode of operation +when a working element fails. +In addition, if the element works for a very long time, its reliability decreases and the likelihood of its failure in +the near future increases. +In addition, if both elements are faulty, then they are repairing at a faster rate than in another situation. +The probability of failure of a working element or the probability of repair of a repaired element is determined +by intensities that depend only on the full state of the system under study. +The full state of this reliability system at the time t is a vector Xt +def +== (n1, x1; n2, x2) +� += (n1(t), x1(t); n2(t), x2(t)) +� +, +where: +nj(t) +def +== +� +1, +if j-th element is in working state at the time t; +0, +if j-th element is in failure state at the time t; +the variable xj(t) is the elapsed time of the stay of j-th element in the status nj at the time t. +The behaviour of Xt, its distribution give the full information about this reliability system. +The probability of a status nj(t) change over a time interval (t, t + ∆) is equal to λj(Xt)∆ + o(∆); λj(Xt) is an +intensity of a status nj(t) change at the time t. +Under these assumptions, the process Xt is Markov on the state space X +def +== {(0; 1) × R+ × (0; 1) × R+}. +ˆt0 = 0 +✲ +✲ +r +r +ELEMENT II. +work [1] +ˆt1 +repair [0] +ˆt2work [1] +ˆt3 +repair [0] +ˆt4 +work [1] ˆt5 +repair [0] +ˆt6 q q q q q +t0 = 0 +ELEMENT I. +work [1] +t1 +repair [0] +t2work [1] +t3 +repair [0] +t4 q q q +θ1 +θ1 +✛x1 +✛ +y1 +θ2 +θ2 +✛ +x2 +✛y2 +θ3 +θ3 +✛x3 +✛ +y3 +Figure 1: This scheme is a visualization of the work and repair of a reliability system with a warm reserve. Here +at the time θ1 we have Xθ1 = (0, x1; 1, y1), and the main element is II; at the time θ2 we have Xθ2 = (1, x2; 1, y2), +and the main element is I; at the time θ2 we have Xθ3 = (0, x3; 0, y3), and the system in failure state. +∗V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences +†The work is supported by RFBR grant No 20-01-00575A +1 + +In our notation, the set S0,0 +def +== {(0, x; 0, y), x, y ∈ R+} corresponds to the failure state of the system. And the +set S>0 +def +== {(n1, x1; n2, x2) : n1 + n2 > 0} corresponds to the system operating state, and P{Xt ∈ S>0} is equal +to the availability factor of considered system. +Therefore, knowledge of the probability distribution of the stochastic process Xt is important for calculating the +characteristics of such systems. +1.1 +Reliability system considered earlier +We suppose that the intensities of failure for working elements and the intensities of repair for repaired elements +depend on the process Xt: λ(1) +i += λ(1) +i (Xt) is the intensities of failure of i-th element, and λ(0) +i += λ(0) +i +(Xt) is the +intensities of repair of i-th element. In this situation, the process Xt is Markov on the state space X +def +== ({0; 1}×R+)2 +with a standard σ-algebra. If +λ(n) +i +(Xt) ≡ const, +(1) +then this is a classic case with exponential d.f.’s. +If 0 < c ⩽ λ(n) +i +(Xt) ⩽ C < ∞, then this “quasi-exponential” case, it is studied in the paper (using some +properties of work and repair times following from (1) – see [1]). +In the paper [7], a reliability system with a warm reserve was studied under the following conditions for intensities. +There exist constants γ, Γ > 0 such that for any Z = (i, x; j, y) ∈ X (state space) +0 < +γ +1 + x ≤ λ(Z) ≤ Γ < ∞; +0 < +γ +1 + y ≤ µ(Z) ≤ Γ < ∞. +(2) +If the conditions (2) are satisfied, then the process Xt is recurrent with finite return time to some compact. Therefore, +the process Xt is ergodic. +In both cases ( [1,7]), by default, it is assumed that the switching time (work-repair and repair-work) is instan- +taneous, and the distribution of work and repair time is absolutely continuous. +2 +Our goal +Thus, we will study the situation when switching between operation and repair modes is not instantaneous, and +the distribution of work and repair time can have a discrete part. In addition to this, we will study the case when +all switching times are random variables bounded above by a constant with a probability of 1. +2.1 +Intensities +First, we define the generalized intensity for random variables whose distribution has a discrete component. +In order to explore the situation of mixed random variables (i.e. the situation where the distribution can have +discrete and continuous components), we will use the concept of generalized intensity introduced in the paper [4]. +Recall, that the intensity of the distribution of a continuous positive random variable (r.v.) ξ with distribution +function (d.f.) F(s) is the function λ(s) +def +== +F ′(s) +1 − F(s), and P{ξ ∈ (s, s + ∆)|ξ > s} = λ(s)∆ + o(∆). +The function λ(s) defines d.f. F(s): +F(s) = 1 − e +− +s� +0 +λ(v) d v +. +(3) +If the distribution of ξ is mixed (non-singular!!!), i.e. d.f. F(s) has jumps, then we can put +f(s) = + + + +F ′(s), +if F ′(s) exists; +0, +otherwise; +λ(s) +def +== +f(s) +1 − F(s) − +� +i +δ(s − ai) ln(F(ai + 0) − F(ai − 0)), +(4) +2 + +where {ai} is the set of discontinuity points of F(s), and δ(s) is a standard δ-function; +The formula (1) is true with these notations. +So, the distribution of non-negative r.v.’s – continuous, discrete and mixed – can be defined by intensities. +However, the absolutely arbitrary dependence of the intensity of the change in the base state (work, repair) does +not allow us to study the behavior of the reliability system with a warm reserve. +2.2 +Condition for intensities +In the future, for the time of operation and / or repair, we will use the conditions a–d or a–d. +The following conditions for the (generalized) intensities λ(n) +i +(s) are assumed: +a. The (generalized) measurable non-negative functions ϕ(s) and Q(s) exist such that for all s ⩾ 0, +ϕ(s) ⩽ λ(n) +i +(s) ⩽ Q(s); +b. +∞ +� +0 +ϕ(s) d s = ∞, and +∞ +� +0 +xk−1 exp + +− +x +� +0 +ϕ(s) d s + + d x < ∞ for some k ⩾ 2; +c. There is a neighborhood of zero U +def +== (−ε, ε) such that +� +U +Q(s) d s < 1; +d. There exists the constant T ⩾ 0 such that ϕ(s) > 0 a.s. for all s > T . +Remark 1. Condition (a) holds: G(s) = P{ζ ⩽ s} ⩾ F (n) +i +(s) = P{ξ(n) +i +⩽ s} ⩾ Φ(s) = P{�ζ ⩽ s}, or ζ ≿ ξ(n) +i +≿ �ζ +are ordered in distribution +� +here G(s) has intensity Q(s), Φ(s) has intensity ϕ(s) +� +. +⊲ +Remark 2. Condition (b) holds: there exists E ζk < ∞ ⇒ E �ζk < ∞ and E +� +ξ(n) +i +�k +< ∞. +⊲ +Remark 3. Condition (c) holds: E ζ2 > 0, and Var(ζ) > 0. +⊲ +Remark 4. Condition (d) holds: Φ′(x) > 0 a.s. for x > T , i.e. we may consider delayed switching for the considered +reliability system. +⊲ +2.3 +The studied warm standby systems is non-regenerative +An analysis of the behaviour of both elements of the investigated reliability system shows that the periods of +operation and repair of both elements depend on each other. Consequently, for each individual element of this +reliability system there is no independence between the periods of work and repair. +ˆt0 = 0 +✲ +✲ +r +r +ELEMENT II. +work [1] +ˆt1 +repair [0] +ˆt2 work [1] +ˆt3 +repair [0] +ˆt4 work [1] +ˆt5 +repair [0] +ˆt6 q q q q q +t0 = 0 +ELEMENT I. +work [1] +t1 +repair [0] +t2 work [1] +t3 +repair [0] +t4 q q q +� +� +� +✒� +� +� +✠ +✁ +✁ +✁✁✕ ✁ +✁ +✁ +✁☛ +✁ +✁✁✕ ✁ +✁✁☛ +Figure 2: Dependence between the work and repair periods of warm standby systems. +The sequence of periods (t0; t1), (t1; t2), (t2; t3), (t3; t4), . . . and (ˆt0; ˆt1), (ˆt1; ˆt2), (ˆt2; ˆt3), (ˆt3; ˆt4), . . . “looks like” +an alternating renewal process. +Through the dependence of the periods (t0; t1) and (ˆt0; ˆt1); (t0; t1) and (ˆt1; ˆt2); (t1; t2) and (ˆt2; ˆt3); etc., we have +some dependencies between the lengths of the periods (t0; t1), (t1; t2), (t2; t3), (t3; t4), . . . and between the lengths +of the periods (ˆt0; ˆt1), (ˆt1; ˆt2), (ˆt2; ˆt3), (ˆt3; ˆt4), . . . +For study, these dependencies cannot be arbitrary; below we indicate the conditions for the intensities. +These dependencies are “weak” in some sense. Thus, the behaviour of each of the elements is described by an +alternating quasi-renewal process, which is “close” to the classical renewal alternating process. +3 + +2.4 +How is the behaviour of this non-regenerative system studied. +Thus, the first step of analysis of the behaviour of our system is the proof of the fact: +Theorem 1. The process Xt is quasi-regenerative, i.e. there exists regenerative Markov process ˜Xt (with finite +regeneration time) on the state space X such that at any time the marginal distribution of Xt is equal the marginal +distribution of ˜Xt. +Thus, the process Xt is ergodic. +⊲ +The proof based on the coupling method1. +The next step is the analysis of the behaviour of two processes Xt and X′ +t with different initial state, but with +the same transition probabilities. +By construction of successful coupling given by D. Griffeath [2], an upper bounds for the convergence rate of Xt +can be calculated. +Recall, that our goal is to compute an upper bound for convergence rate of the distribution of the reliability +system in the case when the distributions of operating and recovery times may not be absolutely continuous, and +mode switching may have a random finite delay. The initial state of the system can be arbitrary. +work [1] +repair [0] +work [1] +repair [0] +work [1] +q q q q +q +t = 0 +repair [0] +work [1] +repair [0] +q q q +t = 0 +✲ +✲ +r +r +Figure 3: Initial state of the process Xt is X0 = (0, x; 1, y). Blue bars represent a switching time. +We suppose that the intensities of failure for working elements and the intensities of repair for repaired elements +depend on the process Xt: λ(1) +i += λ(1) +i (Xt) is the intensities of failure of i-th element, and λ(0) +i += λ(0) +i +(Xt) is the +intensities of repair of i-th element. +Remark 5. For greater generality, we assume that all restrictions under the conditions of a–d are different for +different elements. +So, we can give the following modified conditions +a’. For all s ⩾ 0, ϕ(n) +i +(s) ⩽ λ(n) +i +(s) ⩽ Q(n) +i +(s); +d’. There exists the constants T (n) +i +⩾ 0 such that ϕ(n) +i +(s) > 0 a.s. for all s > T (n) +i +. +In this case, the upper bounds for k-th moments of ξ(n) +i +are different, as well as its delay times. +Put C(n) +j +(ℓ) +def +== +∞ +� +0 +sℓ d Φ(n) +j +(s), where Φ(n) +j +(s) +def +== 1 − e +− +s� +0 +ϕ(n) +j +(v) d v +(see formula (3)). +The vector ⃗C(ℓ) = +� +C(1) +1 (ℓ), C(1) +2 (ℓ), C(0) +1 (ℓ), C(0) +2 (ℓ) +� +depends on the vector ⃗ϕ(s) = +� +ϕ(1) +1 (s), ϕ(1) +2 (s), ϕ(0) +1 (s), ϕ(0) +2 (s) +� +. +Also denote ⃗Q(s) = +� +Q(1) +1 (s), Q(1) +2 (s), Q(0) +1 (s), Q(0) +2 (s) +� +. +3 +Main results +Theorem 2. Let the work and repair periods of both elements satisfy the conditions a–b, and the work periods of +both elements satisfy the conditions c–d. +Then the process Xt is ergodic, i.e. the distribution Pt of Xt at the time t, weak converges to the invariant +stationary probabilistic distribution P. +⊲ +Theorem 3. Let the work and repair periods of both elements satisfy the conditions a–b, and the repair periods of +both elements satisfy the conditions c–d. +Then the process Xt is ergodic, i.e. the distribution Pt of Xt at the time t, weak converges to the invariant +stationary probabilistic distribution P. +⊲ +4 + +Theorem 4. In conditions of the Theorems 2 and 3 for all ℓ ∈ (0, k − 1] we give an algorithm of calculation of the +constant K(ℓ, X0, ⃗ϕ(s), ⃗Q(s)) such that for all t, +∥Pt − P∥T V ⩽ K(ℓ, X0, ⃗ϕ(s), ⃗Q(s)) +tℓ +, +where ∥ · − · ∥T V is a total variation metric: +∥Pt − P∥T V +def +== +sup +A∈B(X ) +|Pt(A) − P(A)|. +⊲ +Theorem 5. Let the conditions of the Theorems 2 and 3 are satisfy, and let E exp(α˜ζ) < ∞ for some α > 0. In +this case we give an algorithm of calculation of the number β ∈ (0, α), and the constant ˜K(β, ℓ, X0, ⃗ϕ(s), ⃗Q(s)) such +that for all t, +∥Pt − P∥T V ⩽ exp −βt ˜K(β, ℓ, X0, ⃗ϕ(s), ⃗Q(s)), +where ∥ · − · ∥T V is a total variation metric: +∥Pt − P∥T V +def +== +sup +A∈B(X ) +|Pt(A) − P(A)|. +⊲ +3.1 +1. Ergodicity +The proof of ergodicity of the process Xt based on the notion Generalized Markov Modulated Poisson Process +(GMMPP) and generalized Lorden’s inequality [3,4]. +The consecutive pairs of periods of work and repair make up GMMPP that satisfy the conditions of ergodicity +of such processes. +3.2 +2. Construction of upper bounds for convergence rate +The process Xt is quasi-regenerative, i.e. on some probability space there exists the Markov regenerative process +� +Xt with the same marginal distributions as the process Xt: for all A ∈ B(X ) and for all t ⩾ 0, P{Xt ∈ A} = +P{ � +Xt ∈ A}. +And for this Markov regenerative process � +Xt we can construct the constant K(ℓ, X0, ⃗ϕ(s), ⃗Q(s)). The construc- +tion based on the coupling method. [5,6] +These calculations are not optimal and can be improved by the use some properties of specific intensities. +Remark 6. The schema used in this work, can be used for reliability systems with many warm-reserve elements. +The construction of upper bounds for convergence rate is also possible for the case when at least one period +(work or repair) for any element satisfies the condition d: for all s > T , ϕ(s) > 0. +Naturally, an increase in the number of elements leads to an increase in the constant K(· · · ). +Also it is important to understand that the coupling method does not provide an accurate estimate for stochastic +processes in continuous time. +References +[1] G. Zverkina. A System with Warm Standby / Computer Networks (Proceedings of the 26th International +Conference (CN 2019, Kamie´n ´Sl¸aski, Poland). Cham: Springer, 2019. P. 387-399. +[2] Griffeath, D. A maximal coupling for Markov chains // Zeitschrift f¨ur Wahrscheinlichkeitstheorie und Ver- +wandte Gebiete — 1975 — Volume 31 — Issue 2, P. 95–106. +5 + +[3] G.A. Zverkina. Ergodicity and Polynomial Convergence Rate of Generalized Markov Modulated Poisson Pro- +cesses / Proceedings of the 23rd International Conference on Distributed Computer and Communication Net- +works: Control, Computation, Communications (DCCN-2020, Moscow). Cham: Springer, 2021. Vol.1337. P. +367-381. (See also arXiv:2010.05875) +[4] E.Yu. Kalimulina, G.A. Zverkina. On generalized intensity function and its application to the backward renewal +time estimation for renewal processes / Proceedings of the 5th International Conference on Stochastic Methods +(ICSM-5, 2020). M.: RUDN, 2020. P. 306-310. (See also arXiv:1910.03381.) +[5] Zverkina G., On strong bounds of the rate of convergence for regenerative processes // Communications in +computer and information science, 2016, v.678, P. 381–393. +[6] Zverkina G., Coupling method for backward renewal process and Lorden’s inequality // Communications in +computer and information science, 2017, v.700, P. 368–379. +[7] A.Yu. Veretennikov, On Polynomial Recurrence for Reliability System with a Warm Reserve // Markov Pro- +cesses and Related Fields, 2019, v.25, Issue 4, P. 745–761. +6 + diff --git a/bdE4T4oBgHgl3EQfog0k/content/tmp_files/load_file.txt b/bdE4T4oBgHgl3EQfog0k/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e54aea7daf72280df09b24f5d3b406d0ce8f533c --- /dev/null +++ b/bdE4T4oBgHgl3EQfog0k/content/tmp_files/load_file.txt @@ -0,0 +1,260 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf,len=259 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='05184v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='PR] 12 Jan 2023 On some quasi-regenerative reliability system Galina Zverkina∗† December 2022 1 Introduction We consider the reliability system consisting of two dependent restorable elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Both restorable elements have the same characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' However, the “speed” of work or repair depends on the full state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' At first, the first (main) element works, and the serviceable second element does not work at full capacity - it is in a warm reserve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' If the main element fails, then the reserve element starts working at full speed – it is now the main one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Or if a working element has already been working for a long time, then the second element begins to warm up in order to start working faster when the first element fails – to immediately switch to the active mode of operation when a working element fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' In addition, if the element works for a very long time, its reliability decreases and the likelihood of its failure in the near future increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' In addition, if both elements are faulty, then they are repairing at a faster rate than in another situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' The probability of failure of a working element or the probability of repair of a repaired element is determined by intensities that depend only on the full state of the system under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' The full state of this reliability system at the time t is a vector Xt def == (n1, x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' n2, x2) � = (n1(t), x1(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' n2(t), x2(t)) � , where: nj(t) def == � 1, if j-th element is in working state at the time t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 0, if j-th element is in failure state at the time t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' the variable xj(t) is the elapsed time of the stay of j-th element in the status nj at the time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' The behaviour of Xt, its distribution give the full information about this reliability system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' The probability of a status nj(t) change over a time interval (t, t + ∆) is equal to λj(Xt)∆ + o(∆);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' λj(Xt) is an intensity of a status nj(t) change at the time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Under these assumptions, the process Xt is Markov on the state space X def == {(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 1) × R+ × (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 1) × R+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ˆt0 = 0 ✲ ✲ r r ELEMENT II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' work [1] ˆt1 repair [0] ˆt2work [1] ˆt3 repair [0] ˆt4 work [1] ˆt5 repair [0] ˆt6 q q q q q t0 = 0 ELEMENT I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' work [1] t1 repair [0] t2work [1] t3 repair [0] t4 q q q θ1 θ1 ✛x1 ✛ y1 θ2 θ2 ✛ x2 ✛y2 θ3 θ3 ✛x3 ✛ y3 Figure 1: This scheme is a visualization of the work and repair of a reliability system with a warm reserve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Here at the time θ1 we have Xθ1 = (0, x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 1, y1), and the main element is II;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' at the time θ2 we have Xθ2 = (1, x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 1, y2), and the main element is I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' at the time θ2 we have Xθ3 = (0, x3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 0, y3), and the system in failure state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ∗V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Trapeznikov Institute of Control Sciences of Russian Academy of Sciences †The work is supported by RFBR grant No 20-01-00575A 1 In our notation, the set S0,0 def == {(0, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 0, y), x, y ∈ R+} corresponds to the failure state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' And the set S>0 def == {(n1, x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' n2, x2) : n1 + n2 > 0} corresponds to the system operating state, and P{Xt ∈ S>0} is equal to the availability factor of considered system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Therefore, knowledge of the probability distribution of the stochastic process Xt is important for calculating the characteristics of such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='1 Reliability system considered earlier We suppose that the intensities of failure for working elements and the intensities of repair for repaired elements depend on the process Xt: λ(1) i = λ(1) i (Xt) is the intensities of failure of i-th element, and λ(0) i = λ(0) i (Xt) is the intensities of repair of i-th element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' In this situation, the process Xt is Markov on the state space X def == ({0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 1}×R+)2 with a standard σ-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' If λ(n) i (Xt) ≡ const, (1) then this is a classic case with exponential d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='f.’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' If 0 < c ⩽ λ(n) i (Xt) ⩽ C < ∞, then this “quasi-exponential” case, it is studied in the paper (using some properties of work and repair times following from (1) – see [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' In the paper [7], a reliability system with a warm reserve was studied under the following conditions for intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' There exist constants γ, Γ > 0 such that for any Z = (i, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' j, y) ∈ X (state space) 0 < γ 1 + x ≤ λ(Z) ≤ Γ < ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 0 < γ 1 + y ≤ µ(Z) ≤ Γ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' (2) If the conditions (2) are satisfied, then the process Xt is recurrent with finite return time to some compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Therefore, the process Xt is ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' In both cases ( [1,7]), by default, it is assumed that the switching time (work-repair and repair-work) is instan- taneous, and the distribution of work and repair time is absolutely continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 2 Our goal Thus, we will study the situation when switching between operation and repair modes is not instantaneous, and the distribution of work and repair time can have a discrete part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' In addition to this, we will study the case when all switching times are random variables bounded above by a constant with a probability of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='1 Intensities First, we define the generalized intensity for random variables whose distribution has a discrete component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' In order to explore the situation of mixed random variables (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' the situation where the distribution can have discrete and continuous components), we will use the concept of generalized intensity introduced in the paper [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Recall, that the intensity of the distribution of a continuous positive random variable (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=') ξ with distribution function (d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=') F(s) is the function λ(s) def == F ′(s) 1 − F(s), and P{ξ ∈ (s, s + ∆)|ξ > s} = λ(s)∆ + o(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' The function λ(s) defines d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' F(s): F(s) = 1 − e − s� 0 λ(v) d v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' (3) If the distribution of ξ is mixed (non-singular!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='!!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' F(s) has jumps, then we can put f(s) = \uf8f1 \uf8f2 \uf8f3 F ′(s), if F ′(s) exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 0, otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' λ(s) def == f(s) 1 − F(s) − � i δ(s − ai) ln(F(ai + 0) − F(ai − 0)), (4) 2 where {ai} is the set of discontinuity points of F(s), and δ(s) is a standard δ-function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' The formula (1) is true with these notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' So, the distribution of non-negative r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='v.’s – continuous, discrete and mixed – can be defined by intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' However, the absolutely arbitrary dependence of the intensity of the change in the base state (work, repair) does not allow us to study the behavior of the reliability system with a warm reserve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='2 Condition for intensities In the future, for the time of operation and / or repair, we will use the conditions a–d or a–d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' The following conditions for the (generalized) intensities λ(n) i (s) are assumed: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' The (generalized) measurable non-negative functions ϕ(s) and Q(s) exist such that for all s ⩾ 0, ϕ(s) ⩽ λ(n) i (s) ⩽ Q(s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ∞ � 0 ϕ(s) d s = ∞, and ∞ � 0 xk−1 exp \uf8eb \uf8ed− x � 0 ϕ(s) d s \uf8f6 \uf8f8 d x < ∞ for some k ⩾ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' There is a neighborhood of zero U def == (−ε, ε) such that � U Q(s) d s < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' There exists the constant T ⩾ 0 such that ϕ(s) > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' for all s > T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Condition (a) holds: G(s) = P{ζ ⩽ s} ⩾ F (n) i (s) = P{ξ(n) i ⩽ s} ⩾ Φ(s) = P{�ζ ⩽ s}, or ζ ≿ ξ(n) i ≿ �ζ are ordered in distribution � here G(s) has intensity Q(s), Φ(s) has intensity ϕ(s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ⊲ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Condition (b) holds: there exists E ζk < ∞ ⇒ E �ζk < ∞ and E � ξ(n) i �k < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ⊲ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Condition (c) holds: E ζ2 > 0, and Var(ζ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ⊲ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Condition (d) holds: Φ′(x) > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' for x > T , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' we may consider delayed switching for the considered reliability system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ⊲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='3 The studied warm standby systems is non-regenerative An analysis of the behaviour of both elements of the investigated reliability system shows that the periods of operation and repair of both elements depend on each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Consequently, for each individual element of this reliability system there is no independence between the periods of work and repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ˆt0 = 0 ✲ ✲ r r ELEMENT II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' work [1] ˆt1 repair [0] ˆt2 work [1] ˆt3 repair [0] ˆt4 work [1] ˆt5 repair [0] ˆt6 q q q q q t0 = 0 ELEMENT I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' work [1] t1 repair [0] t2 work [1] t3 repair [0] t4 q q q � � � ✒� � � ✠ ✁ ✁ ✁✁✕ ✁ ✁ ✁ ✁☛ ✁ ✁✁✕ ✁ ✁✁☛ Figure 2: Dependence between the work and repair periods of warm standby systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' The sequence of periods (t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' t1), (t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' t2), (t2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' t3), (t3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' t4), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' and (ˆt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ˆt1), (ˆt1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ˆt2), (ˆt2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ˆt3), (ˆt3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ˆt4), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' “looks like” an alternating renewal process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Through the dependence of the periods (t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' t1) and (ˆt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ˆt1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' (t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' t1) and (ˆt1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ˆt2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' (t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' t2) and (ˆt2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ˆt3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=', we have some dependencies between the lengths of the periods (t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' t1), (t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' t2), (t2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' t3), (t3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' t4), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' and between the lengths of the periods (ˆt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ˆt1), (ˆt1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ˆt2), (ˆt2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ˆt3), (ˆt3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ˆt4), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' For study, these dependencies cannot be arbitrary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' below we indicate the conditions for the intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' These dependencies are “weak” in some sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Thus, the behaviour of each of the elements is described by an alternating quasi-renewal process, which is “close” to the classical renewal alternating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='4 How is the behaviour of this non-regenerative system studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Thus, the first step of analysis of the behaviour of our system is the proof of the fact: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' The process Xt is quasi-regenerative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' there exists regenerative Markov process ˜Xt (with finite regeneration time) on the state space X such that at any time the marginal distribution of Xt is equal the marginal distribution of ˜Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Thus, the process Xt is ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ⊲ The proof based on the coupling method1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' The next step is the analysis of the behaviour of two processes Xt and X′ t with different initial state, but with the same transition probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' By construction of successful coupling given by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Griffeath [2], an upper bounds for the convergence rate of Xt can be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Recall, that our goal is to compute an upper bound for convergence rate of the distribution of the reliability system in the case when the distributions of operating and recovery times may not be absolutely continuous, and mode switching may have a random finite delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' The initial state of the system can be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' work [1] repair [0] work [1] repair [0] work [1] q q q q q t = 0 repair [0] work [1] repair [0] q q q t = 0 ✲ ✲ r r Figure 3: Initial state of the process Xt is X0 = (0, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 1, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Blue bars represent a switching time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' We suppose that the intensities of failure for working elements and the intensities of repair for repaired elements depend on the process Xt: λ(1) i = λ(1) i (Xt) is the intensities of failure of i-th element, and λ(0) i = λ(0) i (Xt) is the intensities of repair of i-th element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' For greater generality, we assume that all restrictions under the conditions of a–d are different for different elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' So, we can give the following modified conditions a’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' For all s ⩾ 0, ϕ(n) i (s) ⩽ λ(n) i (s) ⩽ Q(n) i (s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' d’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' There exists the constants T (n) i ⩾ 0 such that ϕ(n) i (s) > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' for all s > T (n) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' In this case, the upper bounds for k-th moments of ξ(n) i are different, as well as its delay times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Put C(n) j (ℓ) def == ∞ � 0 sℓ d Φ(n) j (s), where Φ(n) j (s) def == 1 − e − s� 0 ϕ(n) j (v) d v (see formula (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' The vector ⃗C(ℓ) = � C(1) 1 (ℓ), C(1) 2 (ℓ), C(0) 1 (ℓ), C(0) 2 (ℓ) � depends on the vector ⃗ϕ(s) = � ϕ(1) 1 (s), ϕ(1) 2 (s), ϕ(0) 1 (s), ϕ(0) 2 (s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Also denote ⃗Q(s) = � Q(1) 1 (s), Q(1) 2 (s), Q(0) 1 (s), Q(0) 2 (s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 3 Main results Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Let the work and repair periods of both elements satisfy the conditions a–b, and the work periods of both elements satisfy the conditions c–d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Then the process Xt is ergodic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' the distribution Pt of Xt at the time t, weak converges to the invariant stationary probabilistic distribution P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ⊲ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Let the work and repair periods of both elements satisfy the conditions a–b, and the repair periods of both elements satisfy the conditions c–d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Then the process Xt is ergodic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' the distribution Pt of Xt at the time t, weak converges to the invariant stationary probabilistic distribution P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ⊲ 4 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' In conditions of the Theorems 2 and 3 for all ℓ ∈ (0, k − 1] we give an algorithm of calculation of the constant K(ℓ, X0, ⃗ϕ(s), ⃗Q(s)) such that for all t, ∥Pt − P∥T V ⩽ K(ℓ, X0, ⃗ϕ(s), ⃗Q(s)) tℓ , where ∥ · − · ∥T V is a total variation metric: ∥Pt − P∥T V def == sup A∈B(X ) |Pt(A) − P(A)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ⊲ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Let the conditions of the Theorems 2 and 3 are satisfy, and let E exp(α˜ζ) < ∞ for some α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' In this case we give an algorithm of calculation of the number β ∈ (0, α), and the constant ˜K(β, ℓ, X0, ⃗ϕ(s), ⃗Q(s)) such that for all t, ∥Pt − P∥T V ⩽ exp −βt ˜K(β, ℓ, X0, ⃗ϕ(s), ⃗Q(s)), where ∥ · − · ∥T V is a total variation metric: ∥Pt − P∥T V def == sup A∈B(X ) |Pt(A) − P(A)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' ⊲ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Ergodicity The proof of ergodicity of the process Xt based on the notion Generalized Markov Modulated Poisson Process (GMMPP) and generalized Lorden’s inequality [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' The consecutive pairs of periods of work and repair make up GMMPP that satisfy the conditions of ergodicity of such processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Construction of upper bounds for convergence rate The process Xt is quasi-regenerative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' on some probability space there exists the Markov regenerative process � Xt with the same marginal distributions as the process Xt: for all A ∈ B(X ) and for all t ⩾ 0, P{Xt ∈ A} = P{ � Xt ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' And for this Markov regenerative process � Xt we can construct the constant K(ℓ, X0, ⃗ϕ(s), ⃗Q(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' The construc- tion based on the coupling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' [5,6] These calculations are not optimal and can be improved by the use some properties of specific intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' The schema used in this work, can be used for reliability systems with many warm-reserve elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' The construction of upper bounds for convergence rate is also possible for the case when at least one period (work or repair) for any element satisfies the condition d: for all s > T , ϕ(s) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Naturally, an increase in the number of elements leads to an increase in the constant K(· · · ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Also it is important to understand that the coupling method does not provide an accurate estimate for stochastic processes in continuous time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' References [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' Zverkina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' A System with Warm Standby / Computer Networks (Proceedings of the 26th 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 745–761.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} +page_content=' 6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdE4T4oBgHgl3EQfog0k/content/2301.05184v1.pdf'} diff --git a/btFJT4oBgHgl3EQfQixG/content/tmp_files/2301.11491v1.pdf.txt b/btFJT4oBgHgl3EQfQixG/content/tmp_files/2301.11491v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9706fec62f638ed3bb2f6510902cb898ce3a51fd --- /dev/null +++ b/btFJT4oBgHgl3EQfQixG/content/tmp_files/2301.11491v1.pdf.txt @@ -0,0 +1,7734 @@ +Change point detection and inference in multivariable +nonparametric models under mixing conditions +Carlos Misael Madrid Padilla1 +Haotian Xu2 +Daren Wang3 +Oscar Hernan Madrid Padilla4 +Yi Yu5 +1Department of Mathematics, University of Notre Dame +2Department of Statistics, Pennsylvania State University +3Department of Statistics, University of Notre Dame +4Department of Statistics, University of California +5Department of Statistics, University of Warwick +January 30, 2023 +Abstract +This paper studies multivariate nonparametric change point localization and inference prob- +lems. The data consists of a multivariate time series with potentially short range dependence. +The distribution of this data is assumed to be piecewise constant with densities in a H¨older +class. The change points, or times at which the distribution changes, are unknown. We derive +the limiting distributions of the change point estimators when the minimal jump size vanishes +or remains constant, a first in the literature on change point settings. We are introducing two +new features: a consistent estimator that can detect when a change is happening in data with +short-term dependence, and a consistent block-type long-run variance estimator. +Numerical +evidence is provided to back up our theoretical results. +1 +Introduction +In this paper, we study the problem of change point detection in nonparametric settings. Our model +assumes that a vector of measurements is collected at every time point following a distribution +that has a probability density function belonging to a H¨older function class. The change point +assumption implies that the probability density functions remain the same across time except for +abrupt changes at the change points. Furthermore, our theory permits temporal dependence of the +measurements, a feature not explored in prior work. +To be more specific, the observations {Xt}T +t=1 ⊂ Rp are assumed to be an α-mixing sequence of +random vectors with unknown distributions {Pt}T +t=1. The α-mixing coefficients, {αk}k∈Z, have an +exponential-decay, +αk ≤ e−2ck, +k ∈ Z, +(1) +for a certain c > 0. The decay rate of αk imposes a temporal dependence between events that are +separated by k time points, as is stated in (1). This is a standard requirement in the literature +(e.g Abadi, 2004; Merlev`ede et al., 2009). To model the nonstationarity of sequentially observed +1 +arXiv:2301.11491v1 [math.ST] 27 Jan 2023 + +multivariate data, we assume that there exists K ∈ N change points, namely {ηk}K +k=1 ⊂ {2, ..., T} +with 1 = η0 < η1 < . . . < ηk ≤ T < ηK+1 = T + 1, such that +Pt ̸= Pt−1 if and only if t ∈ {η1, . . . , ηK}. +(2) +Our primary interest is to accurately estimate {ηk}K +k=1 and perform inference. We refer to Assump- +tion 1 below for detailed technical conditions on the model described by (1) and (2). +Nonstationary multivariate data is frequently encountered in real-world applications, including +biology (e.g. Molenaar et al., 2009; Wolkovich and Donahue, 2021), epidemiology (e.g. Azhar et al., +2021; Nguyen et al., 2021), social science (e.g. Kunitomo and Sato, 2021; Cai et al., 2022), clima- +tology (e.g. Corbella and Stretch, 2012; Heo and Manuel, 2022), finance (e.g. Herzel et al., 2002; +Schmitt et al., 2013), neuroscience (e.g. Frolov et al., 2020; Gorrostieta et al., 2019), among others. +Due to the importance of modeling nonstationary data in various scientific fields, this problem +has received extensive attention in the statistical change point literature, (e.g. Aue et al., 2009b; +Fryzlewicz, 2014; Cho and Fryzlewicz, 2015; Cho, 2016; Wang et al., 2020). However, there are a +few limitations in the existing works in multivariate nonparametric settings. Firstly, to the best +of our knowledge, temporal dependence has not been considered. Secondly, there is no consistent +result for data with the underlying densities being as general as H¨older smooth. Lastly, but most +importantly, the statistical task of deriving limiting distributions of the change point estimators +has reportedly not been treated in the multivariate nonparametric change point literature. +Taking into account the aforestated limitations, this paper examines change point problems in +a fully nonparametric framework, wherein the underlying distributions are only assumed to have +piecewise and H¨older smooth continuous densities and the magnitudes of the distributional changes +are measured by the L2-norm of the differences between the corresponding densities. +The rest of the paper is organized as follows. In Section 2, we explain the model assumptions +for multivariate time series with change points in a nonparametric setting. Section 3 details the +two-step change point estimation procedure, as well as the estimators at each step. Theoretical +results, including the consistency of the preliminary estimator and the limiting distribution of the +final estimator, are presented in Section 4. Section 5 evaluates the practical performance of the +proposed procedure via various simulations and a real data analysis. Finally, Section 6 concludes +with a discussion. +1.1 +Notation +For any function f : Rp → R and for 1 ≤ q < ∞, define ∥f∥Lq = ( +� +Rp |f(x)|qdx)1/q and for q = ∞, +define ∥f∥L∞ = supx∈Rp |f(x)|. Define Lq = {f : Rp → R, ∥f∥q < ∞}. Moreover, for q = 2, define +⟨f, g⟩L2 = +� +Rp f(x)g(x)dx where f, g : +Rp → R. For any vector s = (s1, . . . , sp)⊤ ∈ Np, define +|s| = �p +i=1 si, s! = s1! · · · sp! and the associated partial differential operator Ds = +∂|s| +∂xs1 +1 ···∂x +sp +p . For +α > 0, denote ⌊α⌋ to be the largest integer smaller than α. For any function f : Rp → R that +is ⌊α⌋-times continuously differentiable at point x0, denote by fα +x0 its Taylor polynomial of degree +⌊α⌋ at x0, which is defined as +fα +x0(x) = +� +|s|≤⌊α⌋ +(x − x0)s +s! +Dsf(x0). +For a constant L > 0, let Hα(L, Rp) be the set of functions f : Rp → R such that f is ⌊α⌋-times +differentiable for all x ∈ Rp and satisfy |f(x)−fα +x0(x)| ≤ L|x−x0|α, for all x, x0 ∈ Rp. Here |x−x0| +2 + +is the Euclidean distance between x, x0 ∈ Rp. In nonparametric statistics literature, Hα(L, Rp) is +often referred to as the class of H¨older functions. We refer readers to Rigollet and Vert (2009) for +detailed discussions on H¨older functions. +A process {Xt}t∈Z is said to be α-mixing if +αk = sup +t∈Z +α(σ(Xs, s ≤ t), σ(Xs, s ≥ t + k)) −→k→∞ 0, +where α(A, B) = supA∈A,B∈B |P(A ∩ B) − P(A)P(B)|. for any two σ-fields A and B. +For two +positive sequences {an}n∈N+ and {bn}n∈N+, we write an = O(bn) or an ≲ bn, if an ≤ Cbn with +some constant C > 0 that does not depend on n, and an = Θ(bn) or an ≍ bn, if an = O(bn) +and bn = O(an). For a deterministic or random R-valued sequence an, write that a sequence of +random variable Xn = Op(an), if limM→∞ lim supn→∞ P(|Xn| ≥ Man) = 0. Write Xn = op(an) if +lim supn→∞ P(|Xn| ≥ Man) = 0 for all M > 0. The convergences in distribution and probability +are respectively denoted by D→ and P. +→. +2 +Model setup +Detailed assumptions imposed on the model (2) are collected in Assumption 1. +Assumption 1. The data {Xt}T +t=1 ⊂ Rp is generated based on model (2), satisfying (1), and +a. For t = 1, . . . , T, the distribution Pt has a Lebesgue density function ft : Rp → R. With r, L > 0, +we assume that, ft ∈ Hr(L, X), where X is the union of the supports of all the density functions +ft, with bounded Lebesgue measure. +b. Let gt be the joint density between X1 and Xt+1. It satisfies that +||gt||L∞ < ∞. +(3) +c. The minimal spacing between two consecutive change points ∆ = minK+1 +k=1 (ηk − ηk−1) satisfies +that ∆ = Θ(T). +d. For k ∈ {1, ..., K}, let +κk = ||fηk − fηk+1||L2 +(4) +be the jump size at the kth change point and let +κ = +min +k=1,...,K κk > 0. +The minimal spacing ∆ and the minimal jump size κ are two key parameters characterizing +the change point phenomenon. Assumption 1c. requires that ∆ = Θ(T), which is necessary only +for our inference results in Theorem 2 and Theorem 3. Indeed, this condition may appear strong +compared to the existing literature on localization, such as (Padilla et al., 2019; Wang et al., 2020; +Padilla et al., 2021). For our localization results, we can easily relax this condition to ∆ ≪ T, as +stated in Padilla et al. (2022). To achieve this, consider increasing K in Definition 2 to broaden +the coverage of the seeded intervals in MNSBS, and apply the narrowest over-threshold selection +method, as described in Theorem 3 of (Kov´acs et al., 2020). +Assumption 1d. characterizes the changes in density functions through the function’s L2-norm. +A reason to use the L2-norm is that the L2 space has an inner product structure. +3 + +Revolving the change point estimators, we are to conduct the estimation and inference tasks. +For a sequence of estimators �η1 < . . . < �η � +K ⊂ {1, . . . , T}, we are to show their consistency, i.e. with +probability tending to one as the sample size T grows unbounded, it holds that +�K = K and +max +k=1,..., � +K +|�ηk − ηk| ≤ ϵ, with lim +T→∞ +ϵ +∆ = 0. +(5) +We refer to ϵ as the localization error in the rest of this paper. +With a consistent estimation result, we further refine {�ηk} � +K +k=1 and obtain {�ηk} � +K +k=1 satisfying +that |�ηk − ηk| = Op(1). We are to derive the limiting distribution of +(�ηk − ηk)κ +p +r +2 +k +, +T → ∞. +2.1 +Summary of the results +The contributions of this paper are as follows. +• We develop a multivariate nonparametric seeded change point detection algorithm detailed +Algorithm 1, which is based on the seeded binary segmentation method (SBS), proposed +in Kov´acs et al. (2020), for the univariate Gaussian change in mean setup. As suggested +in Kov´acs et al. (2020), SBS may be adaptable to a wide range of change point detection +problems, such as that found in Padilla et al. (2022) for Functional data. We have innovatively +adapted SBS to the multivariate nonparametric setup. +• Under the model assumptions outlined in Assumption 1 and the signal-to-noise ratio condition +in Assumption 3 that κ2∆ ≳ log(T)T +p +2r+p , we demonstrate that the output of Algorithm 1 is +consistent, with a localization error of κ−2 +k T +p +2r+p log(T), for k ∈ {1, . . . , K}. We note that this +localization error was obtained under the temporal dependence stated in (1) and with a more +general smoothness assumption outlined in Assumption 1a., which is a novel contribution to +the literature. +• Based on the consistent estimators {�η} � +K +k=1, we construct refined estimators {�ηk} � +K +k=1 and derive +their limiting distributions in different regimes, as detailed in Theorem 2. This result is novel +in the literature of nonparametric temporal dependence models, and such two-regime limiting +distributions are rarely seen in the literature, with the exception of mean change under fixed- +dimensional time series (e.g. Yao, 1987; Yao and Au, 1989; Bai, 1994), high-dimensional vector +time series (e.g. Kaul and Michailidis, 2021), functional time series setting (e.g. Aue et al., +2009a), and high-dimensional linear regression (e.g. Xu et al., 2022b). +• Extensive numerical results are presented in Section 5 to corroborate the theoretical findings. +The code used for numerical experiments is available upon request prior to publication. If +the paper is accepted, we will include the code and instructions on how to reproduce the +numerical results in Section 5. +3 +Multivariate nonparametric seeded change point estimators and +their refinement +In this section, we present the initial and refined change point estimators, both of which share the +same building block, namely CUSUM statistics, defined in Definition 1. +4 + +Definition 1 (CUSUM statistics). For any integer triplet 0 ≤ s < t < e ≤ T, let the CUSUM +statistic be +�F (s,e] +t,h (x) = +� +e − t +(e − s)(t − s) +t +� +i=s+1 +Fi,h(x) − +� +t − s +(e − s)(e − t) +e +� +i=t+1 +Fi,h(x), x ∈ Rp +where Ft,h(·) is a kernel estimator, Ft,h(x) = Kh(x − Xt), x ∈ Rp with the kernel function +Kh(x) = 1 +hp K +�x +h +� +, +x ∈ Rp, +accompanied with the bandwidth h > 0. +The CUSUM statistic is a key ingredient of our algorithm and is based on the kernel estimators +Ft,h(·). +We would like to highlight that kernel-based change-point estimation techniques have +been employed in the detection of change-points in nonparametric models in existing literature, as +demonstrated in (e.g. Arlot et al., 2019; Li et al., 2019; Padilla et al., 2021). +Our preliminary estimator is based on SBS. Such an estimator is obtained by combining the +CUSUM statistic in Definition 1 with a modified version of SBS, which is based on a collection of +deterministic intervals defined in Definition 2. +Definition 2 (Seeded intervals). Let K = ⌈CK log(T)⌉, with some sufficiently large absolute constant +CK > 0. For k ∈ {1, . . . , K}, let Jk be the collection of 2k − 1 intervals of length lk = T2−k+1 that +are evenly shifted by lk/2 = T2−k, i.e. +Jk ={(⌊(i − 1)T2−k⌋, ⌈(i − 1)T2−k + T2−k+1⌉], +i = 1, . . . , 2k − 1}. +The overall collection of seeded intervals is denoted as J = ∪K +k=1Jk. +With the CUSUM statistics and the seeded intervals as building blocks, we are now ready to +present our multivariate nonparametric seeded change point detection algorithm. +Algorithm 1 is proposed as a preliminary estimator for multiple change points in sequentially +observed multivariate time series data. It takes advantage of seeded intervals to provide a multi- +scale search system and recursively uses CUSUM statistics to identify potential change points. +Inputs required are observed data {Xt}T +t=1, seeded intervals J , bandwidth h for constructing the +CUSUM statistics, and threshold τ for detecting change points. Theoretical and numerical guidance +for tuning parameters is presented in Sections 4 and 5. +Denote by {�ηk} � +K +k=1 our preliminary estimators provided by Algorithm 1. It has been demon- +strated in various studies, such as (Rinaldo et al., 2021; Xu et al., 2022b; Yu et al., 2022), that +a refinement procedure can likely reduce the localization error of preliminary estimates of change +points. Thus, a refinement step is proposed. First, let +sk = 9 +10 �ηk−1 + 1 +10 �ηk and ek = 9 +10 �ηk+1 + 1 +10 �ηk. +(6) +Then, {�ηk} � +K +k=1 and �h ≍ h produce an estimator of κk as: +�κk = +��� +��� +� +�ηk+1−�ηk +(�ηk+1−�ηk−1)(�ηk−�ηk−1) +��ηk +i=�ηk−1+1 Fi,�h − +� +(�ηk−�ηk−1) +(�ηk+1−�ηk−1)(�ηk+1−�ηk) +��ηk+1 +i=�ηk+1 Fi,�h +� +(�ηk−�ηk−1)(�ηk+1−�ηk) +�ηk+1−�ηk−1 +��� +��� +L2. +(7) +5 + +Algorithm 1 Multivariate Nonparametric Seeded Binary Segmentation. MNSBS ((s, e), J , τ, h) +INPUT: Sample {Xt}e +t=s ⊂ Rp, collection of seeded intervals J , tuning parameter τ > 0 and +bandwidth h > 0. +initialization: If (s, e] = (0, n], set S → ∅ and set ρ → log(T)h−p. +for I = (α, β] ∈ J do +if I = (α, β] ⊆ (s, e] and β − α > 2ρ then +bI ← arg maxα+ρ≤t≤β−ρ || �F (α,β] +t,h +||L2 +aI ← || �F (α,β] +bI,h ||L2 +else +aI ← −1 +end if +end for +I∗ ← arg maxI∈J aI +if aI∗ > τ then +S ← S ∪ {br∗} +MNSBS((s, bI∗), J , τ, h) +MNSBS((bI∗ + 1, e), J , τ, h) +end if +OUTPUT: The set of estimated change points S. +We then propose the final change points estimators, +�ηk = arg min +sk<η 0. +b. The class of functions FK = {K(x − ·)/h : Rp → R+, h > 0} is separable in L∞(Rd) and +is a uniformly bounded VC-class; i.e. there exist constants A, ν > 0 such that for any probability +measure Q on Rp and any u ∈ (0, ∥K∥L∞), it holds that N(FK, L2(Q), u) ≤ (A∥K∥L∞/u)v, where +N(FK, L2(Q), u) denotes the u-covering number of the metric space (FK, L2(Q)). +c. For fixed m > 0, it holds that +� ∞ +0 +tm−1 sup +||x||≥t +|K(x)|m dt < ∞, +� +Rd K(z)||z|| dz ≤ CK, +where CK > 0 is an absolute constant. +Assumption 2 is a standard result in the nonparametric literature (e.g. Gin´e and Guillou, 1999, +2001; Sriperumbudur and Steinwart, 2012; Kim et al., 2019; Padilla et al., 2021), holding for various +kernels, such as uniform, Epanechnikov and Gaussian. +4.1 +Consistency of preliminary estimators +The consistency of the preliminary estimators outputted by Algorithm 1 holds provided the signal +strength is large enough. This is detailed in the following assumption. +Assumption 3 (Signal-to-noise ratio). Assume there exists an arbitrarily slow diverging sequence +γT > 0 such that +κ2∆ > γT log(T)T +p +2r+p . +7 + +Signal-to-noise ratio condition of such a form is ubiquitous in the change point detection +literature, but dealing with H¨older smooth function, nonparametric, and short range temporal +dependence is novel. Combined with Assumption 1c., the SNR in Assumption 3 is reduced to +κ2 ≳ γT log(T)T − +2r +2r+p . +We are now ready to present the main theorem concerning Algorithm 1, showing the consistency +of the proposed MNSBS. +Theorem 1. Let the data be {Xt}T +t=1 ⊂ Rp satisfying Assumption 1. +Let {�ηk} � +K +k=1 be the es- +timated change points by MNSBS detailed in Algorithm 1, with inputs {Xt}, tuning parameter +τ = cτT +p +4r+2p log +1 +2 (T) and bandwidth h = chT +−1 +2r+p , with ch, cτ > 0 being absolute constants. Under +Assumption 2 and Assumption 3, it holds that, +P +� +�K = K, +����ηk − ηk +��� ≤ Cϵκ−2 +k T +p +2r+p log(T), ∀k = 1, . . . , K +� +≥ 1 − 3Cp,KerT −1, +where, Cϵ > 0, and Cp,Ker > 0 depending on the kernel and the dimension p, are absolute constant. +4.2 +Limiting distributions based on refined estimators +As for the refined estimators {�ηk} � +K +k=1 defined in (8), to derive limiting distributions thereof, we +require the stronger signal-to-noise ratio condition below. +Assumption 4 (Stronger signal-to-noise ratio). Assume that there exists an arbitrarily slow di- +verging sequence γT > 0 such +κ +3p +r +3∆ > γT log(T)T +p +2r+p . +Assumption 4 is strictly stronger than Assumption 3, since deriving limiting distributions is +usually a more challenging task than providing a high-probability estimation error upper bound, +see for example Xu et al. (2022b). +Next, we state our main result of this subsection concerning the estimator (8). +Theorem 2. Given data {Xt}T +t=1, suppose that Assumption 1, Assumption 2, and Assumption 3 +hold. Let {�ηk} � +K +k=1 be the change point estimators defined in (8), with +• the intervals {(sk, ek)} � +K +k=1 defined in (6); +• the preliminary estimators {�ηk} � +K +k=1 from MNSBS +� +J , τ, h +� +detailed in Algorithm 1; +• the MNSBS tuning parameters τ = cτT +p +4r+2p log +1 +2 (T) and h = chT +−1 +2r+p ; +• and �κk as in (7). +a. (Non-vanishing regime) For k ∈ {1, . . . , K}, if κk → ϱk, as T → ∞, with ϱk > 0 being an +absolute constant, then the following results hold. +a.1. The estimation error satisfies that +����ηk − ηk +��� = Op(1), as T → ∞. +a.2. When T → ∞, +(�ηk − ηk)κ +p +r +2 +k +D +−→ arg min +�r∈Z +Pk(�r), +(9) +8 + +where +P(�r) = +� +� +� +� +� +� +� +� +� +�0 +t=�r+1 2 +� +Ft,h2 − ft ∗ Kh2, (fηk − fηk+1) ∗ Kh2 +� +L2 + �r||(fηk+1 − fηk) ∗ Kh2||2 +L2 +if �r < 0 +0 +if �r = 0 +��r +t=1 2 +� +Ft,h2 − ft ∗ Kh2, (fηk+1 − fηk) ∗ Kh2 +� +L2 + �r||(fηk+1 − fηk) ∗ Kh2||2 +L2 +if �r > 0 +(10) +with ∗ denoting convolution and h2 = cκkκ +1 +r +k . +b. (Vanishing regime) For k ∈ {1, . . . , K}, if κk → 0, as n → ∞, then the following results hold. +b.1. The estimation error satisfies that +����ηk − ηk +��� = Op(κ +−2− p +r +k +), as T → ∞. +b.2. When T → ∞, +(�ηk − ηk)κ +p +r +2 +k +D +−→ arg min +�r∈Z +�σ∞(k)B(�r) + |�r|, +(11) +with ∗ denoting convolution and h2 = cκkκ +1 +r +k . Here +B(�r) = +� +� +� +� +� +B1(−�r) +if �r < 0 +0 +if �r = 0 +B2(�r) +if �r > 0 +(12) +and +�σ2 +∞(k) = lim +T→∞ +κ +p +r −2 +k +T +V ar +� +T +� +t=1 +� +Ft,h2 − ft ∗ Kh2, (fηk − fηk+1) ∗ Kh2 +� +L2 +� +(13) +with B1(r) and B2(r) being two independent standard Brownian motions. +Theorem 2 considers two regimes of the jump sizes: vanishing and non-vanishing. Notably, the +upper bounds in these regimes on the localization error can be written as +max +1≤k≤K |�ηk − ηk|κ +p +r +2 +k += Op(1). +Therefore, for r = 1 our final estimator {�ηk} attains a minimax optimal rate of convergence, see +Lemma 3 in Padilla et al. (2021). Furthermore, in the setting r = 1 and ∆ = Θ(T), our resulting +rate is shaper than that in Theorem 1 in Padilla et al. (2021), as we are able to remove the +logarithmic factors from the upper bound. Additionally, our method can achieve optimal rates +with choices of tuning parameters that do not depend on κ. +Comparing Theorem 2 with Theorem 1, we observe an improvement in the localization error, +as Theorem 1 showed maxk=1,..., � +K κ2 +k +����ηk − ηk +��� ≤ CϵT +p +2r+p log(T). +Finally, we highlight that Theorem 2 summarizes our derivations of the limiting distributions +associated with the {�ηk} � +K +k=1 estimators. In the non-vanishing case, the resulting limiting distribu- +tion can be approximated by a two-sided random walk distribution. In contrast, in the vanishing +case, the mixing central limit theorem leads to the two-sided Brownian motion distribution in the +limit. +9 + +The limiting distributions in Theorem 2 quantify the asymptotic uncertainty of {�ηk} � +K +k=1, en- +abling inference on change point locations, such as constructing confidence intervals. This is espe- +cially interesting in the vanishing regime, where the estimating error |�ηk − ηk| diverges, as shown +in Theorem 2b.1.. Thus, in the vanishing regime, the limiting distribution can be used to quantify +the uncertainty of our change point estimator. Our result Theorem 2a. also shows that, in the +non-vanishing regime, change points can be accurately estimated within a constant error rate. +4.2.1 +Consistent long-run variance estimation +Next, we discuss aspects of practically performing inference on change point locations using {�ηk} � +K +k=1. +It is crucial to access consistent estimators of the long-run variances {�σ∞(k)} for the limiting distri- +bution in Theorem 2b.2. In this subsection, we propose a block-type long-run variance estimator +and derive its consistency (Algorithm 2). +Theorem 3. Under Assumption 1, Assumption 2, Assumption 3 and with all the notation in +Theorem 2, let +� +�σ2 +∞(k) +� � +K +k=1 be as in (13) and R = O(T +p+r +2r+p /κ +p +2r + 3 +2 +k +). We have that +K +max +k=1 +����σ2 +∞(k) − �σ2 +∞(k) +��� +P +−→ 0, +T → ∞ +with �σ2 +∞(k) the output of Algorithm 2. +Algorithm 2 Long-run variance estimators +INPUT: {Xt}T +t=1, {�ηk} � +K +k=1, {�κk} � +K +k=1, {(sk, ek)} � +K +k=1 and tuning parameter R ∈ N +for k = 1, . . . , �K do +Let h1 = c�κ�κ +1 +r +k +for t ∈ {sk, . . . , ek − 1} do +Yt = �κ +p +2r −1 +k +� +Ft,h1 − ft ∗ Kh1, (f�ηk − f�ηk+1) ∗ Kh1 +� +L2 +end for +S = ⌊ ek−sk +R +⌋ +for r ∈ {1, . . . , R} do +Sr = {sk + (r − 1)S, . . . , sk + rS − 1} +end for +�σ2 +∞(k) = 1 +R +�R +r=1 +� +1 +√ +S +� +i∈Sr Yi +�2 +end for +OUTPUT: {�σ2 +∞(k)} � +K +k=1. +4.3 +Discussions on multivariate nonparametric seeded change point +detection (MNSBS) +Tuning parameters. Our procedure comprises three steps: (1) preliminary estimation, (2) local +refinement, and (3) confidence interval construction, with three key tuning parameters. +For step (1), we specify the density estimator of the sampling distribution to be a kernel estima- +tor with bandwidth h ≍ T −1/(2r+p), which follows from the classical nonparametric literature (e.g. +10 + +Yu, 1993; Tsybakov, 2009). The threshold tuning parameter τ is set to a high-probability upper +bound on the CUSUM statistics when there is no change point, of the form τ = Cτ log1/2(T) +√ +h−p, +which reflects the requirement on the SNR detailed in Assumption 3, κ +√ +∆ ≳ τ. +The bandwidth h1 satisfying h1 ≍ �κ +1 +r +k , inspired by the near minimax rate-optimal bandwidth +choice in Padilla et al. (2021), is chosen for refined estimation in step (2). +The rest of the tuning parameters are the bandwidth �h ≍ h for estimating �κk in both steps (2) +and (3), and �σ2 +∞ and the number of blocks R for long-run variance estimation in (3), which are +specified in Algorithm 2. +Related work. A comparison with Padilla et al. (2021) is presented: the H¨older condition +is milder than their Lipschitz assumption; Assumption 1d specifies changes through L2-norm of +probability density functions, weaker than L∞ distance when X is compact used in Padilla et al. +(2021); our assumptions allow for some dependence captured by α-mixing coefficients, unlike Padilla +et al. (2021) who assume independent observations. +Let us compare Theorem 1 with Theorem 1 in Padilla et al. (2021): recall the consistency +definition stated at (5), and in view of Assumption 3 and Theorem 1, we see that with properly +chosen tuning parameters and probability tending to one as T grows, +K +max +k=1 |�ηk − ηk|/∆ ≲ T +p +2r+p log(T)/(κ2∆) = o(1), +where the equality follows from Assumption 3. This yields the localization consistency guarantee. +Theorem 1 in Padilla et al. (2021) also establishes a consistency. +In terms of the conditions needed, in Assumption 3 we allow r to be arbitrary. To compare +against Padilla et al. (2021) consider r = 1 and recall we impose ∆ = Θ(T). In this case, ignoring +γT , Assumption 3 reduces to, +h ≲ T − +1 +2+p log1/2(T) ≲ κ, +(14) +where we have used our choice of h in Theorem 1. In contrast, in the same setting, the signal-to- +noise ration condition in Padilla et al. (2021) is +κ2+p∆ ≳ γT log1+ϵ(T), +(15) +and with bandwidth being the unknown parameter κ. However, if (14) holds, then +κp+2∆ ≳ κ2κpT ≳ κ2 � +T − +1 +2+p +�p +T = κ2T +2 +2+p +which combined with (14) implies (15). Thus, in Padilla et al. (2021) the signal-to-noise ratio +condition is weaker than Assumption 3, and its localization error is faster. Nevertheless, Theorem 1 +allows for temporal dependence and has practical implications for the choice of the bandwidth. +5 +Numerical Experiments +We refer to MNSBS to the final estimator. +11 + +5.1 +Simulated data analysis +We compare our proposed MNSBS with four competitors – MNP (Padilla et al., 2021), EMNCP +(Matteson and James, 2014), SBS (Cho and Fryzlewicz, 2015) and DCBS (Cho, 2016) – across +a wide range of simulation settings, using corresponding R functions in changepoints (Xu et al., +2022a), ecp (James et al., 2019) and hdbinseg (Cho and Fryzlewicz, 2018) packages. We evaluate L2 +based statistics in Change point estimation and Long-run variance estimation using the Subregion- +Adaptive Vegas Algorithm1 with a maximum of 105 function evaluations. +For MNSBS implementation we use the Gaussian kernel and the false discovery rate control- +based procedure of Padilla et al. (2021) for τ selection. Preliminary estimators are set as h = 2 × +(1/T)1/(2r+p), while the second stage estimator has bandwidths respectively set as �h = 0.05 and h1 = +2×�κ1/r +k . Selection of R = +� � +max � +K +k=1{ek − sk} +�3/5 � +with {(sk, ek)} � +K +k=1 is guided by Theorem 3 using +{(sk, ek)} � +K +k=1 from (6). For the confidence interval construction, we use {�κk} � +K +k=1 and {�σ2 +∞(k)} � +K +k=1 +to estimate the required unknown quantities for the confidence interval construction. +5.1.1 +Localization +We consider four different scenarios with two equally spaced change points. For each scenario, we +set r = 2, and varies T ∈ {150, 300} and p ∈ {3, 5}. Moreover, we consider +{Yt = 1{⌊T/3⌋ < t ≤ ⌊2T/3⌋}Zt + Xt}T +t=1 ⊂ Rp +with with +Xt = 0.3Xt−1 + ϵt. +•Scenario 1 (S1) For any t, Zt = µ ∈ Rp, such that µj = 0 for j ∈ {1, . . . , ⌈p/2⌉} and µj = 2 +otherwise. Moreover, {ϵt} are i.i.d. N(0p, Ip). +•Scenario 2 (S2) For any t, Zt = 0.3Zt−1 + ϵ(1) +t . Moreover, {ϵ(1) +t }, {ϵt} ⊂ Rp are i.i.d. with +entries independently follow Unif(−1, 1) and Unif(− +√ +3, +√ +3), respectively. +•Scenario 3 (S3) Similarly as in S2. +But now, {ϵ(1) +t }, {ϵt} ⊂ Rp are i.i.d. +with entries +independently follow standardized Pareto(3, 1) and Log-Normal(0, 1), respectively. +•Scenario 4 (S4) For any t, Zt|{ut = 1} = 1.5 × 1p, +Zt|{ut = 0} = −1.5 × 1p and Xt = +0.3Xt−1 + ϵt, and {ϵt} ⊂ Rp are i.i.d. N(Op, Ip). +•Scenario 5 (S5) For any t, Zt = 0.3Zt−1 + ϵ(1) +t ++ 0.5 × 1p and {ϵ(1) +t }, {ϵt} ⊂ Rp are i.i.d. with +entries independently follow Unif(− +√ +3, +√ +3) and the standardised Pareto(3, 1), respectively. +S1-S5 encompass a variety of simulation settings including the same type of distributions, +changed mean and constant covariance S1; the same type of distributions, constant mean, changed +covariance S2; different types of distributions, constant mean and covariance S3; mixture of dis- +tributions S4; and change between light-tailed and heavy-tailed distributions S5. +5.1.2 +Inference +We consider the following process +{Yt = 1{⌊T/2⌋ < t ≤ T}µ + Xt}T +t=1, +1The Subregion-Adaptive Vegas Algorithm is available in R package cubature (Narasimhan et al., 2022) +12 + +with +Xt = 0.3Xt−1 + ϵt, +Here, µ = 1p and {ϵt}T +t=1 ⊂ Rp are i.i.d. N(0p, Ip). We vary T ∈ {100, 200, 300} and p ∈ {2, 3}, +and observe that our localization results are robust to the bandwidth parameters, yet sensitive to +the smoothness parameter r. We thus set r = 1000 in our simulations, as the density function of a +multivariate normal distribution belongs to the H¨older function class with r = ∞. +5.1.3 +Evaluation results +For a given set of true change points C = {ηk}K+1 +k=0 , to assess the accuracy of the estimator �C = +{�ηk} +� +K+1 +k=0 with �η0 = 1 and �ηT+1 = T + 1f, we report (1) the proportion of misestimating K and (2) +the scaled Hausdorff distance dH( �C, C), defined by +dH( �C, C) = 1 +T max{max +x∈ �C +min +y∈C {|x − y|}, max +y∈ �C +min +x∈C {|x − y|}}. +The performance of our change point inference is measured by the coverage of ηk, defined as +coverk(1 − α) for significance level α ∈ (0, 1). For, k = 1, . . . , K, +coverk(1 − α) = 1 +� +ηk ∈ +� +�ηk + �qu(α/2) +�κp/r+2 +k +, �ηk + �qu(1 − α/2) +�κp/r+2 +k +�� +, +(16) +with �qu(α/2) and �qu(1− α/2) are the α/2 and 1− α/2 empirical quantiles of the simulated limiting +distribution given in (11), �κk is defined in (7), and k = 1, . . . , K. +We repeat the experiment 200 times for each setting and report simulation results for localisation +in Table 1, 2, 3, 4, and 5. Inference performance is presented in Table 6. To the best of our +knowledge, no competitor exists for change point inference in multivariate nonparametric change +settings. +MNSBS generally performs well in all scenarios considered, among the top two except for S2. +DCBS, designed to estimate change points in mean or second-order structure, performs best in S2, +while MNSBS is comparable to ECP, and significantly better in S1 and S5 for large T. +5.2 +Real data application +We applied our proposed change point inference procedure to analyze stock price data2, which +consisted of daily adjusted close price of the 3 major stock market indices (S&P 500, Dow Jones +and NASDAQ) from Jan-01-2021 to Jan-19-2023. After removing missing values and standardizing +the raw data, the sample size was n = 515 and the dimension p = 3. +We localized 6 estimated change points and performed inference based on them; results are +summarized in Table 7. We also implemented the NMP and ECP methods on the same dataset, +the estimated change points being presented in Appendix A.2. Except for the time point Aug-24- +2022 estimated by ECP, all other estimated change points were located in the constructed 99% +confidence intervals by our proposed method. +2The stock price data are downloaded from https://fred.stlouisfed.org/series. +13 + +6 +Conclusion +We tackle the problem of change point detection for short range dependent multivariable nonpara- +metric data, which has not been studied in the literature. Our two-stage algorithm MNSBS can +consistently estimate the change points in stage one, a novelty in the literature. Then, we derived +limiting distributions of change point estimators for inference in stage two, a first in the literature. +Our theoretical analysis reveals multiple challenging and interesting directions for future explo- +ration. Relaxing the assumption ∆ ≍ T may be of interest. 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We refer to MNSBS to the final estimator. +A.1 +Simulated data +We present the tables containing the results of the simulation study in Section 5 of the main text. +On each table, the mean over 200 repetitions is reported, and the numbers in parenthesis denote the +standard errors. For the purpose of identifying misestimation, we compute the averaged coverage +among all the repetitions whose �K = K. In each setting, we highlight the best result in bold and +the second best result in bold and italic. +Table 1: Localisation results of Scenario 1. +T = 150 +T = 300 +Method +p = 3 +p = 5 +p = 3 +p = 5 +propotion of times �K ̸= K +MNSBS +0.040 +0.025 +0.140 +0.105 +NMP +0.170 +0.325 +0.255 +0.420 +ECP +0.010 +0 +0.570 +0.630 +SBS +0.705 +0.540 +0.115 +0.010 +DCBS +0.210 +0.120 +0.145 +0.115 +average (standard deviation) of dH +MNSBS +0.026 (0.042) +0.012 (0.024) +0.028 (0.045) +0.016 (0.038) +NMP +0.064 (0.060) +0.077 (0.051) +0.045 (0.050) +0.064 (0.059) +ECP +0.017 (0.021) +0.007 (0.009) +0.080 (0.065) +0.087 (0.066) +SBS +0.245 (0.138) +0.190 (0.157) +0.051 (0.096) +0.010 (0.028) +DCBS +0.061 (0.087) +0.024 (0.035) +0.025 (0.036) +0.017 (0.033) +19 + +Table 2: Localisation results of Scenario 2. +T = 150 +T = 300 +Method +p = 3 +p = 5 +p = 3 +p = 5 +propotion of times �K ̸= K +MNSBS +0.815 +0.645 +0.670 +0.590 +NMP +0.800 +0.705 +0.785 +0.515 +ECP +0.515 +0.270 +0.630 +0.665 +SBS +1.000 +0.995 +0.820 +0.625 +DCBS +0.830 +0.490 +0.160 +0.055 +average (standard deviation) of dH +MNSBS +0.265 (0.124) +0.212 (0.154) +0.185 (0.147) +0.161 (0.149) +NMP +0.261 (0.123) +0.235 (0.142) +0.174 (0.151) +0.151 (0.153) +ECP +0.177 (0.132) +0.094 (0.099) +0.102 (0.060) +0.099 (0.060) +SBS +0.333 (0.003) +0.332 (0.019) +0.279 (0.115) +0.222 (0.146) +DCBS +0.280 (0.117) +0.174 (0.156) +0.066 (0.107) +0.020 (0.034) +Table 3: Localisation results of Scenario 3. +T = 150 +T = 300 +Method +p = 3 +p = 5 +p = 3 +p = 5 +propotion of times �K ̸= K +MNSBS +0.840 +0.835 +0.740 +0.755 +NMP +0.835 +0.815 +0.785 +0.750 +ECP +0.750 +0.705 +0.745 +0.800 +SBS +1.000 +1.000 +1.000 +1.000 +DCBS +0.970 +0.975 +0.990 +0.980 +average (standard deviation) of dH +MNSBS +0.274 (0.073) +0.274 (0.078) +0.257 (0.084) +0.251 (0.087) +NMP +0.269 (0.077) +0.275 (0.076) +0.260 (0.082) +0.251 (0.084) +ECP +0.255 (0.099) +0.232 (0.103) +0.247 (0.091) +0.223 (0.094) +SBS +0.333 (0) +0.333 (0.009) +0.333 (0) +0.333 (0) +DCBS +0.330 (0.024) +0.331 (0.018) +0.333 (0.010) +0.333 (0.002) +20 + +Table 4: Localisation results of Scenario 4. +T = 150 +T = 300 +Method +p = 3 +p = 5 +p = 3 +p = 5 +propotion of times �K ̸= K +MNSBS +0.420 +0.055 +0.110 +0.095 +NMP +0.575 +0.405 +0.145 +0.210 +ECP +0.120 +0.050 +0.125 +0.055 +SBS +1.000 +1.000 +0.910 +0.845 +DCBS +0.885 +0.915 +0.150 +0.155 +average (standard deviation) of dH +MNSBS +0.143 (0.153) +0.020 (0.055) +0.019 (0.037) +0.015 (0.037) +NMP +0.202 (0.149) +0.152 (0.141) +0.038 (0.054) +0.048 (0.050) +ECP +0.058 (0.082) +0.024 (0.032) +0.027 (0.045) +0.011 (0.048) +SBS +0.333 (0) +0.333 (0) +0.305 (0.090) +0.285 (0.112) +DCBS +0.295 (0.102) +0.306 (0.089) +0.060 (0.111) +0.059 (0.112) +Table 5: Localisation results of Scenario 5. +T = 150 +T = 300 +Method +p = 3 +p = 5 +p = 3 +p = 5 +propotion of times �K ̸= K +MNSBS +0.110 +0.080 +0.215 +0.230 +NMP +0.155 +0.090 +0.230 +0.215 +ECP +0.100 +0.055 +0.545 +0.655 +SBS +0.995 +0.990 +0.995 +0.960 +DCBS +0.960 +0.965 +0.975 +0.980 +average (standard deviation) of dH +MNSBS +0.057 (0.089) +0.036 (0.054) +0.039 (0.058) +0.035 (0.051) +NMP +0.070 (0.107) +0.041 (0.060) +0.040 (0.057) +0.038 (0.054) +ECP +0.044 (0.051) +0.032 (0.041) +0.083 (0.063) +0.093 (0.061) +SBS +0.332 (0.021) +0.330 (0.031) +0.331 (0.023) +0.321 (0.059) +DCBS +0.316 (0.061) +0.305 (0.075) +0.317 (0.062) +0.313 (0.068) +Table 6: Localisation results of inference. +α = 0.01 +α = 0.05 +n +cover(1 − α) +width(1 − α) +cover(1 − α) +width(1 − α) +p = 2 +100 +0.864 +17.613 (6.712) +0.812 +14.005 (5.639) +200 +0.904 +22.940 (7.740) +0.838 +18.407 (6.541) +300 +0.993 +26.144 (9.027) +0.961 +20.902 (5.936) +p = 3 +100 +0.903 +15.439 (5.792) +0.847 +11.153 (4.361) +200 +0.966 +20.108 (7.009) +0.949 +13.920 (5.293) +300 +0.981 +22.395 (6.904) +0.955 +15.376 (4.763) +21 + +A.2 +Real data example +The transformed real data used on Section 5.2 is illustrated in the figure below. +These data +correspond to the daily adjusted close price, from Jan-01-2021 to Jan-19-2023, of the 3 major stock +market indices, S&P 500, Dow Jones and NASDAQ. Moreover, in Table 7, we present the estimated +change point by our proposed method MNSBS on the data before mentioned, together with their +respective inference. +date +standardized daily close price +0 +100 +200 +300 +400 +500 +−2 +−1 +0 +1 +2 +Figure 1: Plot of the standardized daily close price, from Jan-01-2021 to Jan-19-2023, of the 3 +major stock market indices. +Table 7: Confidence intervals constructed for change point locations in the Real data example. +α = 0.01 +α = 0.05 +�η +Lower bound +Upper bound +Lower bound +Upper bound +April-07-2021 +April-01-2021 +April-12-2021 +April-05-2021 +April-09-2021 +June-30-2021 +June-23-2021 +July-09-2021 +June-25-2021 +July-07-2021 +Oct-19-2021 +Oct-12-2021 +Oct-26-2021 +Oct-14-2021 +Oct-22-2021 +Jan-18-2021 +Jan-12-2021 +Jan-21-2021 +Jan-13-2021 +Jan-20-2021 +April-25-2022 +April-20-2022 +April-28-2022 +April-21-2022 +April-27-2022 +Oct-27-2022 +Oct-24-2022 +Nov-01-2022 +Oct-25-2022 +Oct-31-2022 +The result of the implementation of NMP and ECP methods on the same dataset are {April-01-2021, +July-01-2021, Oct-19-2021, Jan-14-2022, April-21-2022, Oct-26-2022} and {April-08-2021, June-25-2021, +Oct-18-2021, Jan-18-2022, April-28-2022, Aug-24-2022, Oct-27-2022} respectively. +22 + +B +Proof of Theorem 1 +In this section, we present the proof of theorem Theorem 1. +Proof of Theorem 1. For any (s, e] ⊆ (0, T], let +�f(s,e] +t +(x) = +� +e − t +(e − s)(t − s) +t +� +l=s+1 +fl(x) − +� +t − s +(e − s)(e − t) +e +� +l=t+1 +fl(x), x ∈ X. +For any �r ∈ (ρ, T − ρ], we consider +A((s, e], ρ, λ) = +� +e−ρ +max +t=s+ρ+1 sup +x∈Rp | �F s,e +t,h (x) − �fs,e +t +(x)| ≤ λ +� +; +B(�r, ρ, λ) = +� +T−�r +max +N=ρ sup +x∈Rp +���� +1 +√ +N +�r+N +� +t=�r+1 +Ft,h(x) − +1 +√ +N +�r+N +� +t=�r+1 +ft(x) +���� ≤ λ +� � +� +�r +max +N=ρ sup +x∈Rp +���� +1 +√ +N +�r +� +t=�r−N+1 +Ft,h(x) − +1 +√ +N +�r +� +t=�r−N+1 +ft(x) +���� ≤ λ +� +. +From Algorithm 1, we have that +ρ = log(T) +hp +. +Therefore, Proposition 2 imply that with +λ = Cλ +� +2C +� +log T +hp ++ 2C1√p +√ +hp ++ 2C2 +√ +Thr. +� +, +(17) +for some diverging sequence Cλ, it holds that +P +� +Ac((s, e], ρ, λ) +� +≲ 1 +T 2 , +and, +P +� +Bc(�r, ρ, λ +2 ) +� +≲ 1 +T 2 . +Now, we notice that, +K +� +k=1 +�nk = +K +� +k=1 +(2k − 1) ≤ +K +� +k=1 +2k ≤ 2(2⌈CK(log(T))⌉ − 1) = O(T). +In addition, there are K = O(1) number of change points. In consequence, it follows that +P +� +A(I, ρ, λ) for all I ∈ J +� +≥ 1 − 1 +T , +(18) +P +� +B(s, ρ, λ) ∪ B(e, ρ, λ) for all (s, e] = I ∈ J +� +≥ 1 − 1 +T , +(19) +P +� +B(ηk, ρ, λ) for all 1 ≤ k ≤ K +� +≥ 1 − 1 +T . +(20) +23 + +The rest of the argument is made by assuming the events in equations (18), (19) and (20) hold. By +Remark 1, we have that on these events, it is satisfied that +e−ρ +max +t=s+ρ+1 || �F s,e +t,h (x) − �fs,e +t +(x)||L2 ≤ λ. +Denote +Υk = C log(T) +� +T +p +2r+p +� +κ−2 +k +and +Υmax = C log(T) +� +T +p +2r+p +� +κ−2, +where κ = min{κ1, . . . , κK}. Since Υk is the desired localisation rate, by induction, it suffices to +consider any generic interval (s, e] ⊆ (0, T] that satisfies the following three conditions: +ηm−1 ≤ s ≤ ηm ≤ . . . ≤ ηm+q ≤ e ≤ ηm+q+1, +q ≥ −1; +either ηm − s ≤ Υm +or +s − ηm−1 ≤ Υm−1; +either ηm+q+1 − e ≤ Υm+q+1 +or +e − ηm+q ≤ Υm+q. +Here q = −1 indicates that there is no change point contained in (s, e]. +Denote +∆k = ηk−1 − ηk for k = 1, . . . , K + 1 +and +∆ = min{∆1, . . . , ∆K+1}. +Observe that since κk > 0 for all 1 ≤ k ≤ K and that ∆k = Θ(T), it holds that Υmax = o(∆). +Therefore, it has to be the case that for any true change point ηm ∈ (0, T], either |ηm − s| ≤ Υm or +|ηm − s| ≥ ∆ − Υmax ≥ Θ(T). This means that min{|ηm − e|, |ηm − s|} ≤ Υm indicates that ηm is +a detected change point in the previous induction step, even if ηm ∈ (s, e]. We refer to ηm ∈ (s, e] +as an undetected change point if min{ηm − s, ηm − e} = Θ(T). To complete the induction step, it +suffices to show that MNSBS((s, e], h, τ) +(i) will not detect any new change point in (s, e] if all the change points in that interval have been +previously detected, and +(ii) will find a point DI∗ +m∗ in (s, e] such that |ηm −DI∗ +m∗| ≤ Υm if there exists at least one undetected +change point in (s, e]. +In order to accomplish this, we need the following series of steps. +Step 1. We first observe that if ηk ∈ {ηk}K +k=1 is any change point in the functional time series, +by Lemma 5, there exists a seeded interval Ik = (sk, ek] containing exactly one change point ηk +such that +min{ηk − sk, ek − ηk} ≥ 1 +16ζk, +and +max{ηk − sk, ek − ηk} ≤ ζk +where, +ζk = 9 +10 min{ηk+1 − ηk, ηk − ηk−1}. +Even more, we notice that if ηk ∈ (s, e] is any undetected change point in (s, e]. Then it must hold +that +s − ηk−1 ≤ Υmax. +Since Υmax = O(log(T)T +p +2r+p ) and O(loga(T)) = o(T b) for any positive numbers a and b, we have +that Υmax = o(T). Moreover, ηk − sk ≤ ζk ≤ 9 +10(ηk − ηk−1), so that it holds that +sk − ηk−1 ≥ 1 +10(ηk − ηk−1) > Υmax ≥ s − ηk−1 +24 + +and in consequence sk ≥ s. Similarly ek ≤ e. Therefore +Ik = (sk, ek] ⊆ (s, e]. +Step 2. Consider the collection of intervals {Ik = (sk, ek]}K +k=1 in Step 1. In this step, it is shown +that for each k ∈ {1, . . . , K}, it holds that +t=ek−ρ +max +t=sk+ρ || �F (sk,ek] +t,h +||L2 ≥ c1 +√ +Tκk, +(21) +for some sufficient small constant c1. +Let k ∈ {1, . . . , K}. +By Step 1, Ik contains exactly one change point ηk. +Since ft is a one- +dimensional population time series and there is only one change point in Ik = (sk, ek], it holds +that +fsk+1 = ... = fηk ̸= fηk+1 = ... = fek +which implies, for sk < t < ηk +�f(sk,ek] +t += +� +ek − t +(ek − sk)(t − sk) +t +� +l=sk+1 +fηk − +� +t − sk +(ek − sk)(ek − t) +ηk +� +l=t+1 +fηk +− +� +t − sk +(ek − sk)(ek − t) +ek +� +l=ηk+1 +fηk+1 +=(t − sk) +� +ek − t +(ek − sk)(t − sk)fηk − (ηk − t) +� +t − sk +(ek − sk)(ek − t)fηk +−(ek − ηk) +� +t − sk +(ek − sk)(ek − t)fηk+1 += +� +(t − sk)(ek − t) +(ek − sk) +fηk − (ηk − t) +� +t − sk +(ek − sk)(ek − t)fηk +−(ek − ηk) +� +t − sk +(ek − sk)(ek − t)fηk+1 +=(ek − t) +� +t − sk +(ek − t)(ek − sk)fηk − (ηk − t) +� +t − sk +(ek − sk)(ek − t)fηk +−(ek − ηk) +� +t − sk +(ek − sk)(ek − t)fηk+1 +=(ek − ηk) +� +t − sk +(ek − t)(ek − sk)fηk − (ek − ηk) +� +t − sk +(ek − sk)(ek − t)fηk+1 +=(ek − ηk) +� +t − sk +(ek − t)(ek − sk)(fηk − fηk+1). +25 + +Similarly, for ηk ≤ t ≤ ek +f(sk,ek] +t += +� +ek − t +(ek − sk)(t − sk)(ηk − sk)(fηk − fηk+1). +Therefore, +�f(sk,ek] +t += +� +� +� +� +t−sk +(ek−sk)(ek−t)(ek − ηk)(fηk − fηk+1), +sk < t < ηk; +� +ek−t +(ek−sk)(t−sk)(ηk − sk)(fηk − fηk+1), +ηk ≤ t ≤ ek. +(22) +Since ∆ = Θ(T), ρ = O(log(T)T +p +2r+p ) and loga(T) = o(T b) for any positive numbers a and b, we +have that +min{ηk − sk, ek − ηk} ≥ 1 +16ζk ≥ 3 +4c2T > ρ, +(23) +so that ηk ∈ [sk +ρ, ek −ρ]. Then, from (22), (23) and the fact that |ek −sk| < T and |ηk −sk| < T, +|| �f(sk,ek] +ηk +||L2 = +� +ek − ηk +(ek − sk)(ηk − sk)(ηk − sk)||fηk − fηk+1||L2 ≥ c2 +√ +T 3 +4κk. +(24) +Therefore, it holds that +t=ek−ρ +max +t=sk+ρ || �F (sk,ek] +t,h +||L2 ≥|| �F (sk,ek] +ηk,h +||L2 +≥|| �f(sk,ek] +ηk +||L2 − λ +≥c2 +3 +4 +√ +Tκk − λ, +where the first inequality follows from the fact that ηk ∈ [sk + ρ, ek − ρ], the second inequality +follows from the good event in (18) and Remark 2, and the last inequality follows from (24). +Next, we observe that log +1 +2 (T) +� +1 +hp = o( +� +T +2r+p +p )O( +� +T +p +2r+p ) = o( +√ +T), ρ < c2T, and hr = o(1). +In consequence, since κk is a positive constant, by the upper bound of λ on Equation (17), for +sufficiently large T, it holds that +c2 +4 +√ +Tκk ≥ λ. +Therefore, +t=ek−ρ +max +t=sk+ρ || �F (sk,ek] +t,h +||L2 ≥ c2 +2 +√ +Tκk. +Therefore Equation (21) holds with c1 = c2 +2 . +Step 3. In this step, it is shown that SBS((s, e], h, τ) can consistently detect or reject the ex- +istence of undetected change points within (s, e]. +Suppose ηk ∈ (s, e] is any undetected change point. Then by the second half of Step 1, Ik ⊆ (s, e]. +Therefore +AI∗ ≥ +t=ek−ρ +max +t=sk+ρ || �F (sk,ek] +t,h +||L2 ≥ c1 +√ +Tκk > τ, +26 + +where the second inequality follows from Equation (21), and the last inequality follows from the +fact that, loga(T) = o(T b) for any positive numbers a and b implies τ = Cτ +� +log(T) +� +1 +hp +� += o( +√ +T). +Suppose there does not exist any undetected change point in (s, e]. Then for any I = (α, β] ⊆ (s, e], +one of the following situations must hold, +(a) There is no change point within (α, β]; +(b) there exists only one change point ηk within (α, β] and min{ηk − α, β − ηk} ≤ Υk; +(c) there exist two change points ηk, ηk+1 within (α, β] and +ηk − α ≤ Υk +and +β − ηk+1 ≤ Υk+1. +Observe that if (a) holds, then we have +max +α+ρ cT, +(29) +and +|bI∗ − ηk| ≤ max{C3λ2κ−2 +k , ρ} ≤C4 log(T) +� 1 +hp + Th2r +� +κ−2 +k +≤C log(T) +� +T +p +2r+p +� +κ−2 +k +28 + +for sufficiently large constant C, where we have followed the same line of arguments as for the +conclusion of (28). Observe that +i) The change points of {ft}t∈I∗ belong to (s, e] ∩ {ηk}K +k=1; and +ii) Equation (29) and (α∗, β∗] ⊆ (s, e] imply that +min{e − ηk, ηk − s} > cT ≥ Υmax. +As discussed in the argument before Step 1, this implies that ηk must be an undetected change +point of {ft}t∈I∗. +29 + +C +Proof of Theorem 2 +In this section, we present the proof of theorem Theorem 2. +Proof of Theorem 2. Uniform tightness of κ +2+ p +r +k +����ηk − ηk +���. Here we show a.1 and b.1. For this +purpose, we will follow a series of steps. On step 1, we rewrite (8) in order to derive a uniform +bound. Step 2 analyses the lower bound while Step 3 the upper bound. +Step 1: Denote �r = �ηk − ηk. Without loss of generality, suppose �r ≥ 0. Since �ηk = ηk + �r, defined +in (8), is the minimizer of �Qk(η), it follows that +�Qk(ηk + �r) − �Qk(ηk) ≤ 0. +Let +Q∗(η) = +η +� +t=sk+1 +||Ft,h2 − f(sk,ηk] ∗ Kh2||2 +L2 + +ek +� +t=η+1 +||Ft,h2 − f(ηk,ek] ∗ Kh2||2 +L2, +(30) +where, +f(sk,ηk] = +1 +ηk − sk +ηk +� +i=sk+1 +fi, f(ηk,ek] = +1 +ek − ηk +ek +� +i=ηk+1 +fi. +(31) +Observe that, +Q∗(ηk + �r) − Q∗(ηk) ≤ �Qk(ηk) − �Qk(ηk + �r) − Q∗(ηk) + Q∗(ηk + �r). +(32) +If �r ≤ 1/κ +2+ p +r +k +, then there is nothing to show. So for the rest of the argument, for contradiction, +assume that +�r ≥ +1 +κ +2+ p +r +k +. +(33) +Step 2: Finding a lower bound. In this step, we will find a lower bound of the inequality (32). +To this end, we observe that, +Q∗(ηk + �r) − Q∗(ηk) = +ηk+�r +� +t=ηk+1 +||Ft,h2 − f(sk,ηk] ∗ Kh2||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,h2 − f(ηk,ek] ∗ Kh2||2 +L2 += +ηk+�r +� +t=ηk+1 +||f(sk,ηk] ∗ Kh2 − f(ηk,ek] ∗ Kh2||2 +L2 +− 2 +ηk+�r +� +t=ηk+1 +⟨f(sk,ηk] ∗ Kh2 − f(ηk,ek] ∗ Kh2, Ft,h2 − f(ηk,ek] ∗ Kh2⟩L2 += +ηk+�r +� +t=ηk+1 +1 +2||f(sk,ηk] − f(ηk,ek]||2 +L2 − 2||f(sk,ηk] ∗ Kh2 − f(sk,ηk] + f(ηk,ek] ∗ Kh2 − f(ηk,ek]||2 +L2 +30 + +− 2 +ηk+�r +� +t=ηk+1 +⟨f(sk,ηk] ∗ Kh2 − f(ηk,ek] ∗ Kh2, Ft,h2 − f(ηk,ek] ∗ Kh2⟩L2 +≥1 +2�rκ2 +k − 2 +ηk+�r +� +t=ηk+1 +||f(sk,ηk] ∗ Kh2 − f(sk,ηk] + f(ηk,ek] ∗ Kh2 − f(ηk,ek]||2 +L2 +− 2 +ηk+�r +� +t=ηk+1 +⟨f(sk,ηk] ∗ Kh2 − f(ηk,ek] ∗ Kh2, Ft,h2 − f(ηk,ek] ∗ Kh2⟩L2 +We consider, +I1 := 2 +ηk+�r +� +t=ηk+1 +||f(sk,ηk] ∗ Kh2 − f(sk,ηk] + f(ηk,ek] ∗ Kh2 − f(ηk,ek]||2 +L2, and, +I2 := 2 +ηk+�r +� +t=ηk+1 +⟨f(sk,ηk] ∗ Kh2 − f(ηk,ek] ∗ Kh2, Ft,h2 − f(ηk,ek] ∗ Kh2⟩L2. +From above, we have that, +Q∗(ηk + �r) − Q∗(ηk) ≥1 +2�rκ2 +k − I1 − I2. +We now analyze the order of magnitude of term I1. Then, we get a lower bound for the term −I1. +In fact I1, has an upper bound of the form op(�rk +p +r +2), where we use that ||fηk ∗Kh2 −fηk||L2 = o(1) +and ||fηk+1 ∗ Kh2 − fηk+1||L2 = o(1). For the term I2, we consider the random variable, +Yi = ⟨f[sk+1,ηk] ∗ Kh2 − f[ηk+1,ek] ∗ Kh2, Ft,h2 − f[ηk+1,ek] ∗ Kh2⟩L2 +κkE(||Ft,h2 − fηk+1 ∗ Kh2||3 +L2)1/3 +. +In order to use Lemma 3, we need to bound E(|Yi|3). For this, first we use Cauchy Schwartz +inequality, +E(|Yi|3) ≤(||(fηk+1 − fηk) ∗ Kh2||L2)3E(||Ft,h2 − fηk+1 ∗ Kh2||3 +L2) +κ3 +kE(||Ft,h2 − fηk+1 ∗ Kh2||3 +L2) +then, by Minkowski’s inequality, +||(fηk+1 − fηk) ∗ Kh2||L2 = +��� +��� +� +Rp(fηk+1 − fηk)(· − y)Kh2(y)dy +��� +��� +L2 +≤ +� +Rp +��� +���(fηk+1 − fηk)(· − y)Kh2(y) +��� +��� +L2dy += +� � +Rp |Kh2(y)|dy +���� +���(fηk+1 − fηk)(· − y) +��� +��� +L2 +=||fηk+1 − fηk||L2||Kh2||L1. +31 + +Therefore, by Assumption 2, we have +(||(fηk+1 − fηk) ∗ Kh2||L2)3E(||Ft,h2 − fηk+1 ∗ Kh2||3 +L2) +κ3 +kE(||Ft,h2 − fηk+1 ∗ Kh2||3 +L2) +≤(||fηk+1 − fηk||L2||Kh2||L1)3E(||Ft,h2 − fηk+1 ∗ Kh2||3 +L2) +κ3 +kE(||Ft,h2 − fηk+1 ∗ Kh2||3 +L2) +≤CK. +for any t ∈ (ηk, ek]. Moreover, we have that +E(||Ft,h2 − fηk+1 ∗ Kh2||3 +L2) +1 +3 += +� � � � +(Kh2(x − z) − E(Kh2(x − Xt)))2dx +� 3 +2 ft(z)dz +�1/3 +≤ +� � � � +(Kh2(x − z))2dx +� 3 +2 ft(z)dz +� 1 +3 += +1 +κp/2r . +(34) +Therefore, by Lemma 3, we have that I2 = op +�√ +�rκkκ +− p +2r +k +(log(�rκ +p +r +2 +k +) + 1) +� +. Thus, +Q∗(ηk + �r) − Q∗(ηk) ≥1 +2�rκ2 +k − Op +�√ +�rκkκ +− p +2r +k +(log(�rκ +p +r +2 +k +) + 1) +� +− op(�rκ +p +r +2 +k +). +(35) +Step 3: Finding an upper bound. Now, we proceeded to get an upper bound of (32). This is, +an upper bound of the following expression, +�Qk(ηk) − �Qk(ηk + �r) − Q∗(ηk) + Q∗(ηk + �r). +(36) +Observe that, this expression can be written as, +�Qk(ηk) − �Qk(ηk + �r) − Q∗(ηk) + Q∗(ηk + �r) += − +ηk+�r +� +t=ηk+1 +||Ft,h1 − F(sk,ηk],h1||2 +L2 + +ηk+�r +� +t=ηk+1 +||Ft,h1 − F(ηk,ek],h1||2 +L2 ++ +ηk+�r +� +t=ηk+1 +||Ft,h2 − f(sk,ηk] ∗ Kh2||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,h2 − f(ηk,ek] ∗ Kh2||2 +L2 +So that, +�Qk(ηk) − �Qk(ηk + �r) − Q∗(ηk) + Q∗(ηk + �r) = U1 + U2, +where, +U1 = +ηk+�r +� +t=ηk+1 +||Ft,h2 − f(sk,ηk] ∗ Kh2||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,h1 − F(sk,ηk],h1||2 +L2, and, +U2 = +ηk+�r +� +t=ηk+1 +||Ft,h1 − F(ηk,ek],h1||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,h2 − f(ηk,ek] ∗ Kh2||2 +L2. +32 + +Now, we analyze each of the terms above. For U1, observe that +ηk+�r +� +t=ηk+1 +||Ft,h2 − f(sk,ηk] ∗ Kh2||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,h1 − F(sk,ηk],h1||2 +L2 += +ηk+�r +� +t=ηk+1 +||Ft,h2 − f(sk,ηk] ∗ Kh2||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,h2 − F(sk,ηk],h2||2 +L2 ++ +ηk+�r +� +t=ηk+1 +||Ft,h2 − F(sk,ηk],h2||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,h1 − F(sk,ηk],h1||2 +L2 +=I3 + I4, +where, +I3 = +ηk+�r +� +t=ηk+1 +||Ft,h2 − f(sk,ηk] ∗ Kh2||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,h2 − F(sk,ηk],h2||2 +L2, and, +I4 = +ηk+�r +� +t=ηk+1 +||Ft,h2 − F(sk,ηk],h2||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,h1 − F(sk,ηk],h1||2 +L2. +To analyze I3, we rewrite it as follow, +I3 = +ηk+�r +� +t=ηk+1 +||f(sk,ηk] ∗ Kh2 − F(sk,ηk],h2||2 +L2 − 2 +ηk+�r +� +t=ηk+1 +⟨f(sk,ηk] ∗ Kh2 − F(sk,ηk],h2, Ft,h2 − f(sk,ηk] ∗ Kh2⟩L2 +=I3,1 + I3,2, +where, +I3,1 = +ηk+�r +� +t=ηk+1 +||f(sk,ηk] ∗ Kh2 − F(sk,ηk],h2||2 +L2, and, +I3,2 = −2 +ηk+�r +� +t=ηk+1 +⟨f(sk,ηk] ∗ Kh2 − F(sk,ηk],h2, Ft,h2 − f(sk,ηk] ∗ Kh2⟩L2. +Now, we will get an upper bound for each of the terms above. The term I3,1 = Op +� +�r 1 +T +log(T) +κ +p +r +k +� +, +which is followed by the use of Remark 1 and ∆ = Θ(T). Even more, by Assumption 4, we get +I3,1 = op(�rκ +p +r +2 +k +). +(37) +For the term I3,2, by Cauchy Schwartz inequality and triangle inequality, +⟨f(sk,ηk] ∗ Kκ − F(sk,ηk],κ, Ft,h2 − f(sk,ηk] ∗ Kh2⟩L2 +≤ ||f(sk,ηk] ∗ Kh2 − F(sk,ηk],h2||L2||Ft,h2 − f(sk,ηk] ∗ Kh2||L2 +≤ ||f(sk,ηk] ∗ Kh2 − F(sk,ηk],h2||L2 +� +||Ft,h2 − f[ηk+1,ek] ∗ Kh2||L2 + ||f[ηk+1,ek] ∗ Kh2 − f[sk+1,ηk] ∗ Kh2||L2 +� +33 + +for any t ∈ (ηk, ηk + �r]. By the Remark 1, and the fact that ∆ = Θ(T), we have that +||f(sk,ηk] ∗ Kh2 − F(sk,ηk],h2||L2 = Op +� 1 +√ +T +� +� +� +�log(T) +κ +p +r +k +� +and using basic properties of integrals ||f[ηk+1,ek] ∗ Kh2 − f[sk+1,ηk] ∗ Kh2||L2 = O(κk). Therefore, +I3,2 ≤ Op +� 1 +√ +T +� +� +� +�log(T) +κ +p +r +k +�� +O(�rκk) + +ηk+�r +� +t=ηk+1 +||Ft,h2 − f[ηk+1,ek] ∗ Kh2||L2 +� +Now, we need to get a bound of the magnitude of +ηk+�r +� +t=ηk+1 +||Ft,h2 − f[ηk+1,ek] ∗ Kh2||L2, +in order to get an upper for I3,2. This is done similarly to I2. We consider the random variable +�Yi = +⟨Ft,h2 − f(ηk,ek] ∗ Kh2, Ft,h2 − f(ηk,ek] ∗ Kh2⟩ +1 +2 +L2 − E(||Ft,h2 − fηk+1 ∗ Kh2||L2) +E(||Ft,h2 − fηk+1 ∗ Kh2||3 +L2) +1 +3 +. +In order to use Lemma 3, we observe that since ||Ft,h2 − fηk+1 ∗ Kh2||L2 ≥ 0, +E(|�Yi|3) ≤ E(||Ft,h2 − fηk+1 ∗ Kh2||3 +L2) +E(||Ft,h2 − fηk+1 ∗ Kh2||3 +L2) = 1. +Therefore, using Lemma 3 and that E(||Ft,h2 − fηk+1 ∗ Kh2||L2) = O(κ +−p +2r +k ) by (34), we get that +ηk+�r +� +t=ηk+1 +||Ft,h2 − f[ηk+1,ek] ∗ Kh2||2 +L2 = Op( +� +�rκ +− p +r +k +(log(�rκ +p +r +2 +k +) + 1)) + Op(�rκ +−p +2r +k ). +Thus, by Assumption 4 and above, +I3,2 ≤ Op +� 1 +√ +T +� +� +� +�log(T) +κ +p +r +k +�� +O(�rκk) + Op( +� +�rκ +− p +r +k +(log(�rκ +p +r +2 +k +) + 1)) + Op(�rκ +−p +2r +k ) +� += op(�rκ +p +r +2 +k +). +(38) +Consequently, I3 has been bounded, and we only need to go over the term I4, to finalize the analysis +for U1. To analyze I4, we observe that +I4 = +ηk+�r +� +t=ηk+1 +||Ft,h2 − F(sk,ηk],h2||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,h1 − F(sk,ηk],h1||2 +L2 += +ηk+�r +� +t=ηk+1 +� +⟨Ft,h2, Ft,h2⟩L2 − 2⟨Ft,h2, F(sk,ηk],h2⟩L2 + ⟨F(sk,ηk],h2, F(sk,ηk],h2⟩L2 +� ++ +ηk+�r +� +t=ηk+1 +� +− ⟨Ft,h1, Ft,h1⟩L2 + 2⟨Ft,h1, F(sk,ηk],h1⟩L2 − ⟨F(sk,ηk],h1, F(sk,ηk],h1⟩L2 +� +=I4,1 + I4,2 + I4,3, +34 + +where, +I4,1 = +ηk+�r +� +t=ηk+1 +⟨Ft,h2, Ft,h2⟩L2 − ⟨Ft,h1, Ft,h1⟩L2 +I4,2 = +ηk+�r +� +t=ηk+1 +2⟨Ft,h1, F(sk,ηk],h1⟩L2 − 2⟨Ft,h2, F(sk,ηk],h2⟩L2, and, +I4,3 = +ηk+�r +� +t=ηk+1 +⟨F(sk,ηk],h2, F(sk,ηk],h2⟩L2 − ⟨F(sk,ηk],h1, F(sk,ηk],h1⟩L2. +Now, we explore each of the terms I4,1, I4,2, and I4,3. First, I4,1 can be bounded as follows, we add +and subtract ⟨Ft,h1, Ft,h2⟩L2, to get +ηk+�r +� +t=ηk+1 +⟨Ft,h2, Ft,h2⟩L2 − ⟨Ft,h1, Ft,h1⟩L2 += +ηk+�r +� +t=ηk+1 +⟨Ft,h2, Ft,h2⟩L2 − ⟨Ft,h1, Ft,h1⟩L2 + ⟨Ft,h1, Ft,h2⟩L2 − ⟨Ft,h1, Ft,h2⟩L2 += +ηk+�r +� +t=ηk+1 +⟨Ft,h2 − Ft,h1, Ft,h2⟩L2 + ⟨Ft,h1, Ft,h2 − Ft,h1⟩L2 +which, by H¨older’s inequality, is bounded by +ηk+�r +� +t=ηk+1 +||Ft,h2||L2||Ft,h2 − Ft,h1||L2 + ||Ft,h1||L2||Ft,h2 − Ft,h1||L2 = �rOp( T − +r +2r+p +κ +p +2r + 1 +2 + p +2r +k +log +r +2r+p (T))) +since ||Ft,h1 − Ft,h2||L2 = O( |κ−�κ| +1 +2 +κ +p +2r + 1 +2 +k +) = Op( T +− +r +2r+p +κ +p +2r + 1 +2 +k +log +r +2r+p (T)), for any t, see Remark 2 for more +detail. Similarly, for I4,2, we have that adding and subtracting 2⟨Ft,h1, F(sk,ηk],h2⟩L2, +ηk+�r +� +t=ηk+1 +2⟨Ft,h1, F(sk,ηk],h1⟩L2 − 2⟨Ft,h2, F(sk,ηk],h2⟩L2 += +ηk+�r +� +t=ηk+1 +2⟨Ft,h1, F(sk,ηk],h1⟩L2 − 2⟨Ft,h2, F(sk,ηk],h2⟩L2 + 2⟨Ft,h1, F(sk,ηk],h2⟩L2 − 2⟨Ft,h1, F(sk,ηk],h2⟩L2 += +ηk+�r +� +t=ηk+1 +2⟨Ft,h1 − Ft,h2, F(sk,ηk],h2⟩L2 + 2⟨Ft,h1, F(sk,ηk],h2 − F(sk,ηk],h1⟩L2, +and by H¨older’s inequality and Remark 2, it is bounded by +ηk+�r +� +t=ηk+1 +||Ft,h1 − Ft,h2||L2||F(sk,ηk],h2||L2 + ||Ft,h1||L2||F(sk,ηk],h2 − F(sk,ηk],h1||L2 = �rOp( T − +r +2r+p +κ +p +2r + 1 +2 + p +2r +k +log +r +2r+p (T))). +35 + +Finally, for I4,3, we notice that, adding and subtracting ⟨F(sk,ηk],h1, F(sk,ηk],h2⟩L2, it is written as, +ηk+�r +� +t=ηk+1 +⟨F(sk,ηk],h2, F(sk,ηk],h2⟩L2 − ⟨F(sk,ηk],h1, F(sk,ηk],h1⟩L2 += +ηk+�r +� +t=ηk+1 +⟨F(sk,ηk],h2, F(sk,ηk],h2⟩L2 − ⟨F(sk,ηk],h1, F(sk,ηk],h1⟩L2 + ⟨F(sk,ηk],h1, F(sk,ηk],h2⟩L2 − ⟨F(sk,ηk],h1, F(sk,ηk],h2⟩L2 += +ηk+�r +� +t=ηk+1 +⟨F(sk,ηk],h2 − F(sk,ηk],h1, F(sk,ηk],h2⟩L2 + ⟨F(sk,ηk],h1, F(sk,ηk],h2 − F(sk,ηk],h1⟩L2 +which, by H¨older’s inequality and Remark 2, is bounded by +ηk+�r +� +t=ηk+1 +||F(sk,ηk],h2||L2||F(sk,ηk],h2 − F(sk,ηk],h2||L2 + ||F(sk,ηk],h1||L2||F(sk,ηk],h2 − F(sk,ηk],h1||L2 +=�rOp( T − +r +2r+p +κ +p +2r + 1 +2 + p +2r +k +log +r +2r+p (T))) +Then, by above and Assumption 4, we conclude +I4 = op(�rκ +p +r +2 +k +). +(39) +From (37), (38) and (39), we find that U1 has the following upper bound, +ηk+�r +� +t=ηk+1 +||Ft,h2 − f(sk,ηk] ∗ Kh2||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,h1 − F(sk,ηk],h1||2 +L2 = op(�rκ +p +r +2 +k +). +(40) +Now, making an analogous analysis, we have that U2 is upper bounded by, +ηk+�r +� +t=ηk+1 +||Ft,h1 − F(ηk,ek],h1||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,h2 − f(ηk,ek] ∗ Kh2||2 +L2 = op(�rκ +p +r +2 +k +). +(41) +In fact, we observe that +ηk+�r +� +t=ηk+1 +||Ft,h1 − F(ηk,ek],h1||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,h2 − f(ηk,ek] ∗ Kh2||2 +L2 += +ηk+�r +� +t=ηk+1 +||Ft,h2 − F(ηk,ek],h2||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,h2 − f(ηk,ek] ∗ Kh2||2 +L2 ++ +ηk+�r +� +t=ηk+1 +||Ft,h1 − F(ηk,ek],h1||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,h2 − F(ηk,ek],h2||2 +L2 +=I5 + I6, +36 + +where, +I5 = +ηk+�r +� +t=ηk+1 +||Ft,h2 − F(ηk,ek],h2||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,h2 − f(ηk,ek] ∗ Kh2||2 +L2, and, +I6 = +ηk+�r +� +t=ηk+1 +||Ft,h1 − F(ηk,ek],h1||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,h2 − F(ηk,ek],h2||2 +L2. +Then, I5 is bounded as follows +I5 = +ηk+�r +� +t=ηk+1 +||f(ηk,ek] ∗ Kh2 − F(ηk,ek],h2||2 +L2 + 2 +ηk+�r +� +t=ηk+1 +⟨f(ηk,ek] ∗ Kh2 − F(ηk,ek],h2, Ft,h2 − f(ηk,ek] ∗ Kh2⟩L2 +where, +I5,1 = +ηk+�r +� +t=ηk+1 +||f(ηk,ek] ∗ Kh2 − F(ηk,ek],h2||2 +L2, and, +I5,2 = 2 +ηk+�r +� +t=ηk+1 +⟨f(ηk,ek] ∗ Kh2 − F(ηk,ek],h2, Ft,h2 − f(ηk,ek] ∗ Kh2⟩L2. +The term I5,1 = Op +� +�r 1 +T +log(T) +κ +p +r +� +, using Remark 1. Even more, by Assumption 4, we get +I5,1 = op(�rκ +p +r +2 +k +). +(42) +For the term I5,2, by Cauchy Schwartz inequality, +⟨f(ηk,ek] ∗ Kh2 − F(ηk,ek],h2, Ft,h2 − f(ηk,ek] ∗ Kh2⟩L2 +≤ ||f(ηk,ek] ∗ Kh2 − F(ηk,ek],h2||L2||Ft,h2 − f(ηk,ek] ∗ Kh2||L2 +for any t ∈ (ηk, ηk + �r]. By Remark 1, we have that ||f(ηk,ek]∗Kh2−F(ηk,ek],h2||L2 = Op +� +1 +√ +T +� +log(T) +κ +p +r +k +� +. +Therefore, +I5,2 ≤ Op +� 1 +√ +T +� +� +� +�log(T) +κ +p +r +k +�� +ηk+�r +� +t=ηk+1 +||Ft,h2 − f[ηk+1,ek] ∗ Kh2||L2 +� +Now, similarly to the bound for I2, we consider the random variable +¯Yi = +⟨Ft,h2 − f(ηk,ek] ∗ Kκ, Ft,h2 − f(ηk,ek] ∗ Kh2⟩ +1 +2 +L2 − E(||Ft,h2 − f[ηk+1,ek] ∗ Kh2||L2) +E(||Ft,h2 − fηk+1 ∗ Kh2||3 +L2) +1 +3 +. +In order to use Lemma 3, we observe +E(| ¯Yi|3) = E(||Ft,h2 − fηk+1 ∗ Kh2||3 +L2) +E(||Ft,h2 − fηk+1 ∗ Kh2||3 +L2) = 1. +37 + +so that, by Lemma 3, +I5,2 ≤ Op +� 1 +√ +T +� +log(T) +κ +p +r +�� +Op( +� +�rκ +− p +r +k +(log(�rκ +p +r +2 +k +) + 1)) + Op(κ +−p +2r +k ) +� += op(�rκ +p +r +2 +k +). +(43) +To analyze I6, we observe that +I6 = +ηk+�r +� +t=ηk+1 +||Ft,h2 − F(ηk,ek],h2||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,h1 − F(ηk,ek],h1||2 +L2 += +ηk+�r +� +t=ηk+1 +� +⟨Ft,h2, Ft,h2⟩L2 − 2⟨Ft,h2, F(ηk,ek],h2⟩L2 + ⟨F(ηk,ek],h2, F(ηk,ek],h2⟩L2 +� +ηk+�r +� +t=ηk+1 +� +− ⟨Ft,h1, Ft,h1⟩L2 + 2⟨Ft,h1, F(ηk,ek],h1⟩L2 − ⟨F(ηk,ek],h1, F(ηk,ek],h1⟩L2 +� +=I6,1 + I6,2 + I6,3, +where, +I6,1 = +ηk+�r +� +t=ηk+1 +⟨Ft,h2, Ft,h2⟩L2 − ⟨Ft,h1, Ft,h1⟩L2, +I6,2 = +ηk+�r +� +t=ηk+1 +2⟨Ft,h1, F(ηk,ek],h1⟩L2 − 2⟨Ft,h2, F(ηk,ek],h2⟩L2 +I6,3 = +ηk+�r +� +t=ηk+1 +⟨F(ηk,ek],h2, F(ηk,ek],h2⟩L2 − ⟨F(ηk,ek],h1, F(ηk,ek],h1⟩L2. +Then we bound each of these terms. First, we rewrite I6,1, as +ηk+�r +� +t=ηk+1 +⟨Ft,h2, Ft,h2⟩L2 − ⟨Ft,h1, Ft,h1⟩L2 += +ηk+�r +� +t=ηk+1 +⟨Ft,h2, Ft,h2⟩L2 − ⟨Ft,h1, Ft,h1⟩L2 + ⟨Ft,h1, Ft,h2⟩L2 − ⟨Ft,h1, Ft,h2⟩L2 += +ηk+�r +� +t=ηk+1 +⟨Ft,h2 − Ft,h1, Ft,h2⟩L2 + ⟨Ft,h1, Ft,h2 − Ft,h1⟩L2 +which, by H¨older’s inequality, is bounded by +ηk+�r +� +t=ηk+1 +||Ft,h2||L2||Ft,h2 − Ft,h1||L2 + ||Ft,h1||L2||Ft,h2 − Ft,h1||L2 = �rOp( T − +r +2r+p +κ +p +2r + 1 +2 + p +2r +k +log +r +2r+p (T))) +38 + +since ||Ft,κ − Ft,�κ||2 +L2 = O( |κ−�κ| +1 +2 +κ +p +2r + 1 +2 +k +) = Op( T +− +r +2r+p +κ +p +2r + 1 +2 +k +log +r +2r+p (T))), for any t, see Remark 2 for more +detail. Similarly, for I6,2 we have, +ηk+�r +� +t=ηk+1 +2⟨Ft,h1, F(ηk,ek],h1⟩L2 − 2⟨Ft,h2, F(ηk,ek],h2⟩L2 += +ηk+�r +� +t=ηk+1 +2⟨Ft,h1, F(ηk,ek],h1⟩L2 − 2⟨Ft,h2, F(ηk,ek],h2⟩L2 + 2⟨Ft,h1, F(ηk,ek],h2⟩L2 − 2⟨Ft,h1, F(ηk,ek],h2⟩L2 += +ηk+�r +� +t=ηk+1 +2⟨Ft,h1 − Ft,h2, F(ηk,ek],h2⟩L2 + 2⟨Ft,h1, F(ηk,ek],h2 − F(ηk,ek],h1⟩L2 +and by H¨older’s inequality and Remark 2, it is bounded by +ηk+�r +� +t=ηk+1 +||Ft,h1 − Ft,h2||L2||F(ηk,ek],h2||L2 + ||Ft,h1||L2||F(ηk,ek],h2 − F(ηk,ek],h1||L2 +=�rOp( T − +r +2r+p +κ +p +2r + 1 +2 + p +2r +k +log +r +2r+p (T))). +Now for I6,3, we write it as +ηk+�r +� +t=ηk+1 +⟨F(ηk,ek],h2, F(ηk,ek],h2⟩L2 − ⟨F(ηk,ek],h1, F(ηk,ek],h1⟩L2 += +ηk+�r +� +t=ηk+1 +⟨F(ηk,ek],h2, F(ηk,ek],h2⟩L2 − ⟨F(ηk,ek],h1, F(ηk,ek],h1⟩L2 + ⟨F(ηk,ek],h1, F(ηk,ek],h2⟩L2 − ⟨F(ηk,ek],h1, F(ηk,ek],h2⟩L2 += +ηk+�r +� +t=ηk+1 +⟨F(ηk,ek],h2 − F(ηk,ek],h1, F(sk,ηk],h2⟩L2 + ⟨F(ηk,ek],h1, F(ηk,ek],h2 − F(ηk,ek],h1⟩L2 +which, by H¨older’s inequality and Remark 2, is bounded by +ηk+�r +� +t=ηk+1 +||F(ηk,ek],h2||L2||F(ηk,ek],h2 − F(ηk,ek],h2||L2 + ||F(ηk,ek],h1||L2||F(ηk,ek],h2 − F(ηk,ek],h1||L2 +=�rOp( T − +r +2r+p +κ +p +2r + 1 +2 + p +2r +k +log +r +2r+p (T))) +By above and Assumption 4, we conclude +I6 = op(�rκ +p +r +2 +k +). +(44) +From, (42), (43) and (44), we get that U2 is bounded by +ηk+�r +� +t=ηk+1 +||Ft,�κ − F(ηk,ek],�κ||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,κ − f(ηk,ek] ∗ Kκ||2 +L2 = op(�rκ +p +r +2 +k +) +39 + +Therefore, from (40) and (41) +�Qk(ηk) − �Qk(ηk + �r) − Q∗(ηk) + Q∗(ηk + �r) = op(�rκ +p +r +2 +k +) +(45) +Step 4: Combination of all the steps above. Finally, combining (32), (35) and (45), uniformly +for any �r ≥ +1 +κ +p +r +2 +k +we have that +1 +2�rκ2 +k − Op +�√ +�rκkκ +− p +2r +k +(log(�rκ +p +r +2 +k +) + 1) +� +− op(�rκ +p +r +2 +k +) ≤op(�rκ +p +r +2 +k +) +which implies, +�rκ +p +r +2 +k += Op(1) +(46) +and complete the proofs of a.1 and b.1. +Limiting distributions. For any k ∈ {1, . . . , K}, due to the uniform tightness of �rκ +p +r +2 +k +, (32) +and (45), as T → ∞ +Q∗(η) = +η +� +t=sk+1 +||Ft,h2 − f(sk,ηk] ∗ Kh2||2 +L2 + +ek +� +t=η+1 +||Ft,h2 − f(ηk,ek] ∗ Kh2||2 +L2, +satisfies +��� �Q +� +ηk + �r +� +− �Q +� +ηk +� +− +� +Q∗� +ηk + �r +� +− Q∗� +ηk +����� +p→ 0. +Therefore, it is sufficient to find the limiting distributions of Q∗� +ηk + �r +� +− Q∗� +ηk +� +when T → ∞. +Non-vanishing regime. Observe that for �r > 0, we have that when T → ∞, +Q∗(ηk + �r) − Q∗(ηk) = +ηk+�r +� +t=ηk+1 +||Ft,h2 − f(sk,ηk] ∗ Kh2||2 +L2 − +ηk+�r +� +t=ηk+1 +||Ft,h2 − f(ηk,ek] ∗ Kh2||2 +L2 += +ηk+�r +� +t=ηk+1 +||f(sk,ηk] ∗ Kh2 − f(ηk,ek] ∗ Kh2||2 +L2 +− 2 +ηk+�r +� +t=ηk+1 +⟨f(sk,ηk] ∗ Kh2 − f(ηk,ek] ∗ Kh2, Ft,h2 − f(ηk,ek] ∗ Kh2⟩L2 +D +−→ +�r +� +t=1 +2 +� +Fh2,t − ft ∗ Kh2, (fηk+1 − fηk) ∗ Kh2 +� +L2 + �r||(fηk+1 − fηk) ∗ Kh2||2 +L2. +40 + +When �r < 0 and T → ∞, we have that +Q∗(ηk + �r) − Q∗(ηk) = +ηk−1 +� +t=ηk+�r +||Ft,h2 − f(sk,ηk] ∗ Kh2||2 +L2 − +ηk−1 +� +t=ηk+�r +||Ft,h2 − f(ηk,ek] ∗ Kh2||2 +L2 += +ηk−1 +� +t=ηk+�r +||f(sk,ηk] ∗ Kh2 − f(ηk,ek] ∗ Kh2||2 +L2 +− 2 +ηk−1 +� +t=ηk+�r +⟨f(sk,ηk] ∗ Kh2 − f(ηk,ek] ∗ Kh2, Ft,h2 − f(ηk,ek] ∗ Kh2⟩L2 +D +−→ +0 +� +t=�r+1 +2 +� +Fh2,t − ft ∗ Kh2, (fηk − fηk+1) ∗ Kh2 +� +L2 + �r||(fηk+1 − fηk) ∗ Kh2||2 +L2. +Therefore, using Slutsky’s theorem and the Argmax (or Argmin) continuous mapping theorem (see +3.2.2 Theorem van der Vaart and Wellner, 1996) we conclude +(�ηk − ηk)κ +p +r +2 +k +D +−→ arg min +�r∈Z +Pk(�r) +(47) +Vanishing regime. Vanishing regime. Let m = κ +−2− p +r +k +, and we have that m → ∞ as T → ∞. +Observe that for �r > 0, we have that +Q∗ +k +� +ηk + �rm +� +− Q∗ +k +� +ηk +� += +ηk+�rm−1 +� +t=ηk +||f(sk,ηk] ∗ Kh2 − f(ηk,ek] ∗ Kh2||2 +L2 +−2 +ηk+�rm−1 +� +t=ηk +⟨f(sk,ηk] ∗ Kh2 − f(ηk,ek] ∗ Kh2, Ft,h2 − f(ηk,ek] ∗ Kh2⟩L2 +Following the Central Limit Theorem for α−mixing, see Lemma 4, we get +1 +√m +ηk+rm−1 +� +t=ηk +⟨f(sk,ηk] ∗ Kh2 − f(ηk,ek] ∗ Kh2, Ft,h2 − f(ηk,ek] ∗ Kh2⟩L2 +κ +p +2r +1 +k +D→ κ +− p +r +k +�σ∞(k)B(�r), +where B(�r) is a standard Brownian motion and �σ(k) is the long-run variance given in (13). There- +fore, it holds that when T → ∞ +Q∗ +k +� +ηk + �rm +� +− Q∗ +k +� +ηk +� D→ κ +− p +r +k +�σ∞(k)B1(r) + �rκ +− p +r −2 +k +||f(sk,ηk] ∗ Kh2 − f(ηk,ek] ∗ Kh2||2 +L2. +Similarly, for �r < 0, we have that when n → ∞ +Q∗ +k +� +ηk + rm +� +− Q∗ +k +� +ηk +� D→ κ +− p +r +k +�σ∞(k)B1(−�r) − �rκ +− p +r −2 +k +||f(sk,ηk] ∗ Kh2 − f(ηk,ek] ∗ Kh2||2 +L2.. +Then, using Slutsky’s theorem and the Argmax (or Argmin) continuous mapping theorem (see +3.2.2 Theorem in van der Vaart and Wellner (1996)), and the fact that, E(||f(sk,ηk] ∗ Kh2 − f(ηk,ek] ∗ +Kh2||2 +L2) = O(κ2 +k), we conclude that +κ +2+ p +r +k +� +�ηk − ηk +� +D +−→ arg min +r∈Z +�σ∞(k)B(�r) + |�r|, +which completes the proof of b.2. +41 + +D +Proof of Theorem 3 +In this section, we present the proof of theorem Theorem 3. +Proof of Theorem 3. First, letting h2 = cκκ +1 +r +k and R = O( T +p+r +2r+p +κ +p +2r + 3 +2 +k +), we consider +˘σ2 +∞(k) = 1 +R +R +� +r=1 +� 1 +√ +S +� +i∈Sr +˘Yi +�2 +, where, ˘Yi = κ +p +2r −1 +k +� +Fh2,i − fi ∗ Kh2, (fηk − fηk+1) ∗ Kh2 +� +L2. +(48) +We will show that +(i) +����σ2 +∞(k) − ˘σ2 +∞(k) +��� +P +−→ 0, +T → ∞, and +(ii) +���˘σ2 +∞(k) − �σ2 +∞(k) +��� +P +−→ 0, +T → ∞ +in order to conclude the result. For (i), we use a2 − b2 = (a + b)(a − b), to write, +����σ2 +∞(k) − ˘σ2 +∞(k) +��� = +��� 1 +R +R +� +r=1 +� 1 +√ +S +� +i∈Sr +˘Yi +�2 +− 1 +R +R +� +r=1 +� 1 +√ +S +� +i∈Sr +Yi +�2��� += +��� 1 +R +R +� +r=1 +� 1 +√ +S +� +i∈Sr +˘Yi − Yi +�� 1 +√ +S +� +i∈Sr +˘Yi + Yi +���� += +��� 1 +R +R +� +r=1 +I1I2 +��� +Then, we bound each of the terms I1 and I2. For I1, we observe that, +I1 = +��� 1 +√ +S +� +i∈Sr +˘Yi − Yi +��� ≤ +1 +√ +S +� +i∈Sr +��� ˘Yi − Yi +���. +Then, adding and subtracting, �κ +p +2r −1 +k +� +Fh1,i − fi ∗ Kh1, (fηk − fηk+1) ∗ Kh2 +� +L2 and +�κ +p +2r −1 +k +� +Fh2,i − fi ∗ Kh2 − Fh1,i + fi ∗ Kh1, (fηk − fηk+1) ∗ Kh1 +� +L2, +we get that, +��� ˘Yi − Yi +��� += +���κ +p +2r −1 +k +� +Fh2,i − fi ∗ Kh2, (fηk − fηk+1) ∗ Kh2 +� +L2 − �κ +p +2r −1 +k +� +Fh1,i − fi ∗ Kh1, (fηk − fηk+1) ∗ Kh1 +� +L2 +��� += +���κ +p +2r −1 +k +� +Fh2,i − fi ∗ Kh2, (fηk − fηk+1) ∗ Kh2 +� +L2 − �κ +p +2r −1 +k +� +Fh1,i − fi ∗ Kh1, (fηk − fηk+1) ∗ Kh2 +� +L2 ++�κ +p +2r −1 +k +� +Fh1,i − fi ∗ Kh1, (fηk − fηk+1) ∗ Kh2 +� +L2 − �κ +p +2r −1 +k +� +Fh1,i − fi ∗ Kh1, (fηk − fηk+1) ∗ Kh1 +� +L2 +��� ++�κ +p +2r −1 +k +� +Fh2,i − fi ∗ Kh2 − Fh1,i + fi ∗ Kh1, (fηk − fηk+1) ∗ Kh1 +� +L2 +−�κ +p +2r −1 +k +� +Fh2,i − fi ∗ Kh2 − Fh1,i + fi ∗ Kh1, (fηk − fηk+1) ∗ Kh1 +� +L2 +42 + +which can be written as, +��� +� +κ +p +2r −1 +k +(Fh2,i − fi ∗ Kh2) − �κ +p +2r −1 +k +(Fh1,i − fi ∗ Kh1), (fηk − fηk+1) ∗ Kh2 − (fηk − fηk+1) ∗ Kh1 +� +L2 ++ +� +�κ +p +2r −1 +k +(Fh1,i − fi ∗ Kh1) − κ +p +2r −1 +k +(Fh2,i − fi ∗ Kh2), (fηk − fηk+1) ∗ Kh1 − (fηk − fηk+1) ∗ Kh2 +� +L2 ++ +� +κ +p +2r −1 +k +(Fh2,i − fi ∗ Kh2), (fηk − fηk+1) ∗ Kh2 − (fηk − fηk+1) ∗ Kh1 +� +L2 ++ +� +�κ +p +2r −1 +k +(Fh1,i − fi ∗ Kh1) − κ +p +2r −1 +k +(Fh2,i − fi ∗ Kh2), (fηk − fηk+1) ∗ Kh1 +� +L2 +���. +Now, we bound the expression above. For this purpose, by triangle inequality, it is enough to bound +each of the terms above. Then, we use H¨older’s inequality. First, +��� +� +κ +p +2r −1 +k +(Fh2,i − fi ∗ Kh2) − �κ +p +2r −1 +k +(Fh1,i − fi ∗ Kh1), (fηk − fηk+1) ∗ Kh2 − (fηk − fηk+1) ∗ Kh1 +� +L2 +��� +≤ |κ +p +2r −1 +k +− �κ +p +2r −1 +k +|||Fh2,i − fi ∗ Kh2 − Fh1,i + fi ∗ Kh1||L2||(fηk − fηk+1) ∗ Kh2 − (fηk − fηk+1) ∗ Kh1||L2. +Then, using (60), we have that |κ +p +2r −1 +k +− �κ +p +2r −1 +k +| = Op( T +− +2r +2r+p +κ +2− p +2r +k +log +2r +2r+p (T)), and using Remark 2, it +follows that +||Fh2,i − fi ∗ Kh2 − Fh1,i + fi ∗ Kh1||L2 ≤||Fh2,i − Fh1,i||L2 + ||fi ∗ Kh1 − fi ∗ Kh2||L2 +=Op(T − +r +2r+p +κ +p +2r + 1 +2 +k +log +r +2r+p (T)) +and, +||(fηk − fηk+1) ∗ Kh2 − (fηk − fηk+1) ∗ Kh1||L2 +≤||fηk ∗ Kh2 − fηk ∗ Kh1||L2 + ||fηk+1 ∗ Kh1 − fηk+1 ∗ Kh2||L2 = Op(T − +r +2r+p +κ +p +2r + 1 +2 +k +log +r +2r+p (T)). +So that, +��� +� +κ +p +2r −1 +k +(Fh2,i − fi ∗ Kh2) − �κ +p +2r −1 +k +(Fh1,i − fi ∗ Kh1), (fηk − fηk+1) ∗ Kh2 − (fηk − fηk+1) ∗ Kh1 +� +L2 +��� += Op(T − +2r +2r+p +κ +2− p +2r +k +log +2r +2r+p (T))Op(T − +r +2r+p +κ +p +2r + 1 +2 +k +log +r +2r+p (T))Op(T − +r +2r+p +κ +p +2r + 1 +2 +k +log +r +2r+p (T)). +Now, in a similar way, we observe that +� +κ +p +2r −1 +k +(Fh2,i − fi ∗ Kh2), (fηk − fηk+1) ∗ Kh2 − (fηk − fηk+1) ∗ Kh1 +� +L2 +≤||κ +p +2r −1 +k +(Fh2,i − fi ∗ Kh2)||L2||(fηk − fηk+1) ∗ Kh2 − (fηk − fηk+1) ∗ Kh1||L2 +=Op(κ +p +2r −1 +k +κ +− p +2r +k +)Op(T − +r +2r+p +κ +p +2r + 1 +2 +k +log +r +2r+p (T)) +43 + +where equality is followed by noticing that +||Fh2,i − fi ∗ Kh2||L2 ≤||Fh2,i||L2 + ||fi ∗ Kh2||L2 +(49) +=O(κ +− p +2r +k +) + O(1), +(50) +and then using Remark 2 and Assumption 1. Finally, +� +�κ +p +2r −1 +k +(Fh1,i − fi ∗ Kh1) − κ +p +2r −1 +k +(Fh2,i − fi ∗ Kh2), (fηk − fηk+1) ∗ Kh1 +� +L2 +≤|�κ +p +2r −1 +k +− κ +p +2r −1 +k +|||(Fh1,i − fi ∗ Kh1) − (Fh2,i − fi ∗ Kh2)||L2||(fηk − fηk+1) ∗ Kh1||L2 +=Op(T − +2r +2r+p +κ +2− p +2r +k +log +2r +2r+p (T))Op(T − +r +2r+p +κ +p +2r + 1 +2 +k +log +r +2r+p (T))κk +where equality is followed by Remark 2, Assumption 1 and Minkowski’s inequality. Therefore, +I1 ≤ +1 +√ +S +� +i∈Sr +��� ˘Yi − Yi +��� +√ +S +� +Op(T − +2r +2r+p +κ +2− p +2r +k +log(T) +2r +2r+p )Op(T − +r +2r+p +κ +p +2r + 1 +2 +k +log(T) +r +2r+p )Op(T − +r +2r+p +κ +p +2r + 1 +2 +k +log(T) +r +2r+p ) ++Op(κ +p +2r −1 +k +κ +− p +2r +k +)Op(T − +r +2r+p +κ +p +2r + 1 +2 +k +log +r +2r+p (T)) + Op(T − +r +4r+2p +κ +2− p +2r +k +log +r +2r+p (T))Op(T − +r +2r+p +κ +p +2r + 1 +2 +k +log +r +2r+p (T))κk +� += +√ +S +� +Op(T − +2r +2r+p +κ +2− p +2r +k +log(T) +2r +2r+p )Op(T − +2r +2r+p +κ +p +r +1 +k +log(T) +2r +2r+p ) + Op(κ−1 +k )Op(T − +r +2r+p +κ +p +2r + 1 +2 +k +log(T) +r +2r+p ) +� +. +To bound the I2 term, we add and subtract κ +p +2r −1 +k +� +Fh2,i − fi ∗ Kh2, (fηk − fηk+1) ∗ Kh1 +� +L2, to get +��� ˘Yi + Yi +��� += +���κ +p +2r −1 +k +� +Fh2,i − fi ∗ Kh2, (fηk − fηk+1) ∗ Kh2 +� +L2 + �κ +p +2r −1 +k +� +Fh1,i − fi ∗ Kh1, (fηk − fηk+1) ∗ Kh1 +� +L2 +��� += +���κ +p +2r −1 +k +� +Fh2,i − fi ∗ Kh2, (fηk − fηk+1) ∗ Kh2 +� +L2 − κ +p +2r −1 +k +� +Fh2,i − fi ∗ Kh2, (fηk − fηk+1) ∗ Kh1 +� +L2 ++κ +p +2r −1 +k +� +Fh2,i − fi ∗ Kh2, (fηk − fηk+1) ∗ Kh1 +� +L2 + �κ +p +2r −1 +k +� +Fh1,i − fi ∗ Kh1, (fηk − fηk+1) ∗ Kh1 +� +L2 +��� += +���κ +p +2r −1 +k +� +Fh2,i − fi ∗ Kh2, (fηk − fηk+1) ∗ Kh2 +� +L2 + κ +p +2r −1 +k +� +Fh2,i − fi ∗ Kh2, (fηk − fηk+1) ∗ Kh1 +� +L2 ++ +� +�κ +p +2r −1 +k +(Fh1,i − fi ∗ Kh1) − κ +p +2r −1 +k +(Fh2,i − fi ∗ Kh2), (fηk − fηk+1) ∗ Kh1 +� +L2 +���. +Then, as before, we bound each of the terms above using H¨older’s inequality. We start with the +term +���κ +p +2r −1 +k +� +Fh2,i − fi ∗ Kh2, (fηk − fηk+1) ∗ Kh2 +� +L2 +��� +≤κ +p +2r −1 +k +||Fh2,i − fi ∗ Kh2||L2||(fηk − fηk+1) ∗ Kh2||L2 +≤κ +p +2r −1 +k +Op(κ +− p +2r +k +)κk = Op(1) +44 + +where the second inequality is followed by (49). Similarly, +���κ +p +2r −1 +k +� +Fh2,i − fi ∗ Kh2, (fηk − fηk+1) ∗ Kh1 +� +L2 +��� +≤κ +p +2r −1 +k +||Fh2,i − fi ∗ Kh2||L2||(fηk − fηk+1) ∗ Kh1||L2 +≤κ +p +2r −1 +k +Op(κ +− p +2r +k +)κk = Op(1) +where the second inequality is followed by the (49). Finally, the term +��� +� +�κ +p +2r −1 +k +(Fh1,i − fi ∗ Kh1) − κ +p +2r −1 +k +(Fh2,i − fi ∗ Kh2), (fηk − fηk+1) ∗ Kh1 +� +L2 +��� +was previously bounded by, +Op(T − +2r +2r+p +κ +2− p +2r +k +log +2r +2r+p (T))Op(T − +r +2r+p +κ +p +2r + 1 +2 +k +log +r +2r+p (T))κk. +Therefore, +I2 ≤ +1 +√ +S +� +i∈Sr +��� ˘Yi + Yi +��� = +√ +S +� +Op(1) + Op(T − +2r +2r+p +κ +2− p +2r +k +log +2r +2r+p (T))Op(T − +r +2r+p +κ +p +2r + 1 +2 +k +log +r +2r+p (T))κk +� +. +In consequences, +����σ2 +∞(k) − ˘σ2 +∞(k) +��� = +��� 1 +R +R +� +r=1 +� 1 +√ +S +� +i∈Sr +˘Yi +�2 +− 1 +R +R +� +r=1 +� 1 +√ +S +� +i∈Sr +Yi +�2��� += +��� 1 +R +R +� +r=1 +I1I2 +��� +=S +� +Op(T − +4r +2r+p +κ +4− p +r +k +log +4r +2r+p (T))Op(T − +2r +2r+p +κ +p +r +1 +k +log +2r +2r+p (T))Op(T − +r +2r+p +κ +p +2r + 1 +2 +k +log +r +2r+p (T))κk ++Op(T − +2r +2r+p +κ +2− p +2r +k +log +2r +2r+p (T))Op(T − +r +2r+p +κ +p +2r + 1 +2 +k +log +r +2r+p (T))Op(T − +r +2r+p +κ +p +2r + 1 +2 +k +log +r +2r+p (T)) ++Op(κ−1 +k )Op(T − +r +2r+p +κ +p +2r + 1 +2 +k +log +r +2r+p (T)) ++Op(κ−1 +k )Op(T − +r +2r+p +κ +p +2r + 1 +2 +k +log +r +2r+p (T))Op(T − +2r +2r+p +κ +2− p +2r +k +log +2r +2r+p (T))Op(T − +r +2r+p +κ +p +2r + 1 +2 +k +log +r +2r+p (T))κk +� +. +In order to conclude (i), we notice that by Assumption 4 and that S = O(T +r +2r+p κ +p +2r + 3 +2 +k +), which +implies, +����σ2 +∞(k) − ˘σ2 +∞(k) +��� =op(1). +45 + +Now, we are going to see that +���˘σ2 +∞(k) − �σ2 +∞(k) +��� +P +−→ 0, +T → ∞. To this end, we will show that +the estimator is asymptotically unbiased, and its variance → 0 as T → ∞. First, we notice that, +by H¨older’s inequality and Minkowsky’s inequality, +| ˘Yi| =|κ +p +2r −1 +k +� +Fh2,i − fi ∗ Kh2, (fηk − fηk+1) ∗ Kh2 +� +L2| +≤κ +p +2r −1 +k +||Fh2,i − fi ∗ Kh2||L2||(fηk − fηk+1) ∗ Kh2||L2 +≤κ +p +2r −1 +k +κ +− p +2r +k +κk = 1. +Now, we analyze the Bias. We observe that, +E(˘σ2 +∞(k)) = 1 +R +R +� +r=1 +E +�� 1 +√ +S +� +i∈Sr +˘Yi +�2� += 1 +S E +�� � +i∈Sr +˘Yi +�2� += +S+1 +� +l=−S+1 +S − l +S +E( ˘Yi ˘Yi+l) +and, +�σ2 +∞(k) = +∞ +� +l=−∞ +E( ˘Yi ˘Yi+l). +so that, the bias has the following form, +�σ2 +∞(k) − E(˘σ2 +∞(k)) = 2 +∞ +� +l=S +E( ˘Yi ˘Yi+l) + 2 +S +� +l=1 +l +S E( ˘Yi ˘Yi+l). +Now, we show that each of the above terms vanishes as T → ∞. We have that, by condition (1) +and covariance inequality +2 +∞ +� +l=S +E( ˘Yi ˘Yi+l) ≤8 +∞ +� +l=S +|| ˘Yi||2 +L∞αl ≤ 8 +∞ +� +l=S +αl → 0, as T → ∞ +where αl is the mixing coefficient. Then, +2 +S +� +l=1 +l +S E( ˘Yi ˘Yi+l) ≤ 8 +S +� +l=1 +l +S || ˘Yi||2 +L∞αl ≤ C +S → 0, +by condition (1), choice of S and Assumption 4. Therefore, we conclude that the Bias vanishes as +T → ∞. To analyze the Variance, we observe that, if Yr = 1 +S +� � +i∈Sr ˘Yi +�2 +V ar(˘σ2 +∞(k)) = E((˘σ2 +∞(k) − E(˘σ2 +∞(k)))2) += 1 +R2 E +�� +R +� +r=1 +Yr − E(Yr) +�2� += 1 +R +R−1 +� +l=−R+1 +R − l +R +cov(Yr, Yl+r) +≤ 8 +R||Yr||2 +L∞ +∞ +� +l=0 +�αl ≤ 8CS +R +→ 0, as, T → ∞. +where, �αl are the mixing coefficients of {Yr}r∈Z, which is bounded by the mixing coefficient αl. +From here, we conclude the result (ii). +46 + +E +Large probability events +In this section, we deal with all the large probability events that occurred in the proof of Theorem 1. +Recall that, for any (s, e] ⊆ (0, T], +�fs,e +t +(x) = +� +e − t +(e − s)(t − s) +t +� +l=s+1 +fl(x) − +� +t − s +(e − s)(e − t) +e +� +l=t+1 +fl(x), x ∈ X. +Proposition 1. For any x, +P +� +max +ρ⩽k≤T−�r +��� 1 +√ +k +�r+k +� +t=�r+1 +� +Kh(x − Xt) − +� +Kh(x − z)dFt(z) +���� ≥ C +� +log T +hp +� +⩽ T −p−3. +Proof. We have that the random variables {Zt = Kh +� +x − Xt +� +}T +t=1 satisfies +σ +� +Kh +� +x − Xt +�� +⊂ σ +� +Xt +� +and, +���Kh +� +x − Xt +���� ⩽ 1 +hp CK. +Moreover, let +V 2 = sup +t>0 +� +var(Kh(x − Xt) + 2 +� +j>t +|cov(Zt, Zj)|) +� +. +We observe that, +var(Kh(x − Xt)) ≤E(( 1 +hp K(x − Xt +h +))2) +≤ +� +1 +h2p K(x − z +h +)dFt(z) +making µ = x−z +h , the last inequality is equal to +� +1 +h2p K(x − z +h +)dFt(z) ≤ 1 +hp +� +K2(u)dFt(z) +≤ 1 +hp CkCf. +Then, by proposition 2.5 on Fan and Yao (2008), |cov(Z1, Z1 + t)| ≤ Cα(t) 1 +h2p C2 +K. On the other +hand, +cov(Z1, Z1 + t) =|E(Z1Zt+1) − E(Z1)2| +≤ +� � +Kh(x − z1)Kh(x − z2)gt(z1, z2)dz1dz2 + E(Z1)2 +≤||gt||L∞ + E(Z1)2. +47 + +Since by assumption, equation (3), ||gt||L∞ < ∞ and, +E(Z1) = E(Kh(x − X1)) = +� +1 +hp K(x − z +h +)dFt(z) = +� +K(u)ft(x − hu)du = O(1), +we obtain that |cov(Z1, Z1+t)|. +Therefore, � 1 +h −1 +t=1 |cov(Z1, Zt+1)| ≤ C 1 +h and, using the mixing +condition bound, inequality (1), +T−1 +� +t= 1 +hp +|cov(Z1, Z1 + t)| ≤D +∞ +� +t= 1 +h +e−2Ct +h2p +≤De−2C 1 +hp +h2p +≤ �D 1 +h2p hp = �Dhp +where the last inequity is followed by the fact e−x < 1 +x for x > −1. In consequence, +V 2 = sup +t>0 +� +var(Kh(x − Xt) + 2 +� +j>t +|cov(Zt, Zj)|) +� += �C 1 +hp + �D 1 +hp = ��C 1 +hp . +Then, by Bernstein inequality for mixing dependence, see Merlev`ede et al. (2009) for more details, +letting +λ = Cp +�� +k log(T) +hp ++ +� +log(T) +h2p ++ +� +log(T) log2(k) +hp +� +we get that, +P +���� +�r+k +� +t=�r+1 +� +Kh(x − Xt) − +� +Kh(x − z)dFt(z) +���� > λ +� +≤ T −p−3. +in consequence, +P +���� 1 +√ +k +�r+k +� +t=�r+1 +� +Kh(x − Xt) − +� +Kh(x − z)dFt(z) +���� > λ +√ +k +� +≤ T −p−3. +Since khp ≥ log(T) if k > ρ, and log2(k) = O(k), +λ +√ +k += +Cp +�� +k log(T) +hp ++ +� +log(T) +h2p ++ +� +log(T) log2(k) +hp +� +√ +k +=Cp +�� +log(T) +hp ++ +� +log(T) +kh2p + +� +log(T) log2(k) +khp +� +≤Cp +�� +log(T) +hp ++ +� +1 +hp + +� +log(T) +hp +� +≤C1 +� +log(T) +hp +. +48 + +It follows that, +P +���� 1 +√ +k +�r+k +� +t=�r+1 +� +Kh(x − Xt) − +� +Kh(x − z)dFt(z) +���� > C1 +� +log(T) +hp +� +≤ T −p−3. +Proposition 2. Define the events +A1 = +� +e−ρ +max +t=s+ρ+1 sup +x∈Rp +���� �F s,e +t,h (x) − �fs,e +t +(x) +���� ≥ 2C +� +log T +hp ++ 2C1√p +hp ++ 2C2 +√ +Thr� +and, +A2 = +� +max +ρ⩽k≤T−�r sup +x∈Rp +��� 1 +√ +k +�r+k +� +t=�r+1 +� +Kh(x − Xt) − ft(x) +���� ≥ C +� +log T +hp ++ C1√p +hp ++ C2 +√ +Thr� +. +Then +P +� +A1 +� +⩽ 2RpT −2 +(51) +P +� +A2 +� +⩽ RpT −2 +(52) +where R is a positive constant. +Proof. First, we notice that +max +ρ⩽k⩽T−�r sup +x∈Rp +��� 1 +√ +k +�r+k +� +t=�r+1 +Kh +� +x − Xt +� +− ft(x) +��� +≤ +max +ρ⩽k≤T−�r sup +x∈Rp +��� 1 +√ +k +�r+k +� +t=�r+1 +� +Kh(x − Xt) − +� +Kh(x − z)dFt(z) +���� ++ +max +ρ⩽k⩽T−�r sup +x∈Rp +��� 1 +√ +k +r+k +� +t=�r+1 +� � +Kh(x − z)dFz(z) − ft(x) +���� = I1 + I2 +Now we will bound each of the terms I1, I2. For I1, we consider +A = {x1, ..., x(R +√ +T/ +√ +hph)p} +with ∪xi∈{x1,...,x(R +√ +T / +√ +hph)p}Rec(xi, +√ +hph +√ +T ) ⊃ D, where D is the support of K and Rec(xi, +√ +hph +√ +T ) are +boxes centered at xi of size +√ +hph +√ +T +and R is the size of the boxe containing D. Then by Proposition 1, +for any xi, +P +� +max +ρ⩽k≤T−�r +��� 1 +√ +k +�r+k +� +t=�r+1 +� +Kh(xi − Xt) − +� +Kh(xi − z)dFt(z) +���� ≥ C +� +log T +hp +� +⩽ T −p−3, +(53) +49 + +by an union bound argument, +P +� +max +ρ⩽k≤T−�r sup +x∈A +��� 1 +√ +k +�r+k +� +t=�r+1 +� +Kh(x − Xt) − +� +Kh(x − z)dFt(z) +���� ≥ C +� +log T +hp +� +⩽ T −p−3|A|. +(54) +Let I1,1 = {maxρ⩽k≤T−�r supx∈A +��� 1 +√ +k +��r+k +t=�r+1 +� +Kh(x − Xt) − +� +Kh(x − z)dFt(z) +����}. For any x ∈ Rp, +there exist xi ∈ A such that +max +ρ⩽k≤T−�r +��� 1 +√ +k +�r+k +� +t=�r+1 +� +Kh(x − Xt) − +� +Kh(x − z)dFt(z) +���� +≤ +max +ρ⩽k≤T−�r +��� 1 +√ +k +�r+k +� +t=�r+1 +� +Kh(xi − Xt) − +� +Kh(xi − z)dFt(z) +���� ++ +max +ρ⩽k≤T−�r +��� 1 +√ +k +�r+k +� +t=�r+1 +� +Kh(x − Xt) − Kh(xi − Xt) +���� ++ +max +ρ⩽k≤T−�r +��� 1 +√ +k +�r+k +� +t=�r+1 +� � +Kh(xi − z)dFt(z) − +� +Kh(x − z)dFt(z) +���� +≤ +max +ρ⩽k≤T−�r sup +x∈A +��� 1 +√ +k +�r+k +� +t=�r+1 +� +Kh(xi − Xt) − +� +Kh(xi − z)dFt(z) +���� ++ +max +ρ⩽k≤T−�r +��� 1 +√ +k +�r+k +� +t=�r+1 +� +Kh(x − Xt) − Kh(xi − Xt) +���� ++ +max +ρ⩽k≤T−�r +��� 1 +√ +k +�r+k +� +t=�r+1 +� � +Kh(xi − z)dFt(z) − +� +Kh(x − z)dFt(z) +���� +=I1,1 + I1,2 + I1,3 +The term I1,2 is bounded as followed. +max +ρ⩽k≤T−�r +��� 1 +√ +k +�r+k +� +t=�r+1 +� +Kh(x − Xt) − Kh(xi − Xt) +���� +≤ +max +ρ⩽k≤T−�r +1 +√ +k +�r+k +� +t=�r+1 +���Kh(x − Xt) − Kh(xi − Xt) +��� +≤ +max +ρ⩽k≤T−�r +1 +√ +k +�r+k +� +t=�r+1 +|x − xi| +hp+1 +≤ +max +ρ⩽k≤T−�r +1 +√ +k +�r+k +� +t=�r+1 +√ +hph√p +√ +Thp+1 ≤ +√p +√ +hp +50 + +For the term I1,3, since the random variables {Kh(x − Xt)}T +t=1 have bounded expected value for +any x ∈ Rp +max +ρ⩽k≤T−�r +��� 1 +√ +k +�r+k +� +t=�r+1 +� � +Kh(xi − z)dFt(z) − +� +Kh(x − z)dFt(z) +���� +≤ +max +ρ⩽k≤T−�r +��� 1 +√ +k +�r+k +� +t=�r+1 +�√ +hph2C√p +hp+1√ +T +���� ≤ 2C√p +√ +hp . +Thus, +max +ρ⩽k≤T−�r sup +x∈Rp +��� 1 +√ +k +�r+k +� +t=�r+1 +� +Kh(x − Xt) − +� +Kh(x − z)dFt(z) +���� +≤I1,1 + C2 +√p +√ +hp +From here, +P +� +I1 > C1 +� +log T +hp ++ C2√p +√ +hp +� +≤P +� +I1,1 + I1,2 + I1,3 > C1 +� +log T +hp ++ C2√p +√ +hp +� +(55) +≤P +� +I1,1 +� +≤ T −p−3|A| = T −p−2(R +√ +T/ +√ +hph)p +(56) +≤T −p−3(R +√ +T +√ +TT +1 +p )p = RpT −2 +(57) +Finally, we analyze the term I2. By the adaptive assumption, the following is satisfied, +max +ρ⩽k⩽T−�r sup +x∈Rp +��� 1 +√ +k +r+k +� +t=�r+1 +� � +Kh(x − z)dFz(z) − ft(x) +���� +≤ +max +ρ⩽k⩽T−�r +1 +√ +k +r+k +� +t=�r+1 +sup +x∈Rp +��� +� +Kh(x − z)dFz(z) − ft(x) +��� +≤ +max +ρ⩽k⩽T−�r +1 +√ +k +r+k +� +t=�r+1 +C2hr +≤C2 +√ +Thr +We conclude the bound for event A2. We conclude the bound for event A2. Next, to derive the +bound for event A1, by definition of �F s,e +t,h and �fs,e +t +, we have that +���� �F s,e +t,h (x) − �fs,e +t +(x) +���� ≤ +���� +� +e − t +(e − s)(t − s) +t +� +l=s+1 +(Fl,h(x) − fl,h(x)) +���� ++ +���� +� +t − s +(e − s)(e − t) +e +� +l=t+1 +(Fl,h(x) − fl,h(x)) +����. +51 + +Then, we observe that, +� +e − t +(e − s)(t − s) ≤ +� +1 +t − s if s ≤ t, and +� +t − s +(e − s)(e − t) ≤ +� +1 +e − t if t ≤ e. +Therefore, +X = +e−ρ +max +t=s+ρ+1 +���� �F (s,e +t,h (x) − �fs,e +t +(x) +���� ≤ +e−ρ +max +t=s+ρ+1 +���� +� +1 +t − s +t +� +l=s+1 +� +Fl,h(x) − {fl,h(x) +����� ++ +e−ρ +max +t=s+ρ+1 +���� +� +1 +e − t +e +� +l=t+1 +� +Fl,h(x) − fl,h(x) +����� = X1 + X2. +Finally, letting λ = 2C1 +� +log T +hp ++ 2C2√p +√ +hp ++ 2C2 +√ +Thr, we get that +P(X ≥ λ) ≤P(X1 + X2 ≥ λ +2 + λ +2 ) +≤P(X1 ≥ λ +2 ) + P(X2 ≥ λ +2 ) +≤2RpT −2, +where the last inequality follows from above. This concludes the bound for A1. +Remark 1. On the events (A1)c and, (A2)c, by Assumption 1, we have that +e−ρ +max +t=s+ρ+1 || �F s,e +t,h (x) − �fs,e +t +(x)||L2 ≤ +e−ρ +max +t=s+ρ+1 +�CX sup +x∈Rp +���� �F s,e +t,h (x) − �fs,e +t +(x) +���� +≤2 �CX C +� +log T +hp ++ 2 �CX C1√p +hp ++ 2 �CX C2 +√ +Thr +where �CX is the volume of the set X. Moreover, using inequality (55), we have that +max +ρ⩽k≤T−�r +��� +��� 1 +√ +k +�r+k +� +t=�r+1 +� +Kh(· − Xt) − +� +Kh(· − z)dFt(z) +���� +��� +L2 +≤CX +max +ρ⩽k≤T−�r sup +x∈Rp +��� 1 +√ +k +�r+k +� +t=�r+1 +� +Kh(x − Xt) − +� +Kh(x − z)dFt(z) +���� +=Op +�� +log T +hp +� +. +52 + +F +α-mixing condition +A process (Xt, t ∈ Z) is said to be α-mixing if +αk = sup +t∈Z +α(σ(Xs, s ≤ t), σ(Xs, s ≥ t + k)) −→k→∞ 0. +The strong mixing, or α-mixing coefficient between two σ-fields A and B is defined as +α(A, B) = +sup +A∈A,B∈B +|P(A ∩ B) − P(A)P(B)|. +Suppose X and Y are two random variables. Then for positive numbers p−1 + q−1 + r−1 = 1, it +holds that +| Cov(X, Y )| ≤ 4∥X∥Lp∥Y ∥Lq{α(σ(X), σ(Y ))}1/r. +Let +� +Zt +�∞ +t=−∞ be a stationary time series vectors. Denote the alpha mixing coefficients of k to be +α(k) = α +� +σ +� +. . . , Zt−1, Zt +� +, σ +� +Zt+k, Zt+k+1, . . . +�� +. +Note that the definition is independent of t. +F.1 +Maximal Inequality +The unstationary version of the following lemma is in Lemma B.5. of Kirch (2006). +Lemma 1. Suppose +� +yi +�∞ +i=1 is a stationary alpha-mixing time series with mixing coefficient α(k) +and that E +� +yi +� += 0. Suppose that there exists δ, ∆ > 0 such that +E +����yi +��� +2+δ+∆� +≤ D1 +and +∞ +� +k=0 +(k + 1)δ/2α(k)∆/(2+δ+∆) ≤ D2. +Then +E +� +max +k=1,...,n +��� +k +� +i=1 +yi +��� +2+δ� +≤ Dn(2+δ)/2, +where D only depends on δ and the joint distribution of +� +yi +�∞ +i=1. +Proof. This is Lemma B.8. of Kirch (2006). +Lemma 2. Suppose that there exists δ, ∆ > 0 such that +E +����yi +��� +2+δ+∆� +≤ D1 +53 + +and +∞ +� +k=0 +(k + 1)δ/2α(k)∆/(2+δ+∆) ≤ D2. +Then it holds that for any d > 0, 0 < ν < 1 and x > 0, +P +� +max +k∈[νd,d] +��� �k +i=1 yi +��� +√ +k +≥ x +� +≤ Cx−2−δ, +where C is some constant. +Proof. Let +S∗ +d = max +k=1,...,d +��� +k +� +i=1 +yi +���. +Then Lemma 1 implies that +���S∗ +d +��� +L2+δ +≤ C1d1/2 +Therefore it holds that +P +���� S∗ +d +√ +d +��� ≥ x +� += P +���� S∗ +d +√ +d +��� +2+δ +≥ x2+δ� +≤ C1x−2−δ. +Observe that +���S∗ +d +��� +√ +d += max +k=1,...,d +��� �k +i=1 yi +��� +√ +d +≥ max +k∈[νd,d] +��� �k +i=1 yi +��� +√ +d +≥ max +k∈[νd,d] +��� �k +i=1 yi +��� +� +k/ν +Therefore +P +� +max +k∈[νd,d] +��� +�k +i=1 yi +��� +√ +k +≥ x/√ν +� +≤ P +���� S∗ +d +√ +d +��� ≥ x +� +≤ C1x−2−δ, +which gives +P +� +max +k∈[νd,d] +��� �k +i=1 yi +��� +√ +k +≥ x +� +≤ C2x−2−δ. +Lemma 3. Let ν > 0 be given. Under the same assumptions as in Lemma 1, for any 0 < a < 1 it +holds that +P +���� +r +� +i=1 +yi +��� ≤ C +a +√r{log(rν) + 1} for all r ≥ 1/ν +� +≥ 1 − a2, +where C is some absolute constant. +Proof. Let s ∈ Z+and Ts = +� +2s/ν, 2s+1/ν +� +. By Lemma 3, for all x ≥ 1, +P +� +sup +r∈Ts +��� +�r +i=1 yi +��� +√r +≥ x +� +≤ C1x−2−δ ≤ C1x−2. +54 + +Therefore by a union bound, for any 0 < a < 1, +P +� +∃s ∈ Z+ : sup +r∈Ts +��� �r +i=1 yi +��� +√r +≥ +√C1 +a +(s + 1) +� +≤ +∞ +� +s=0 +a2 +(s + 1)2 = a2π2/6. +For any r ∈ +� +2s/ν, 2s+1/ν +� +, s ≤ log(rν)/ log(2), and therefore +P +� +∃s ∈ Z+ : sup +r∈Ts +��� +�r +i=1 yi +��� +√r +≥ +√C1 +a +�log(rν) +log(2) + 1 +�� +≤ a2π2/6. +Equation (2) directly gives +P +� +sup +r∈Ts +��� �r +i=1 yi +��� +√r +≥ C +a {log(rν) + 1} +� +≤ a2. +F.2 +Central Limit theorem +Below is the central limit theorem for α-mixing random variable. We refer to Doukhan (1994) for +more details. +Lemma 4. Let +� +Zt +� +be a centred α-mixing stationary time series. Suppose for the mixing coeffi- +cients and moments, for some δ > 0 it holds +∞ +� +k=1 +αδ/(2+δ) +k +< ∞, +E +����Z1 +��� +2+δ +< ∞ +� +. +Denote Sn = �n +t=1 Zt and σ2 +n = E +����Sn +��� +2� +. Then +S⌊nt⌋ +σn +→ W(t), +where convergence is in Skorohod topology and W(t) is the standard Brownian motion on [0, 1]. +55 + +G +Additional Technical Results +Lemma 5. Let J be defined as in Definition 2 and suppose Assumption 1 e holds. Denote +ζk = 9 +10 min{ηk+1 − ηk, ηk − ηk−1} k ∈ {1, ..., K}. +Then for each change point ηk there exists a seeded interval Ik = (sk, ek] such that +a. Ik contains exactly one change point ηk; +b. min{ηk − sk, ek − ηk} ≥ 1 +16ζk; and +c. max{ηk − sk, ek − ηk} ≤ ζk; +Proof. These are the desired properties of seeded intervals by construction. The proof is the same +as theorem 3 of Kov´acs et al. (2020) and is provided here for completeness. +Since ζk = Θ(T), by construction of seeded intervals, one can find a seeded interval (sk, ek] = +(ck − rk, ck + rk] such that (ck − rk, ck + rk] ⊆ (ηk − ζk, ηk + ζk], rk ≥ ζk +4 and |ck − ηk| ≤ 5rk +8 . So +(ck − rk, ck + rk] contains only one change point ηk. In addition, +ek − ηk = ck + rk − ηk ≥ rk − |ck − ηk| ≥ 3rk +8 +≥ 3ζk +32 , +and similarly ηk − sk ≥ 3ζk +32 , so b holds. Finally, since (ck − rk, ck + rk] ⊆ (ηk − ζk, ηk + ζk], it holds +that ck + rk ≤ ηk + ζk and so +ek − ηk = ck + rk − ηk ≤ ζk. +Lemma 6. Let {Xi}T +i=1 be random grid points sampled from a common density function ft : Rp → +R, satisfying Assumption 1-a and -b. Under Assumption (2), the density estimator of the sampling +distribution µ, +�ft(x) = 1 +T +T +� +t=1 +Kh(x − Xi), +x ∈ Rp, +satisfies, +|| �fT − ft||L∞ = Op +��log(T) +T +� +2r +2r+p � +. +(58) +The verification of these bounds can be found in many places in the literature. See for example +Yu (1993) and Tsybakov (2009). +Remark 2. Even more, by Assumption 1, +|| �fT − ft||L2 ≤ CX || �fT − ft||L∞ = O +��log(T) +T +� +2r +2r+p � +(59) +with high probability. Therefore, given that +κ = +|| +� +ηk+1−ηk +(ηk+1−ηk−1)(ηk−ηk−1) +�ηk +i=ηk−1+1 fi − +� +(ηk−ηk−1) +(ηk+1−ηk−1)(ηk+1−η k) �ηk+1 +i=ηk+1 fi||L2 +� +(ηk−ηk−1)(ηk+1−ηk) +ηk+1−ηk−1 +(60) +56 + +and (7), by triangle inequality, (59) and the fact that ∆ = Θ(T), +|κ − �κ| = Op +��log(T) +T +� +2r +2r+p � +. +From here, and Assumption 2, if h1 = O(κ +1 +r ) and h2 = O(�κ +1 +r ), we conclude that +||Ft,h1 − Ft,h2||2 +L2 = O +�|κ − �κ| +κ +p +r +1 +� +. +In fact, +||Ft,h1 − Ft,h2||2 +L2 += +� +Rp( 1 +hp +1 +K(x − Xt +h1 +) − 1 +hp +2 +K(x − Xt +h2 +))2dx += +� +Rp +� 1 +hp +1 +K(x − Xt +h1 +) +�2 +− 2 1 +hp +1 +K(x − Xt +h1 +) 1 +hp +2 +K(x − Xt +h2 +) + +� 1 +hp +2 +K(x − Xt +h2 +) +�2 +dx. +Now, we analyze the two following terms, +I1 = +� +Rp +� 1 +hp +1 +K(x − Xt +h1 +) +�2 +− 1 +hp +1 +K(x − Xt +h1 +) 1 +hp +2 +K(x − Xt +h2 +)dx +and +I2 = +� +Rp +� 1 +hp +2 +K(x − Xt +h2 +) +�2 +− 1 +hp +1 +K(x − Xt +h1 +) 1 +hp +2 +K(x − Xt +h2 +)dx. +For I1, letting u = x−Xt +h1 , we have that +� +Rp +� 1 +hp +1 +K(x − Xt +h1 +) +�2 +dx = +� +Rp +1 +hp +1 +� +K(u) +�2 +du +and, letting v = x−Xt +h2 , we have that +� +Rp +1 +hp +1 +K(x − Xt +h1 +) 1 +hp +2 +K(x − Xt +h2 +)dx = +� +Rp +1 +hp +1 +K(vh2 +h1 +)K(v)dv. +Therefore, by Assumption 2 and the Mean Value Theorem, +I1 = 1 +hp +1 +� +Rp K(v) +� +K(v) − K(vh2 +h1 +) +� +dv ≤C 1 +hp +1 +���1 − h2 +h1 +��� +� +Rp K(v)||v||dv +≤C1 +|h1 − h2| +hp+1 +1 +=O +�|κ − �κ| +κ +p+1 +r +κ +1 +r −1� += O +�|κ − �κ| +κ +p +r +1 +� +. +Similarly, we have, +I2 = +� +Rp +� 1 +hp +2 +K(x − Xt +h2 +) +�2 +− 1 +hp +1 +K(x − Xt +h1 +) 1 +hp +2 +K(x − Xt +h2 +)dx += 1 +hp +2 +� +Rp K(v) +� +K(v) − K(vh1 +h2 +) +� +dv ≤ C 1 +hp +2 +���1 − h1 +h2 +��� +� +Rp K(v)||v||dv = O +�|κ − �κ| +κ +p +r +1 +� +. +57 + +G.1 +Multivariate change point detection lemmas +We present some technical results corresponding to the generalization of the univariate CUSUM to +the Multivariate case. For more details, we refer the interested readers to Padilla et al. (2021) and +Wang et al. (2020). +Let {Xt}T +t=1 ⊂ Rp a process with unknown densities {ft}T +t=1. +Assumption 5. We assume there exist {ηk}K +k=1 ⊂ {2, ..., T} with 1 = η0 < η1 < ... < ηk ≤ T < +ηK+1 = T + 1, such that +ft ̸= ft+1 if and only if t ∈ {η1, ..., ηK}, +(61) +Assume +min +k=1,...,K+1(ηk − ηk−1) ≥ ∆ > 0, +0 < ||fηk+1 − fηk||L∞ = κk for all k = 1, . . . , K. +In the rest of this section, we use the notation +�f(s,e] +t +(x) = +� +e − t +(e − s)(t − s) +t +� +j=s+1 +fj(x) − +� +t − s +(e − s)(e − t) +e +� +j=t+1 +fj(x), +for all 0 ≤ s < t < e ≤ T and x ∈ Rp. +Lemma 7. If [s, e] contain two and only two change points ηr and ηr+1, then +sup +s≤t≤e +|| �fs,e +t +||L2 ≤ √e − ηr+1||fr+1 − fr||L2 + √ηr − s||fr − fr−1||L2. +Proof. This is Lemma 15 in Wang et al. (2020). Consider the sequence +� +gt +�e +t=s+1 be such that +gt = +� fηk, +if s + 1 ≤ t < ηk, +ft, +if +ηk ≤ t ≤ e. +For any t ≥ ηk, +�fs,e +t +− �gs,e +t += +� +e − t +(e − s)(t − s) +� +t +� +i=s+1 +fi − +ηk +� +i=s+1 +fηk − +t +� +i=ηk+1 +fi +� +− +� +t − s +(e − s)(e − t) +� +e +� +i=t+1 +fi − +e +� +i=t+1 +fi +� += +� +e − t +(e − s)(t − s) +� +ηk − s +�� +fηk − fηk−1 +� +. +So for t ≥ ηk, || �fs,e +t +− �gs,e +t ||L2 ≤ √ηk − sκk. Since sups≤t≤e || �fs,e +t +||L2 = max +� +|| �fs,e +ηk ||L2, || �fs,e +ηk+1||L2 +� +, +and that +max +� +|| �fs,e +ηk ||L2, || �fs,e +ηk+1||L2 +� +≤ sup +s≤t≤e +||�gs,e +t ||L2 + √ηk − sκk +≤ +� +e − ηk+1κk+1 + √ηr − sκk +where the last inequality follows form the fact that gt has only one change point in [s, e]. +58 + +Lemma 8. Suppose e − s ≤ CR∆, where CR > 0 is an absolute constant, and that +ηk−1 ≤ s ≤ ηk ≤ . . . ≤ ηk+q ≤ e ≤ ηk+q+1, +q ≥ 0 +Denote +κs,e +max = max +� +sup +x∈Rp +���fηp(x) − fηp−1(x) +��� : k ≤ p ≤ k + q +� +. +Then for any k − 1 ≤ p ≤ k + q, it holds that +sup +x∈Rp +��� +1 +e − s +e +� +i=s+1 +fi(x) − fηp(x) +��� ≤ CRκs,e +max. +Proof. This is Lemma 18 in Wang et al. (2020). Since e − s ≤ CR∆, the interval [s, e] contains at +most CR + 1 change points. Observe that +��� +��� +1 +e − s +e +� +i=s +fi − fηp +��� +��� +L∞ += +1 +e − s +��� +��� +ηk +� +i=s +� +fηk−1 − fηp +� ++ +ηk+1 +� +i=ηk+1 +� +fηk − fηp +� ++ . . . + +e +� +i=ηk+q+1 +� +fηk+q − fηp +���� +��� +L∞ +≤ +1 +e − s +ηk +� +i=s +|p − k|κs,e +max + +ηk+1 +� +i=ηk+1 +|p − k − 1|κs,e +max + . . . + +e +� +i=ηk+q+1 +|p − k − q − 1|κs,e +max +≤ +1 +e − s +e +� +i=s +� +CR + 1 +� +κs,e +max, +where +���p1 − p2 +��� ≤ CR + 1 for any ηp1, ηp2 ∈ [s, e] is used in the last inequality. +Lemma 9. Let (s, e) ⊂ (0, n) contains two or more change points such that +ηk−1 ≤ s ≤ ηk ≤ . . . ≤ ηk+q ≤ e ≤ ηk+q+1, +q ≥ 1 +If ηk − s ≤ c1∆, for c1 > 0, then +��� +��� �fs,e +ηk +��� +��� +L∞ ≤ √c1 +��� +��� �fs,e +ηk+1 +��� +��� +L∞ + 2κk +√ηk − s +Proof. This is Lemma 20 in Wang et al. (2020). Consider the sequence +� +gt +�e +t=s+1 be such that +gt = +� +fηr+1, +s + 1 ≤ t ≤ ηk, +ft, +ηk + 1 ≤ t ≤ e +For any t ≥ ηr, it holds that +|| �fs,e +ηk − �gs,e +ηk ||L∞ = +��� +��� +� +(e − s) − t +(e − s)(t − s) +� +ηk − s +�� +fηk+1 − fηk +���� +��� +L∞ ≤ √ηk − sκk. +59 + +Thus, +|| �fs,e +ηk ||L∞ ≤ ||�gs,e +ηk ||L∞ + √ηk − sκk ≤ +� +� +� +� +� +� +ηk − s +�� +e − ηk+1 +� +� +ηk+1 − s +�� +e − ηk +�||�gs,e +ηk+1||L∞ + √ηk − sκk +≤ +� +c1∆ +∆ ||�gs,e +ηk+1||L∞ + √ηk − sκk ≤ √c1|| �fs,e +ηk+1||L∞ + 2√ηk − sκk, +where the first inequality follows from the observation that the first change point of gt in (s, e) is +at ηk+1. +Lemma 10. Under Assumption 5, for any interval (s, e) ⊂ (0, T) satisfying +ηk−1 ≤ s ≤ ηk ≤ . . . ≤ ηk+q ≤ e ≤ ηk+q+1, +q ≥ 0. +Let +b ∈ arg max +t=s+1,...,e +sup +x∈Rp +��� �f(s,e] +t +(x) +���. +Then b ∈ +� +η1, . . . , ηK +� +. For any fixed z ∈ Rp, if �f(s,e] +t +(z) > 0 for some t ∈ (s, e), then �f(s,e] +t +(z) is ei- +ther strictly monotonic or decreases and then increases within each of the interval +� +s, ηk +� +, +� +ηk, ηk+1 +� +, . . . , +� +ηk+q, e +� +. +Proof. We prove this by contradiction. Assume that b /∈ {η1, . . . , ηK}. Let z1 ∈ arg max +x∈Rp +��� ¯fs,e +b (x) +���. +Due to the definition of b, we have +b ∈ arg max +t=s+1,...,e +��� �f(s,e] +t +� +z1 +����. +It is easy to see that the collection of change points {ft(z1)}e +t=s+1 is a subset of the change points +of {f}e +t=s+1. Then, from Lemma 2.2 in Venkatraman (1992) that +�f(s,e] +b +� +z1 +� +< +max +j∈{k,...,k+q} +�f(s,e] +ηj +� +z1 +� +≤ +max +t=s+1,...,e sup +x∈Rp +��� �f(s,e] +t +(x) +��� +which is a contradiction. +Recall that in Algorithm 1, when searching for change points in the interval (s, e), we actually +restrict to values t ∈ +� +s + ρ, e − ρ +� +. We now show that for intervals satisfying condition SE from +Lemma 1, taking the maximum of the CUSUM statistic over +� +s+ρ, e−ρ +� +is equivalent to searching +on (s, e), when there are change points in +� +s + ρ, e − ρ +� +. +Lemma 11. Let z0 ∈ Rp, (s, e) ⊂ (0, T). Suppose that there exists a true change point ηk ∈ (s, e) +such that +min +� +ηk − s, e − ηk +� +≥ c1∆, +(62) +and +��� �f(s,e] +ηk +� +z0 +���� ≥ +� +c1/2 +� +κ∆ +√e − s, +(63) +60 + +where c1 > 0 is a sufficiently small constant. In addition, assume that +max +t=s+1,...,e +��� �f(s,e] +t +� +z0 +���� − +��� �f(s,e] +ηk +� +z0 +���� ≤ c2∆4(e − s)−7/2κ, +(64) +where c2 > 0 is a sufficiently small constant. Then for any d ∈ (s, e) satisfying +���d − ηk +��� ≤ c1∆/32, +(65) +it holds that +��� �f(s,e] +ηk +� +z0 +���� − +��� �f(s,e] +d +� +z0 +���� > c +���d − ηk +���∆ +��� �f(s,e] +ηk +� +z0 +����(e − s)−2, +(66) +where c > 0 is a sufficiently small constant, depending on all the other absolute constants. +Proof. Without loss of generality, we assume that d ≥ ηk and �fηk +� +z0 +� +≥ 0. Following the arguments +in Lemma 2.6 in Venkatraman (1992), it suffices to consider two cases: (i) ηk+1 > e and (ii) ηk+1 ≤ e +Case (i). Note that +�f(s,e] +ηk +� +z0 +� += +� +� +� +� +� +e − ηk +�� +ηk − s +� +e − s +� +fηk +� +z0 +� +− fηk+1 +� +z0 +�� +and +�f(s,e] +d +� +z0 +� += +� +ηk − s +�� +e − d +(e − s)(d − s) +� +fηk +� +z0 +� +− fηk+1 +� +z0 +�� +. +Therefore, it follows from (62) that +�f(s,e] +ηk +� +z0 +� +− �f(s,e] +d +� +z0 +� += +� +1 − +� +� +� +� +� +(e − d) +� +ηk − s +� +(d − s) +� +e − ηk +� +� +�f(s,e] +ηk +� +z0 +� +≥ c∆ +���d − ηk +���(e − s)−2 �f(s,e] +ηk +� +z0 +� +. (67) +The inequality follows from the following arguments. Let u = ηk − s, v = e − ηk and w = d − ηk. +Then +1 − +� +� +� +� +� +(e − d) +� +ηk − s +� +(d − s) +� +e − ηk +� − c∆ +���d − ηk +���(e − s)2 +=1 − +� +(v − w)u +(u + w)v − c +∆w +(u + v)2 += +w(u + v) +� +(u + w)v( +� +(v − w)u + +� +(u + w)v) +− c +∆w +(u + v)2 . +The numerator of the above equals +w(u + v)3 − c∆w(u + w)v − c∆w +� +uv(u + w)(v − w) +≥2c1∆w +� +(u + v)2 − c(u + w)v +2c1 +− c +� +uv(u + w)(v − w) +2c1 +� +≥2c1∆w +�� +1 − c/ +� +2c1 +�� +(u + v)2 − 2−1/2c/c1uv +� +> 0 +61 + +as long as +c < +√ +2c1 +4 + 1/ +�√ +2c1 +�. +Case (ii). Let g = c1∆/16. We can write +�f(s,e] +ηk +� +z0 +� += a +� +� +� +� +e − s +� +ηk − s +�� +e − ηk +�, +�f(s,e] +ηk+g +� +z0 +� += (a + gθ) +� +� +� +� +e − s +� +e − ηk − g +�� +ηk + g − s +�, +where +a = +ηk +� +j=s+1 +� +fj +� +z0 +� +− +1 +e − s +e +� +j=s+1 +fj +� +z0 +�� +θ = +a +�� +ηk + g − s +�� +e − ηk − g +� +g +� +1 +�� +ηk − s +�� +e − ηk +� − +1 +� +ηk + g − s +�� +e − ηk − g +� + +b +a√e − s +� +, +and b = �f(s,e] +ηk+g +� +z0 +� +− �f(s,e] +ηk +� +z0 +� +. To ease notation, let d − ηk = l ≤ g/2, N1 = ηk − s and N2 = +e − ηk − g. We have +El = �f(s,e] +ηk +� +z0 +� +− �f(s,e] +d +� +z0 +� += E1l +� +1 + E2l +� ++ E3l, +(68) +where +E1l = +al(g − l)√e − s +� +N1 +� +N2 + g +��� +N1 + l +�� +g + N2 − l +���� +N1 + l +�� +g + N2 − l +� ++ +� +N1 +� +g + N2 +��, +E2l = +� +N2 − N1 +�� +N2 − N1 − l +� +��� +N1 + l +�� +g + N2 − l +� ++ +�� +N1 + g +� +N2 +��� +N1 +� +g + N2 +� ++ +�� +N1 + g +� +N2 +�, +and +E3l = −bl +g +� +� +� +� +� +� +N1 + g +� +N2 +� +N1 + l +�� +g + N2 − l +�. +Next, we notice that g − l ≥ c1∆/32. It holds that +E1l ≥ c1l +���d − ηk +���∆ �f(s,e] +ηk +� +z0 +� +(e − s)−2, +(69) +where c1l > 0 is a sufficiently small constant depending on c1. As for E2l, due to (65), we have +E2l ≥ −1/2. +(70) +62 + +As for E3l, we have +E3l ≥ −c3l,1b +���d − ηk +���(e − s)∆−2 ≥ −c3l,2b +���d − ηk +���∆−3(e − s)3/2 �f(s,e] +ηk +� +z0 +� +κ−1 +(71) +≥ −c1l/2 +���d − ηk +���∆ �f(s,e] +ηk +� +z0 +� +(e − s)−2, +(72) +where the second inequality follows from (63) and the third inequality follows from (64), c3l,1, c3l,2 > +0 are sufficiently small constants, depending on all the other absolute constants. Combining (68), +(69), (70) and (71), we have +�f(s,e] +ηk +� +z0 +� +− �f(s,e] +d +� +z0 +� +≥ c +���d − ηk +���∆ �f(s,e] +ηk +� +z0 +� +(e − s)−2, +(73) +where c > 0 is a sufficiently small constant. In view of (67) and (73), the proof is complete. +Consider the following events +A((s, e], ρ, γ) = +� +max +t=s+ρ+1,...,e−ρ sup +z∈Rp | �F s,e +t,h (z) − �fs,e +t +(z)| ≤ γ +� +; +B(r, ρ, γ) = +� +max +N=ρ,...,T−r sup +z∈Rp +���� +1 +√ +N +r+N +� +t=r+1 +(Ft,h − ft) +���� ≤ γ +� +� � +max +N=ρ,...,r +���� +1 +√ +N +r +� +t=r−N+1 +sup +z∈Rp(Ft,h(z) − ft(z)) +���� ≤ γ +� +. +Lemma 12. Suppose Assumption 5 holds. Let [s, e] be an subinterval of [1, T] and contain at least +one change point ηr with min{ηr − s, e − ηr} ≥ cT for some constant c > 0. Let κs,e +max = max{κp : +min{ηp − s, e − ηp} ≥ cT}. Let +b ∈ arg +max +t=s+ρ,...,e−ρ || �F s,e +t,h ||L2. +For some c1 > 0, λ > 0 and δ > 0, suppose that the following events hold +A((s, e], ρ, γ), +(74) +B(s, ρ, γ) ∪ B(e, ρ, γ) ∪ +� +η∈{ηk}K +k=1 +B(η, ρ, γ) +(75) +and that +max +t=s+ρ,...,e−ρ || �F s,e +t,h ||L2 = || �F s,e +b,h||L2 ≥ c1κs,e +max +√ +T +(76) +If there exists a sufficiently small c2 > 0 such that +γ ≤ c2κs,e +max +√ +T +and that +ρ ≤ c2T, +(77) +then there exists a change point ηk ∈ (s, e) such that +min{e − ηk, ηk − s} > c3T +and +|ηk − b| ≤ C3 max{γ2κ−2 +k , ρ}, +where c3 is some sufficiently small constant independent of T. +63 + +Proof. Let z1 ∈ arg maxz∈Rp +��� �f(s,e] +b +(z) +���. Without loss of generality, assume that �f(s,e] +b +� +z1 +� +> 0 and +that �f(s,e] +b +� +z1 +� +as a function of t is locally decreasing at b. Observe that there has to be a change +point ηk ∈ (s, b), or otherwise �f(s,e] +b +� +z1 +� +> 0 implies that �f(s,e] +t +� +z1 +� +is decreasing, as a consequence +of Lemma 10. Thus, there exists a change point ηk ∈ (s, b) satisfying that +sup +z∈Rp +��� �f(s,e] +ηk +(z) +��� ≥ +��� �f(s,e] +ηk +� +z1 +���� > +��� �f(s,e] +b +� +z1 +���� ≥ sup +z∈Rp +��� �F (s,e] +b +(z) +��� − γ ≥ cκk +√ +∆ +(78) +where the second inequality follows from Lemma 10, the third because of the good event A, and +fourth inequalities by (76) and Assumption 1, and c > 0 is an absolute constant. Observe that +(s, e) has to contain at least one change point or otherwise supz∈R +��� �f(s,e] +ηk +(z) +��� = 0 which contradicts +(78). +Step 1. In this step, we are to show that +min +� +ηk − s, e − ηk +� +≥ min +� +1, c2 +1 +� +∆/16 +(79) +Suppose that ηk is the only change point in (s, e). Then (79) must hold or otherwise it follows from +(22) that +sup +z∈Rp +��� �fs,e +ηk (z) +��� ≤ κk +c1 +√ +∆ +4 +, +which contradicts (78). +Suppose (s, e) contains at least two change points. Then arguing by contradiction, if ηk − s < +min +� +1, c2 +1 +� +∆/16, it must be the cast that ηk is the left most change point in (s, e). Therefore +sup +z∈Rp +��� �fs,e +ηk (z) +��� ≤ c1/4 sup +z∈Rp +��� �fs,e +ηk+1(z) +��� + 2κk +√ηk − s +(80) +< c1/4 +max +s+ρ 0 +to be specified, +ηk + C3λ2 +Aκ−2 +k +< b. +(86) +We will show that this leads to the bound +���F s,e − Ps,e +b +� +F s,e���� +2 +> +���F s,e − Ps,e +ηk +� +fs,e���� +2 +, +(87) +which is a contradiction. If we can show that +2 +� +F s,e − fs,e, Ps,e +b +� +F s,e� +− Ps,e +ηk +� +fs,e�� +< +���fs,e − Ps,e +b +� +fs,e���� +2 +− +���fs,e − Ps,e +ηk +� +fs,e���� +2 +, +(88) +then (87) holds. To derive (88) from (86), we first note that min +� +e−ηk, ηk −s +� +≥ min +� +1, c2 +1 +� +∆/16 +and that +���b − ηk +��� ≤ c1∆/32 implies that +min{e − b, b − s} ≥ min +� +1, c2 +1 +� +∆/16 − c1∆/32 ≥ min +� +1, c2 +1 +� +∆/32 +(89) +As for the right-hand side of (88), we have +���fs,e − Ps,e +b +� +fs,e���� +2 +− +���fs,e − Ps,e +ηk +� +fs,e���� +2 += +� +�fs,e +ηk +� +X +� +j∗���2 +− +� +�fs,e +b +� +X +� +j∗���2 +(90) +≥ +� +�fs,e +ηk +� +X +� +j∗�� +− �fs,e +b +� +X +� +j∗������ �fs,e +ηk +� +X +� +j∗����� +(91) +On the event A ∩ B, we are to use Lemma 11. Note that (63) holds due to the fact that here we +have +��� �fs,e +ηk +� +X +� +j∗����� ≥ +��� �fs,e +b +� +X +� +j∗����� ≥ +��� �F s,e +b +� +X +� +j∗����� − γ ≥ c1κk +√ +∆ − γ ≥ +� +c1 +� +/2κk +√ +∆, +(92) +65 + +where the first inequality follows from the fact that ηk is a true change point, the second inequality +holds due to the event A, the third inequality follows from (76), and the final inequality follows +from (77). Towards this end, it follows from Lemma 11 that +| �fs,e +ηk +� +X +� +j∗����� − | �fs,e +b +� +X +� +j∗�����| > c|b − ηk|∆| �fs,e +ηk +� +X +� +j∗��� +| (e − s)−2. +(93) +Combining (90), (92) and (93), we have +���fs,e − Ps,e +b +� +fs,e���� +2 +− +���fs,e − Ps,e +ηk +� +fs,e���� +2 +≥ cc2 +1 +4 ∆2κkA2(e − s)−2���b − ηk +���. +(94) +The left-hand side of (88) can be decomposed as follows. +2 +� +F s,e − fs,e, Ps,e +b +� +F s,e� +− Ps,e +ηk +� +fs,e�� +(95) +=2 +� +F s,e − fs,e, Ps,e +b +� +F s,e� +− Ps,e +b +� +fs,e�� ++ 2 +� +Y s,e − fs,e, Ps,e +b +� +fs,e� +− Ps,e +ηk +� +fs,e�� +(96) +=(I) + 2 +� ηk−s +� +i=1 ++ +b−s +� +i=ηk−s+1 ++ +e−s +� +i=b−s+1 +�� +F s,e − fs,e� +i +� +Ps,e +b +� +fs,e� +− Ps,e +ηk +� +fs,e�� +i +(97) +=(I) + (II.1) + (II.2) + (II.3). +(98) +As for the term (I), we have +(I) ≤ 2γ2. +(99) +As for the term (II.1), we have +(II.1) = 2√ηk − s +� +1 +√ηk − s +ηk−s +� +i=1 +� +F s,e − fs,e� +i +�� +1 +b − s +b−s +� +i=1 +� +fs,e� +i − +1 +ηk − s +ηk−s +� +i=1 +� +fs,e� +i +� +. +In addition, it holds that +��� +1 +b − s +b−s +� +i=1 +� +fs,e� +i − +1 +ηk − s +ηk−s +� +i=1 +� +fs,e� +i +��� = b − ηk +b − s +��� − +1 +ηk − s +ηk−s +� +i=1 +fi +� +X +� +j∗�� ++ fηk+1 +� +X +� +j∗����� +≤ b − ηk +b − s +� +CR + 1 +� +κmax +s0,e0, +where the inequality is followed by Lemma 8. Combining with the good events, +(II.1) ≤ 2√ηk − sb − ηk +b − s +� +CR + 1 +� +κmax +s0,e0γ +(100) +≤ 2 +4 +min +� +1, c2 +1 +�∆−1/2γ +���b − ηk +��� +� +CR + 1 +� +κmax +s0,e0 +(101) +As for the term (II.2), it holds that +(II.2) ≤ 2 +����b − ηk +���γ +� +2CR + 3 +� +κmax +s0,e0 +(102) +66 + +As for the term (II.3), it holds that +(II.3) ≤ 2 +4 +min +� +1, c2 +1 +�∆−1/2γ +���b − ηk +��� +� +CR + 1 +� +κmax +s0,e0 +(103) +Therefore, combining (100), (102), (103), (94), (95) and (99), we have that (88) holds if +∆2κ2 +k(e − s)−2���b − ηk +��� ≳ max +� +γ2, ∆−1/2γ +���b − ηk +���κk, +����b − ηk +���γκk +� +(104) +The second inequality holds due to Assumption 3, the third inequality holds due to (85) and the +first inequality is a consequence of the third inequality and Assumption 3. +67 + diff --git a/btFJT4oBgHgl3EQfQixG/content/tmp_files/load_file.txt b/btFJT4oBgHgl3EQfQixG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4280d61e28058a9956c9b4160765a729eaf9274b --- /dev/null +++ b/btFJT4oBgHgl3EQfQixG/content/tmp_files/load_file.txt @@ -0,0 +1,2788 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf,len=2787 +page_content='Change point detection and inference in multivariable nonparametric models under mixing conditions Carlos Misael Madrid Padilla1 Haotian Xu2 Daren Wang3 Oscar Hernan Madrid Padilla4 Yi Yu5 1Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' University of Notre Dame 2Department of Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Pennsylvania State University 3Department of Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' University of Notre Dame 4Department of Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' University of California 5Department of Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' University of Warwick January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' 2023 Abstract This paper studies multivariate nonparametric change point localization and inference prob- lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' The data consists of a multivariate time series with potentially short range dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' The distribution of this data is assumed to be piecewise constant with densities in a H¨older class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' The change points, or times at which the distribution changes, are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' We derive the limiting distributions of the change point estimators when the minimal jump size vanishes or remains constant, a first in the literature on change point settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' We are introducing two new features: a consistent estimator that can detect when a change is happening in data with short-term dependence, and a consistent block-type long-run variance estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Numerical evidence is provided to back up our theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' 1 Introduction In this paper, we study the problem of change point detection in nonparametric settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Our model assumes that a vector of measurements is collected at every time point following a distribution that has a probability density function belonging to a H¨older function class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' The change point assumption implies that the probability density functions remain the same across time except for abrupt changes at the change points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Furthermore, our theory permits temporal dependence of the measurements, a feature not explored in prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' To be more specific, the observations {Xt}T t=1 ⊂ Rp are assumed to be an α-mixing sequence of random vectors with unknown distributions {Pt}T t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' The α-mixing coefficients, {αk}k∈Z, have an exponential-decay, αk ≤ e−2ck, k ∈ Z, (1) for a certain c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' The decay rate of αk imposes a temporal dependence between events that are separated by k time points, as is stated in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' This is a standard requirement in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='g Abadi, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Merlev`ede et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' To model the nonstationarity of sequentially observed 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='11491v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='ST] 27 Jan 2023 multivariate data, we assume that there exists K ∈ N change points, namely {ηk}K k=1 ⊂ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', T} with 1 = η0 < η1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' < ηk ≤ T < ηK+1 = T + 1, such that Pt ̸= Pt−1 if and only if t ∈ {η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' , ηK}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' (2) Our primary interest is to accurately estimate {ηk}K k=1 and perform inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' We refer to Assump- tion 1 below for detailed technical conditions on the model described by (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Nonstationary multivariate data is frequently encountered in real-world applications, including biology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Molenaar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Wolkovich and Donahue, 2021), epidemiology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Azhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2021), social science (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Kunitomo and Sato, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2022), clima- tology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Corbella and Stretch, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Heo and Manuel, 2022), finance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Herzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Schmitt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2013), neuroscience (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Frolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Gorrostieta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2019), among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Due to the importance of modeling nonstationary data in various scientific fields, this problem has received extensive attention in the statistical change point literature, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Aue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2009b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Fryzlewicz, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Cho and Fryzlewicz, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Cho, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' However, there are a few limitations in the existing works in multivariate nonparametric settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Firstly, to the best of our knowledge, temporal dependence has not been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Secondly, there is no consistent result for data with the underlying densities being as general as H¨older smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Lastly, but most importantly, the statistical task of deriving limiting distributions of the change point estimators has reportedly not been treated in the multivariate nonparametric change point literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Taking into account the aforestated limitations, this paper examines change point problems in a fully nonparametric framework, wherein the underlying distributions are only assumed to have piecewise and H¨older smooth continuous densities and the magnitudes of the distributional changes are measured by the L2-norm of the differences between the corresponding densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' In Section 2, we explain the model assumptions for multivariate time series with change points in a nonparametric setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Section 3 details the two-step change point estimation procedure, as well as the estimators at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Theoretical results, including the consistency of the preliminary estimator and the limiting distribution of the final estimator, are presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Section 5 evaluates the practical performance of the proposed procedure via various simulations and a real data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Finally, Section 6 concludes with a discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='1 Notation For any function f : Rp → R and for 1 ≤ q < ∞, define ∥f∥Lq = ( � Rp |f(x)|qdx)1/q and for q = ∞, define ∥f∥L∞ = supx∈Rp |f(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Define Lq = {f : Rp → R, ∥f∥q < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Moreover, for q = 2, define ⟨f, g⟩L2 = � Rp f(x)g(x)dx where f, g : Rp → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' For any vector s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' , sp)⊤ ∈ Np, define |s| = �p i=1 si, s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' = s1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' · · · sp!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' and the associated partial differential operator Ds = ∂|s| ∂xs1 1 ···∂x sp p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' For α > 0, denote ⌊α⌋ to be the largest integer smaller than α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' For any function f : Rp → R that is ⌊α⌋-times continuously differentiable at point x0, denote by fα x0 its Taylor polynomial of degree ⌊α⌋ at x0, which is defined as fα x0(x) = � |s|≤⌊α⌋ (x − x0)s s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Dsf(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' For a constant L > 0, let Hα(L, Rp) be the set of functions f : Rp → R such that f is ⌊α⌋-times differentiable for all x ∈ Rp and satisfy |f(x)−fα x0(x)| ≤ L|x−x0|α, for all x, x0 ∈ Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Here |x−x0| 2 is the Euclidean distance between x, x0 ∈ Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' In nonparametric statistics literature, Hα(L, Rp) is often referred to as the class of H¨older functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' We refer readers to Rigollet and Vert (2009) for detailed discussions on H¨older functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' A process {Xt}t∈Z is said to be α-mixing if αk = sup t∈Z α(σ(Xs, s ≤ t), σ(Xs, s ≥ t + k)) −→k→∞ 0, where α(A, B) = supA∈A,B∈B |P(A ∩ B) − P(A)P(B)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' for any two σ-fields A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' For two positive sequences {an}n∈N+ and {bn}n∈N+, we write an = O(bn) or an ≲ bn, if an ≤ Cbn with some constant C > 0 that does not depend on n, and an = Θ(bn) or an ≍ bn, if an = O(bn) and bn = O(an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' For a deterministic or random R-valued sequence an, write that a sequence of random variable Xn = Op(an), if limM→∞ lim supn→∞ P(|Xn| ≥ Man) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Write Xn = op(an) if lim supn→∞ P(|Xn| ≥ Man) = 0 for all M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' The convergences in distribution and probability are respectively denoted by D→ and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' →.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' 2 Model setup Detailed assumptions imposed on the model (2) are collected in Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' The data {Xt}T t=1 ⊂ Rp is generated based on model (2), satisfying (1), and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' For t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' , T, the distribution Pt has a Lebesgue density function ft : Rp → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' With r, L > 0, we assume that, ft ∈ Hr(L, X), where X is the union of the supports of all the density functions ft, with bounded Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Let gt be the joint density between X1 and Xt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' It satisfies that ||gt||L∞ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' (3) c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' The minimal spacing between two consecutive change points ∆ = minK+1 k=1 (ηk − ηk−1) satisfies that ∆ = Θ(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' For k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', K}, let κk = ||fηk − fηk+1||L2 (4) be the jump size at the kth change point and let κ = min k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=',K κk > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' The minimal spacing ∆ and the minimal jump size κ are two key parameters characterizing the change point phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Assumption 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' requires that ∆ = Θ(T), which is necessary only for our inference results in Theorem 2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Indeed, this condition may appear strong compared to the existing literature on localization, such as (Padilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Padilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' For our localization results, we can easily relax this condition to ∆ ≪ T, as stated in Padilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' To achieve this, consider increasing K in Definition 2 to broaden the coverage of the seeded intervals in MNSBS, and apply the narrowest over-threshold selection method, as described in Theorem 3 of (Kov´acs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Assumption 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' characterizes the changes in density functions through the function’s L2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' A reason to use the L2-norm is that the L2 space has an inner product structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' 3 Revolving the change point estimators, we are to conduct the estimation and inference tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' For a sequence of estimators �η1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' < �η � K ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' , T}, we are to show their consistency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' with probability tending to one as the sample size T grows unbounded, it holds that �K = K and max k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', � K |�ηk − ηk| ≤ ϵ, with lim T→∞ ϵ ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' (5) We refer to ϵ as the localization error in the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' With a consistent estimation result, we further refine {�ηk} � K k=1 and obtain {�ηk} � K k=1 satisfying that |�ηk − ηk| = Op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' We are to derive the limiting distribution of (�ηk − ηk)κ p r +2 k , T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='1 Summary of the results The contributions of this paper are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' We develop a multivariate nonparametric seeded change point detection algorithm detailed Algorithm 1, which is based on the seeded binary segmentation method (SBS), proposed in Kov´acs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' (2020), for the univariate Gaussian change in mean setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' As suggested in Kov´acs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' (2020), SBS may be adaptable to a wide range of change point detection problems, such as that found in Padilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' (2022) for Functional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' We have innovatively adapted SBS to the multivariate nonparametric setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Under the model assumptions outlined in Assumption 1 and the signal-to-noise ratio condition in Assumption 3 that κ2∆ ≳ log(T)T p 2r+p , we demonstrate that the output of Algorithm 1 is consistent, with a localization error of κ−2 k T p 2r+p log(T), for k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' , K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' We note that this localization error was obtained under the temporal dependence stated in (1) and with a more general smoothness assumption outlined in Assumption 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', which is a novel contribution to the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Based on the consistent estimators {�η} � K k=1, we construct refined estimators {�ηk} � K k=1 and derive their limiting distributions in different regimes, as detailed in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' This result is novel in the literature of nonparametric temporal dependence models, and such two-regime limiting distributions are rarely seen in the literature, with the exception of mean change under fixed- dimensional time series (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Yao, 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Yao and Au, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Bai, 1994), high-dimensional vector time series (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Kaul and Michailidis, 2021), functional time series setting (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Aue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2009a), and high-dimensional linear regression (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Extensive numerical results are presented in Section 5 to corroborate the theoretical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' The code used for numerical experiments is available upon request prior to publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' If the paper is accepted, we will include the code and instructions on how to reproduce the numerical results in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' 3 Multivariate nonparametric seeded change point estimators and their refinement In this section, we present the initial and refined change point estimators, both of which share the same building block, namely CUSUM statistics, defined in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' 4 Definition 1 (CUSUM statistics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' For any integer triplet 0 ≤ s < t < e ≤ T, let the CUSUM statistic be �F (s,e] t,h (x) = � e − t (e − s)(t − s) t � i=s+1 Fi,h(x) − � t − s (e − s)(e − t) e � i=t+1 Fi,h(x), x ∈ Rp where Ft,h(·) is a kernel estimator, Ft,h(x) = Kh(x − Xt), x ∈ Rp with the kernel function Kh(x) = 1 hp K �x h � , x ∈ Rp, accompanied with the bandwidth h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' The CUSUM statistic is a key ingredient of our algorithm and is based on the kernel estimators Ft,h(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' We would like to highlight that kernel-based change-point estimation techniques have been employed in the detection of change-points in nonparametric models in existing literature, as demonstrated in (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Arlot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Padilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Our preliminary estimator is based on SBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Such an estimator is obtained by combining the CUSUM statistic in Definition 1 with a modified version of SBS, which is based on a collection of deterministic intervals defined in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Definition 2 (Seeded intervals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Let K = ⌈CK log(T)⌉, with some sufficiently large absolute constant CK > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' For k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' , K}, let Jk be the collection of 2k − 1 intervals of length lk = T2−k+1 that are evenly shifted by lk/2 = T2−k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Jk ={(⌊(i − 1)T2−k⌋, ⌈(i − 1)T2−k + T2−k+1⌉], i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' , 2k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' The overall collection of seeded intervals is denoted as J = ∪K k=1Jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' With the CUSUM statistics and the seeded intervals as building blocks, we are now ready to present our multivariate nonparametric seeded change point detection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Algorithm 1 is proposed as a preliminary estimator for multiple change points in sequentially observed multivariate time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' It takes advantage of seeded intervals to provide a multi- scale search system and recursively uses CUSUM statistics to identify potential change points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Inputs required are observed data {Xt}T t=1, seeded intervals J , bandwidth h for constructing the CUSUM statistics, and threshold τ for detecting change points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Theoretical and numerical guidance for tuning parameters is presented in Sections 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Denote by {�ηk} � K k=1 our preliminary estimators provided by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' It has been demon- strated in various studies, such as (Rinaldo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=', 2022), that a refinement procedure can likely reduce the localization error of preliminary estimates of change points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' Thus, a refinement step is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' First, let sk = 9 10 �ηk−1 + 1 10 �ηk and ek = 9 10 �ηk+1 + 1 10 �ηk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' (6) Then, {�ηk} � K k=1 and �h ≍ h produce an estimator of κk as: �κk = ��� ��� � �ηk+1−�ηk (�ηk+1−�ηk−1)(�ηk−�ηk−1) ��ηk i=�ηk−1+1 Fi,�h − � (�ηk−�ηk−1) (�ηk+1−�ηk−1)(�ηk+1−�ηk) ��ηk+1 i=�ηk+1 Fi,�h � (�ηk−�ηk−1)(�ηk+1−�ηk) �ηk+1−�ηk−1 ��� ��� L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' (7) 5 Algorithm 1 Multivariate Nonparametric Seeded Binary Segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' MNSBS ((s, e), J , τ, h) INPUT: Sample {Xt}e t=s ⊂ Rp, collection of seeded intervals J , tuning parameter τ > 0 and bandwidth h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' initialization: If (s, e] = (0, n], set S → ∅ and set ρ → log(T)h−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' for I = (α, β] ∈ J do if I = (α, β] ⊆ (s, e] and β − α > 2ρ then bI ← arg maxα+ρ≤t≤β−ρ || �F (α,β] t,h ||L2 aI ← || �F (α,β] bI,h ||L2 else aI ← −1 end if end for I∗ ← arg maxI∈J aI if aI∗ > τ then S ← S ∪ {br∗} MNSBS((s, bI∗), J , τ, h) MNSBS((bI∗ + 1, e), J , τ, h) end if OUTPUT: The set of estimated change points S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFJT4oBgHgl3EQfQixG/content/2301.11491v1.pdf'} +page_content=' We then propose the final change points estimators, �ηk = arg min sk<η 0 in the exterior, d < 0 in the interior of M. Let +us denote the outward unit normal of M with n(x) = ∇d(x), where ∇ is the standard gradient in R3 +(see [Dem09] for more details). Let M ⊂ R3, the disjoint union of tangent spaces to M is the tangent +bundle of M: +TM := ⨿x∈MTxM = {(x, y) | x ∈ M and y ∈ TxM}. +The normal space of M in R3 at x is the orthogonal complement of the tangent space TxM, namely +NxM = +� +TxR3�⊥ . The normal bundle is a smooth embedded submanifold of R3 × R3 of dimension 3, +NM := ⨿x∈MNxM = {(x, z) | x ∈ M and z ∈ NxM}. +Let δ : M −→ R+ be a positive, continuous function. Consider the following open tubular neighborhood +of the normal bundle: +Nδ = {(x, yx) | ∥yx∥ < δ (x)}, +where yx is normal vector attached at x. The map F : NM −→ R3 (x, y) �→ x + y is smooth and there +exists a δ such that the restriction F|Nδ becomes a diffeomorphism onto its image [Lee13]. Consequently, +2 + +Nδ = F (Nδ) is a 3−dimensional, open, smooth, embedded submanifold of R3 that forms a tubular +neighborhood of M. Since a global parameterization of the whole surface is computationally difficult. +We will assume that we have a polyhedral surfaces Mh in Euclidean three-space which is defined to be +a compact subset Mh ⊆ R3 and is homeomorphic to a smooth connected, orientable two-dimensional +closed hypersurface M. This discrete surface is composed of finitely many triangles such that each edge +is contained in a certain (affine) line and each face is contained in a certain (affine) plane: +Mh = +n� +i=1 +Ti, +T 1 +h = +n� +i=1 +{Ti}, +where each triangle Ti is parameterized over a reference and T 1 +h is a collection of flat triangles with +mesh size h = maxTi∈T 1 +h diam +� +Ti +� +. The collection T 1 +h is called a conforming triangulation if for any +Ti, Tj ∈ T 1 +h with Ti ̸= Tj the intersection Ti ∩ Tj is either empty or a proper k-sub-simplex of Ti (k < 2). +Let assume that Mh is contained in the tubular neighborhood Nδ. Under these conditions, we define a +unique nonlinear closest point projection map: +π : Nδ ⊇ Mh −→ M ⊂ R3 +(2.1) +of the form +π (x) = x − d (x) n (x) +which assigns to every x ∈ Mh the closest point on M, so that (x − π(x)) ⊥ Tπ(x)M, ∀x ∈ Mh. We +assume that Nδ has the additional property that for each point x ∈ Mh ⊆ Nδ there is a unique point +π(x) on M that minimizes the distance from M to x. In other words, the computation of the closest +point projection (2.1) is a local minimizer problem [BV21] in the sense: +π : Mh ∋ x �→ argmin +y∈M +∥y − x∥ , +(2.2) +where the nonlinear projection maps a point x ∈ Mh to the point on M that minimizes the distance to x. +The accuracy of standard surface integration methods is limited to only first or second order due to +piecewise linear approximations of the surface geometry and integrand. To obtain high order accuracy, +we must construct a high order approximation both to the geometry of the surface and to the integrand. +By relying on [PS22] we give now the construction of a surface Mk +h which is locally parametrised over +the reference simplex σ by polynomials of degree k and interpolates the smooth surface M. As we stated +earlier we first consider a piecewise flat representation Mh of the smooth surface having triangular faces +with vertices lying on M. To simplify the notation, the subscripts from the transformation map (1.1) +and the triangulation of Mh are dropped. For any regular simplex T ∈ T 1 +h with vertices q1, q2, q3 ∈ R3 +we define the following map +ξ : σ −→ T, +qi = ξ ( ˆpi) , +1 ≤ i ≤ 3, +with +ξ (s, t) = q1 + +� +q3 − q1 +� +s + +� +q2 − q1 +� +t +to be the affine linear parametrization which maps each vertex ˆpi of σ to the vertex qi of T. We de- +note the images of the non-vertex nodes of σ on the simplex T by ¯qi = ξ ( ˆpi) , 3 < i ≤ N (2, k), +where N (2, k) is the dimension of the vector space P2,k (σ) of bivariate polynomials of degree k. +Let Lk +1 (ˆp) , Lk +2 (ˆp) , . . . , Lk +N(2,k) (ˆp) be the local Lagrange basis functions of degree k on σ corresponding +to the nodal points ˆp1, . . . ˆpN(2,k). Set +pi := π (¯qi) = +� +π ◦ ξ +� +( ˆpi) = ψ ( ˆpi) , +where +ψ := π ◦ ξ, +3 < i ≤ N (2, n) +3 + +and define Qψ,k to be a k−th order polynomial interpolation of the mapping ψ: +Ik : C0� +σ, R +� +−→ Pk (σ) , +ψ �→ Qψ,k, +Qψ,k (ˆp) := +N(2,n) +� +i=1 +piLk +i (ˆp) +(2.3) +Figure 1: Construction of the second order approximation of the smooth surface M2 +h (blue line). A simplex of a ‘base’ +triangulation Mh (green line) is shown. The interpolation nodes, here the center ¯qi of an edge, are projected (grey line) +onto the smooth surface M (red line) via the projection π. The projected nodal points π (¯qi) and the vertices of Mh are +then interpolated, giving the second order approximation of the smooth surface M2 +h. +Now, +Qψ,k : σ → ˜T k := Qψ,k +� +σ +� +defines a polynomial mapping Qψ,k by interpolating the points pi ∈ M with the Lagrange-polynomials +Lk +1 (ˆp) , Lk +2 (ˆp) , . . . , Lk +N(2,k) (ˆp) . Thus, for every simplex T ∈ T 1 +h we compute the projection π(¯qi) and +define an isoparametric simplex ˜T k by applying Lagrange interpolation of order k to the coordinates of +the projected equidistant nodes (see Fig (1)). Furthermore, if the base-triangulation T 1 +h is fine enough, +then the map Qψ,k is a diffeomorphism. By differentiating the interpolation polynomial of the map ψ, +we obtain: +∂sQψ,k (ˆp) := +N(2,n) +� +i=1 +pi∂sLk +i (ˆp) , +∂tQψ,k (ˆp) := +N(2,n) +� +i=1 +pi∂tLk +i (ˆp) . +(2.4) +The Jacobian of the transformation is calculated using the equation (2.4). Given that Qψ,k is a diffeo- +morphism ∀ T ∈ T 1 +h , then by union of non-overlapping mapped elements: +Mk +h := +� +T∈T 1 +h +Qψ,k (T) = +� +T∈T 1 +h , ˆp∈σ +N(2,n) +� +i=1 +π +� +ξ ( ˆpi) +� +Lk +i (ˆp) . +(2.5) +Thus, we have obtained a k−th order discrete approximation Mk +h of the smooth surface M. An illus- +tration of this process for a torus and a sphere is presented in Fig (2). +3 +Accuracy of Integration +In this section, we will prove our main theorem about the accuracy of integration when replacing M by +its k−th order polygonal approximation Mk +h and show that symmetric triangles of the mesh combined +with even-degree polynomials improve global errors. This motivates the following. +4 + +pi =π(qi) +M +Mh +Mn +qi(a) Mh +(b) M4 +h +(c) Mh +(d) M3 +h +Figure 2: Lagrange parametrization for a torus and a sphere using equidistant nodes on vertices and edges. +Definition 3.1. Given a collection of flat triangles T 1 +h , a pair of triangles Tx1,x2,x3, Tx1,x4,x5 ∈ T 1 +h as in +Fig (3) are called symmetric, if they satisfy the following property +(x3 − x1) = − (x5 − x1) , +(x2 − x1) = − (x4 − x1) . +(3.1) +Figure 3: A pair of symmetric triangles +Next, we state one of our main results, which explains why even degree polynomials combined with +symmetric triangles perform better. +Theorem 3.2. Let M be a smooth closed embedded hypersurface and f ∈ Ck+2 (M, R). Consider a +piecewise linear triangulation Mh with mesh size h of the smooth surface having vertices lie on M and +let Mk +h be the k−th order approximation of the smooth surface constructed using local fittings and assume +π ∈ Ck+3 � +Mh, R3� +. Assume that the triangulation mesh is composed of symmetric triangles, then +����� +� +M +fdS − +� +Mk +h +Qf,kdS +����� +L∞ +≤ +� +Chk+2, +k ≡ 0 (mod 2) +Chk+1, +k ≡ 1 (mod 2), +(3.2) +where Qf,k : Mk +h → R is a k− th order polynomial approximating the integrand f. +5 + +X4 +X3 +x1 +X5 +X2As we approach the proof of Theorem 3.2, we need three lemmata. +Lemma 3.3. Let T be a triangle in T 1 +h and assume that ψ ∈ Ck+1 (T) and k ≥ 0, then we have +∥ψ − Qψ,k∥L∞(T) ≤ chk+1 +l,p,q≥0 +max +l+p+q=k+1 +max +(x,y,z)∈T +���∂k+1ψ (x, y, z) +∂xl∂yp∂zq +��� +(3.3) +with h = diam(T). The constant c depends on k, but it is independent of both ψ and T. +Proof. In order to prove this lemma, we will use Taylor’s formula in several variables. Let’s denote with +α = (α1, α2), where |α| = |α1| + |α2|, h := (h1, h2) = (s − 0, t − 0) and ∂α1 = ∂s, ∂α2 = ∂t. +As a first step, let’s consider the k− order interpolation of the map ψ +Qψ,k (ˆp) := +N(2,n) +� +i=1 +piLk +i (ˆp) . +Using Taylor’s formula for ψ around the point (0, 0) we obtain: +ψ (ˆp) = +� +|α|=k +�∂α +α! ψ (0) hα� ++ 1 +k! +� 1 +0 +(1 − µ)k dk+1ψ (sµ, tµ) +dk+1µ +dµ. +(3.4) +Let us denote H (ˆp) = 1 +k! +� 1 +0 (1 − µ)k dk+1ψ(sµ,tµ) +dk+1µ +dµ, by interpolating each term in the Eq. (3.4) using a +polynomial of order k and using the fact that the interpolation of the first term is exact because it is a +polynomial of degree ≤ k,i.e +� +Ik +� +|α|=k +� +∂α +α! ψ (0) hα� += � +|α|=k +� +∂α +α! ψ (0) hα�� +, we obtain +Qψ,k (ˆp) = +� +|α|=k +�∂α +α! ψ (0) hα� ++ +N(2,n) +� +i=1 +H (ˆpi) Lk +i (ˆp) . +(3.5) +Subtracting Eq.(3.5) from Eq.(3.4), we have +ψ (ˆp) − Qψ,k (ˆp) = H (ˆp) − +N(2,n) +� +i=1 +H (ˆpi) Lk +i (ˆp) . +(3.6) +The right hand side of Eq.(3.6) can be written +H (ˆp) − +N(2,n) +� +i=1 +H (ˆpi) Lk +i (ˆp) = 1 +k! +� 1 +0 +(1 − µ)kE (µ; s, t) dµ, +(3.7) +where E (µ; s, t) has the following form +E (µ; s, t) := dk+1ψ (sµ, tµ) +dk+1µ +− +N(2,n) +� +i=1 +Lk +i (ˆp) dk+1ψ (siµ, tiµ) +dk+1µ +. +(3.8) +It is important to note that the left-hand side of Eq. (3.6) is affected by the behavior of the term +E (µ; s, t), which itself is influenced by dk+1ψ(sµ,tµ) +dk+1µ +. For k = 0, we have +max +0≤µ≤1, ˆp∈σ +���� +dψ (sµ, tµ) +dµ +���� ≤ ch max +����� +∂ψ +∂x +���� +L∞ , +���� +∂ψ +∂y +���� +L∞ , +���� +∂ψ +∂z +���� +L∞ +� +, +(3.9) +6 + +where +���� +∂ψ +∂x +���� +L∞ := +max +(x,y,z)∈T +���� +∂ψ +∂x +���� +and analogously for ∂ψ +∂y , ∂ψ +∂z . Following the same line of reasoning with the higher-order derivatives of ψ, +we obtain +max +0≤µ≤1, ˆp∈σ +���� +dk+1ψ (sµ, tµ) +dk+1µ +���� ≤ chk+1 max +����� +∂ψ +∂x +���� +L∞ , +���� +∂ψ +∂y +���� +L∞ , +���� +∂ψ +∂z +���� +L∞ +� +. +(3.10) +Using (3.7), yields +H (ˆp) − +N(2,n) +� +i=1 +H (ˆpi) Lk +i (ˆp) = O +� +hk+1� +. +(3.11) +Combining Eq. (3.6) with Eq. (3.11), yields Eq. (3.3). +Next, we recall the following Lemma. +Lemma 3.4 ([Chi93]). Let k be an even integer. Let ψ (ˆp) ∈ Pk+1 be a polynomial of degree k + 1 on +σ, and let Qψ,k (ˆp) be its interpolant of degree k. Then, for each ˆp ∈ σ +� +σ +∂s (ψ (ˆp) − Qψ,k (ˆp)) dsdt = 0, +� +σ +∂t (ψ (ˆp) − Qψ,k (ˆp)) dsdt = 0. +(3.12) +Lemma 3.5. Let M be a smooth closed hypersurface and consider a piecewise linear triangulation +Mh with mesh size h of the smooth surface having vertices lie in M and let Mk +h be the k−th order +approximation of the smooth surface constructed using local fittings and assume π ∈ Ck+3 � +Mh, R3� +. +Assume that the triangulation mesh is composed of symmetric triangles, then +����� +� +M +dS − +� +Mk +h +dS +����� +L∞ +≤ +� +Chk+1, +k ≡ 1 (mod 2) +Chk+2, +k ≡ 0 (mod 2). +(3.13) +Proof. First, let us write every term in Eq. (3.13) over a reference simplex +� +M +dS = +n +� +i=1 +� +σ +gidsdt, +� +Mk +h +dS = +n +� +i=1 +� +σ +˜gidsdt. +(3.14) +In the same manner, as in Lemma 3.3, we obtain +ψ (ˆp) − Qψ,k (ˆp) = Rk+1 + Rk+2 + O +� +hk+3� +, +(3.15) +where +Rk+1 := +� +|α|=k+1 +�∂α +α! ψ (0) hα − +N(2,n) +� +i=1 +hα ∂α +α! ψ (0) Lk +i (ˆp) +� +Rk+2 := +� +|α|=k+2 +�∂α +α! ψ (0) hα − +N(2,n) +� +i=1 +hα ∂α +α! ψ (0) Lk +i (ˆp) +� +. +Analogously, an expansion can be given for the errors in the partial derivatives of ∂α1ψ (ˆp) , ∂α2ψ (ˆp), +amounting to apply the derivative on (3.15). Therefore, we have +∂α1ψ (ˆp) − ∂α1Qψ,k (ˆp) = ∂α1Rk+1 + ∂α1Rk+2 + O +� +hk+2� +(3.16) +7 + +∂α2ψ (ˆp) − ∂α2Qψ,k (ˆp) = ∂α2Rk+1 + ∂α2Rk+2 + O +� +hk+2� +. +(3.17) +Using Taylor’s theorem and expanding about (s = 0, t = 0), we obtain +∥˜gi − gi∥L∞ := +���� +� +det(∂sQT +ψ,n (ˆp) ∂tQψ,n (ˆp)) − +� +det(∂sψT (ˆp) ∂tψ (ˆp)) +���� +L∞ +(3.18) += E +� +hk+2; (xk − xj) , (xℓ − xj) +� ++ E +� +hk+3; (xk − xj) , (xℓ − xj) +� ++ O +� +hk+4� +, +where E +� +hk+2; (xk − xj) , (xℓ − xj) +� +and E +� +hk+3; (xk − xj) , (xℓ − xj) +� +, describes the collection of terms +with order k+2, k+3 in h, whose coefficients are combinations of the vertices of triangles with appropriate +indices j, ℓ, and k. For example for a triangle Tx1,x2,x3 the coefficients are combination of (x3 − x1) and +(x2 − x1) both for E (k + 2) and E (k + 3). From above computation we see that ∥˜gi −gi∥L∞ is at least of +order O +� +hk+2� +, confirming also the result obtained in [RWJG12]. However, this result can be improved +under certain conditions, i.e., an approximation of order O +� +hk+4� +can be obtained. Thus, integrating +Eq. (3.18) over a reference simplex, we have +� +σ +��� +˜gi − gi +��� +L∞ dsdt = +� +σ +E (k + 2) dsdt + +� +σ +E (k + 3) dsdt + O +� +hk+4� +. +(3.19) +At this point we notice that if k is even, then +� +σ +E (k + 2) dsdt = 0. +This is due to the Lemma 3.4 because E (k + 2) represents errors when integrating a polynomial of order +k + 1. Thus, we have +� +σ +��� +˜gi − gi +��� +L∞ dsdt = +� +σ +E (k + 3) dsdt + O +� +hk+4� +. +(3.20) +This shows that the left hand side of the Eq. (3.20) is at least of order O +� +hk+3� +for each triangle. At +this point using that the number of triangles in the surface mesh is inversely proportional to the average +area of triangle n = O +� +h−2� +, yields the first part of inequality(3.13). While if k is odd, yields +� +σ +E (k + 2) dsdt ̸= 0 +then, the left hand side of the Eq. (3.19) is at least of order O +� +hk+2� +for every T ∈ T 1 +h . It should be noted +that this theorem can be generalized and proved for d− dimensional simplices. Assuming that k is even +and writing Eq. (3.20) with respect to a pair of symmetric triangles, we obtain Tx1,x2,x3 and Tx1,x4,x5 +� +σ +��� +˜gi − gi +��� +L∞ dsdt +���� +Tx1,x2,x3 += +� +σ +E (k + 3) dsdt + O +� +hk+4� +(3.21) +� +σ +��� +˜gi − gi +��� +L∞ dsdt +���� +Tx1,x4,x5 += +� +σ +E (k + 3) dsdt + O +� +hk+4� +. +(3.22) +Now, if k is even and triangles are pairwise symmetric then the error contributed is +� +σ +��� +˜gi − gi +��� +L∞ dsdt = O +� +hk+4� +. +(3.23) +8 + +This is due to the fact that the left hand side integrand of the Eq. (3.21) and Eq. (3.22), are the +collection of terms which are of order k + 3 in h, where the coefficients are the combination of the +vertices of triangles, thus each integrand is an odd function. In general for a triangle Tx1,x2,x3 we may +write: +E +� +hk+3; − (x3 − x1) , − (x2 − x1) +� += −E +� +hk+3; (x3 − x1) , (x2 − x1) +� +. +(3.24) +In the same manner for Tx1,x4,x5 +E +� +hk+3; − (x4 − x1) , − (x5 − x1) +� += −E +� +hk+3; (x4 − x1) , (x5 − x1) +� +. +(3.25) +But using the property (3.1), Eq. (3.24) and Eq. (3.25), we obtain +E +� +hk+3; (x3 − x1) , (x2 − x1) +� ++ E +� +hk+3; (x4 − x1) , (x5 − x1) +� += 0. +Again, using the fact that the number of triangles in the surface mesh is inversely proportional to the +average area of triangle n = O +� +h−2� +and Eq. (3.23), yields the second part of inequality(3.13). +We can now prove Theorem 3.2 +Poof of theorem 3.2. As in Lemma 3.5, expanding the function f using Taylor’s formula around the +point (0, 0), yields +f (ψ (ˆp)) − Qf,k (ψ (ˆp)) = Rk+1 + O +� +hk+2� +, +(3.26) +where Rk+1 = O +� +hk+1� +and it has the form +Rk+1 := +� +|α|=k+1 +�∂α +α! f (0) hα − +N(2,n) +� +i=1 +hα ∂α +α! f (0) Lk +i (ˆp) +� +. +Integrate both side of the Eq. (3.26), we have +� +σ +(f (ψ (ˆp)) − Qf,k (ψ (ˆp))) gidsdt = +� +σ +Rk+1gidsdt + O +� +hk+4� +. +(3.27) +Using ∥gi∥L∞ = O +� +h2� +, the integral term on the right hand side is at least of order O +� +hk+3� +. If k is +even due to the presence of symmetric triangles the first term on the right-hand side integral is 0, so +we can compute the left-hand integral with an accuracy of order O +� +hk+4� +, while for k odd, we have the +following +� +σ +∥(f (ψ (ˆp)) − Qf,k (ψ (ˆp))) gi∥L∞ dsdt ≤ Chk+3. +(3.28) +Assume that Mk +h is composed of triangles ˜T k, i.e Mk +h = �n +i=1 ˜T k +i and the smooth closed surface M = +�n +i=1 Ti where n is the number of triangles. Then +� +M +fdS = +n +� +i=1 +� +Ti +fdS, +� +Mk +h +Qf,kdS = +n +� +i=1 +� +˜T k +i +Qf,kdS. +(3.29) +Let us rewrite Eq. (3.29) over a reference simplex where the quadrature rules are defined +� +M +fdS = +n +� +i=1 +� +σ +f (ψ (ˆp)) gidsdt, +� +Mk +h +Qf,kdS = +n +� +i=1 +� +σ +Qf,k (ψ (ˆp)) ˜gidsdt. +(3.30) +9 + +By making use of Lemma 3.5 and the fact that the number of triangles in the surface mesh is inversely +proportional to the average area of triangle n = O +� +h−2� +, we have +����� +� +M +fdS − +� +Mk +h +Qf,kdS +����� +L∞ +≤ +����� +n +� +i=1 +� +σ +� +f (ψ (ˆp)) gi − Qf,k (ψ (ˆp)) ˜gi +� +dsdt +����� +L∞ +≤ +n +� +i=1 +���� +� +σ +(f (ψ (ˆp)) − Qf,k (ψ (ˆp))) gi + Qf,k (ψ (ˆp)) (gi − ˜gi) dsdt +���� +L∞ +≤ +n +� +i=1 +�� +σ +∥(f (ψ (ˆp)) − Qf,k (ψ (ˆp))) gi∥L∞ dsdt + +� +σ +∥Qf,k (ψ (ˆp)) (gi − ˜gi)∥L∞ dsdt +� +≤ +� +Chk+2, +k ≡ 0 (mod 2) +Chk+1, +k ≡ 1 (mod 2). +For the last inequality, we have used Eq. (3.27) and Eq. (3.28). +4 +Results and Discussion +In order to validate our findings, we developed tests employing the Gauss-Bonnet theorem [PPCT01], +[Spi99] for a series of classic smooth closed surfaces given by the following equations: +1) Ellipsoid +x2 +a2 + y2 +b2 + z2 +c2 = 1, +a, b, c ∈ R \ {0}. +2) Torus +(x2 + y2 + z2 + R2 − r2)2 − 4R2(x2 + y2) = 0, +0 < r < R ∈ R +3) Sphere +x2 + y2 + z2 = R2, +R ∈ R \ {0}. +where the analytic expression for the Gaussian Curvature are +1) Ellipsoid +KGauss = +1 +(abc)2 +� +x2 +a4 + y2 +b4 + z2 +c4 +�2 , +a, b, c ∈ R \ {0}. +2) Torus +KGauss = +cos v +r(R+r cos v), where we used toric coordinates: +(x, y, z) = +� +(R + r cos θ) cos ϕ, (R + r cos θ) sin ϕ, r sin θ +� +, ϕ, θ ∈ [0, 2π) +. +3) Sphere +KGauss = +1 +R2 , +r ∈ R \ {0}. +In all experiments, we applied a grade-12 Gaussian quadrature rule on each triangle, with 32 quadrature +points. +In Fig. 4 and Fig 5 we show the relative errors under mesh refinement. In order to calculate the +relative error, we integrate the Gaussian curvature on the manifold and compare it with the predictions +of the Gauss-Bonnet Theorem. The convergence rates shown on the plots confirm our theoretical results. +However, as can be seen from Fig. 6, the approach presented in [PS22] has some limitations, it runs into +Runge’s phenomenon. This is due to the fact that the Lebesgue constant Λ for the set of equidistant +points grows exponentially [MS92]. To mitigate this effect, we propose an alternative approach. It is +well known that under a proper map the operations (e.g., interpolation, numerical differentiation, and +quadrature) on a triangular element can be performed on the reference square. Below we will state and +prove a theorem that we hope will pave the way for developing another powerful numerical integration +method for closed surfaces. +10 + +10 +1 +h +10 +10 +10 +8 +10 +6 +10 +4 +10 +2 +error +fdS +k +hfkdS +, k = 2 +fdS +k +hfkdS +, k = 4 +h4 +h6 +(a) Even order +10 +1 +h +10 +11 +10 +9 +10 +7 +10 +5 +10 +3 +error +fdS +k +hfkdS +, k = 3 +fdS +k +hfkdS +, k = 5 +h4 +h6 +(b) Odd order +Figure 4: Relative errors by integrating the Gaussian curvature over the torus with radii R = 2, r = 1 with the ideal +convergence lines hn. +10 +1 +h +10 +10 +10 +8 +10 +6 +10 +4 +10 +2 +error +fdS +k +hfkdS +, k = 2 +fdS +k +hfkdS +, k = 4 +h4 +h6 +(a) Even order +10 +1 +h +10 +10 +10 +8 +10 +6 +10 +4 +10 +2 +error +fdS +k +hfkdS +, k = 3 +fdS +k +hfkdS +, k = 5 +h4 +h6 +(b) Odd order +Figure 5: Relative errors by integrating the Gaussian curvature over the unit sphere with the ideal convergence lines hn. +Theorem 4.1. Let f ∈ Ck � +[−1, 1]2� +, we have the estimate +∥f (x, y) − Qf,n (x, y)∥L∞([−1,1]2) ≤ ˜C ln2 (n) n−k +� +ω +� +∂k +xf (x, y) ; 1 +n, 0 +� ++ ω +� +∂k +yf (x, y) ; 0, 1 +n +�� +. (4.1) +Remark 4.2. Based on Theorem 4.1 in future work we plan to incorporate recent advances in mul- +tivariate interpolation [HGM+20], providing a stable approach suppressing Runge’s phenomenon by +interpolating the map ψ in proper chosen, transformed Chebyshev-Lobatto nodes. +In order to prove the theorem we recall the modulus of continuity from [TIM63]. For δ1, δ2 the +modulus of continuity ω (f; δ1, δ2) is defined as: +ω (f; δ1, δ2) = +sup +|x1−x2|≤δ1, x1,x2∈[−1,1] +sup +|y1−y2|≤δ2, y1,y2∈[−1,1] +|f (x1, x2) − f (y1, y2) |. +(4.2) +The function ω (f; δ1, δ2) is semi-additive, i.e, +ω (δ1 + δ2, λ1 + λ2) ≤ ω (δ1 + λ1) + ω (δ2 + λ2) . +11 + +(a) Visualization of Gaussian Curvature for the torus. +(b) Visualization of Gaussian Curvature for the ellipsoid. +3 +4 +5 +6 +7 +8 +9 +10 +Degree of Polynomial +10 +11 +10 +10 +10 +9 +10 +8 +10 +7 +10 +6 +10 +5 +10 +4 +error +(c) torus with radii R = 2, r = 1 +3 +4 +5 +6 +7 +8 +9 +10 +Degree of Polynomial +10 +12 +10 +10 +10 +8 +10 +6 +error +(d) ellipsoid with a = b = 1, c = 0.6. +Figure 6: Relative errors by integrating the Gaussian curvature over the torus and the ellipsoid using N∆ = 2528, N∆ = 6152 +respectively. +Poof of Theorem 4.1. We consider the interpolation operator +I[−1,1]2 : C0� +[−1, 1]2, R +� +→ P2,n +with P2,n being the space of bivariate polynomials of degree n. +Let us denote with Q∗ +f,n (x, y) the +best polynomial approximation of degree n. The identity theorem for polynomials yields Q∗ +f,n (x, y) = +I[−1,1]2 +� +Q∗ +f,n (x, y) +� +, by making use of Lemma 7.4 in [HGM+20], we have +∥f (x, y) − Qf,n (x, y)∥L∞([−1,1]2) +≤ +��Q∗ +f,n (x, y) − f (x, y) +�� +L∞([−1,1]2) + +��Q∗ +f,n (x, y) − Qf,n (x, y) +�� +L∞([−1,1]2) +≤ +��Q∗ +f,n (x, y) − f (x, y) +�� +L∞([−1,1]2) + +��I[−1,1]2 +� +Q∗ +f,n (x, y) − f (x, y) +��� +L∞([−1,1]2) +≤ +� +1 + CΛ ln2 (n) +� ��f (x, y) − Q∗ +f,n (x, y) +�� +L∞([−1,1]2) . +(4.3) +At this point the multivariate version of Jackson’s inequality and semi-additivity of the modulus [TIM63], +gives +��f (x, y) − Q∗ +f,n (x, y) +�� +L∞([−1,1]2) ≤ C +� +2kn−kω +� +∂k +xf (x, y) ; 2 +n, 0 +� ++ 2kn−kω +� +∂k +yf (x, y) ; 0, 2 +n +�� +≤ C2k+1n−k +� +ω +� +∂k +xf (x, y) ; 1 +n, 0 +� ++ ω +� +∂k +yf (x, y) ; 0, 1 +n +�� +. +(4.4) +12 + +3.3e-01 +21 +K_Gauss +-1.0e+002.8e+00 +2 +1.5 +_Gauss +1 +0.5 +K +0 +-0.5 +-1.0e+00Combining inequality (4.4) with (4.3), we obtain (4.1). Additionally, it is easy to see that, for f ∈ Ck(T), +we have +∥f (x, y) − Qf,n (x, y)∥L∞(T) ≤ c ∥f (x, y) − Qf,n (x, y)∥L∞([−1,1]2) = O +� +n−k� +. +(4.5) +In light of the multivariate extension of Jackson’s theorem [BBL02], we have the following result. +Corollary 4.3. Let f ∈ Ck � +[−1, 1]2� +, we have the estimate +∥∂xf (x, y) − ∂xQf,n (x, y)∥L∞([−1,1]2) ≤ C (f; n) ln2 (n) n−(k−1) +(4.6) +and similarly +∥∂yf (x, y) − ∂yQf,n (x, y)∥L∞([−1,1]2) ≤ C (f; n) ln2 (n) n−(k−1), +(4.7) +where C is a suitable constant (with n), dependent on f(x, y) and k. +Remark 4.4. If f ∈ Ck � +[−1, 1]2� +satisfy the Dini–Lipschitz criterion in the sense that +lim +n−→∞ ln2 (n) ω +� +f (x, y) ; 1 +n, 1 +n +� += 0 +then Eq. (4.1) guarantees uniform convergence +lim +n−→∞ ∥f (x, y) − Qf,n (x, y)∥L∞([−1,1]2) = 0. +It is anticipated that with this initiative, we can develop an efficient numerical integration method +for closed surfaces. This will be discussed in a forthcoming work. +Acknowledgement +We deeply acknowledge Paul Breiding and Michael Hecht for many inspiring comments and helpful +suggestions. The research of Gentian Zavalani was partially funded by the Center of Advanced Systems +Understanding (CASUS) which is financed by Germany’s Federal Ministry of Education and Research +(BMBF) and by the Saxon Ministry for Science, Culture, and Tourism (SMWK) with tax funds on +the basis of the budget approved by the Saxon State Parliament. The research of Elima Shehu was +funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Projektnummer +445466444. +References +[BBL02] +Thomas Bagby, Len Bos, and Norman Levenberg. Multivariate simultaneous approximation. +Constructive approximation, 18(4):569–577, 2002. +[BV21] +Paul Breiding and Nick Vannieuwenhoven. The condition number of riemannian approxima- +tion problems. SIAM Journal on Optimization, 31(1):1049–1077, jan 2021. +[Chi93] +David Da-Kwun Chien. +Piecewise polynomial collocation for integral equations with a +smooth kernel on surfaces in three dimensions. The Journal of Integral Equations and Ap- +plications, 5:315–44, 1993. +13 + +[Dem09] +Alan Demlow. Higher-order finite element methods and pointwise error estimates for elliptic +problems on surfaces. SIAM Journal on Numerical Analysis, 47(2):805–827, 2009. +[Geo98] +Kurt Georg. Approximation of integrals for boundary element methods. SIAM Journal on +Scientific and Statistical Computing, 12, 08 1998. +[HGM+20] Michael Hecht, Krzysztof Gonciarz, Jannik Michelfeit, Vladimir Sivkin, and Ivo F Sbalzarini. +Multivariate interpolation in unisolvent nodes–lifting the curse of dimensionality. +arXiv +preprint arXiv:2010.10824, 2020. +[Lee13] +John M Lee. Smooth manifolds. In Introduction to smooth manifolds, pages 1–31. Springer, +2013. +[Lim88] +Elon L. Lima. The jordan-brouwer separation theorem for smooth hypersurfaces. The Amer- +ican Mathematical Monthly, 95(1):39–42, 1988. +[MS92] +T. M. Mills and Simon Jeffrey Smith. The lebesgue constant for lagrange interpolation on +equidistant nodes. Numerische Mathematik, 61:111–115, 1992. +[PPCT01] L.M.A. Pressley, A. Pressley, M. Chaplain, and J.F. Toland. Elementary Differential Geom- +etry. Springer undergraduate mathematics series. Springer, 2001. +[PS22] +Simon Praetorius and Florian Stenger. +Dune-curvedgrid - a dune module for surface +parametrization. Archive of Numerical Software, page Vol. 1 No. 1 (2022), 2022. +[RWJG12] Navamita Ray, Duo Wang, Xiangmin Jiao, and James Glimm. High-order numerical in- +tegration over discrete surfaces. +SIAM Journal on Numerical Analysis, 50(6):3061–3083, +2012. +[Spi99] +M. Spivak. +A Comprehensive Introduction to Differential Geometry. +Number v. 1 in A +Comprehensive Introduction to Differential Geometry. Publish or Perish, Incorporated, 1999. +[TIM63] +A. TIMAN. Theory of Approximation of Functions of a Real Variable. International Series +of Monographs on Pure and Applied Mathematics. Pergamon, 1963. +[YQ09] +Pinghai Yang and Xiaoping Qian. A general, accurate procedure for calculating molecular +interaction force. Journal of Colloid and Interface Science, 337(2):594–605, 2009. +14 + diff --git a/itE1T4oBgHgl3EQfNAN9/content/tmp_files/load_file.txt b/itE1T4oBgHgl3EQfNAN9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bde2b872eae0c2743ff3cc8d11fcc41b94589fc4 --- /dev/null +++ b/itE1T4oBgHgl3EQfNAN9/content/tmp_files/load_file.txt @@ -0,0 +1,449 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf,len=448 +page_content='A note on the rate of convergence of integration schemes for closed surfaces Gentian Zavalani1 and Elima Shehu2 1Center for Advanced Systems Understanding (CASUS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' D-02826 G¨orlitz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Germany 1Technische Universit¨at Dresden 2Max Planck Institute for Mathematics in the Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Leipzig,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Germany 2Osnabr¨uck University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Osnabr¨uck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Germany Abstract In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' we issue an error analysis for integration over discrete surfaces using the surface parametrization presented in [PS22] as well as prove why even-degree polynomials exhibit a higher convergence rate than odd-degree polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Additionally, we provide some numerical examples that illustrate our findings and propose a potential approach that overcomes the problems associated with the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 1 Introduction Many applications, including mathematical physics and mathematical biology [YQ09], require accurate approximation of integrals on curved surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' The concept of surface integration is a fundamental procedure in a wide range of numerical methods, including the boundary integral method, the finite element method, the surface finite element method, and the finite volume method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' As a result of piecewise linear approximations of the surface and integrand, integration over discrete surfaces (such as surface triangulation) is typically only of first- or second-order accuracy [Geo98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' To improve the order of accuracy, using the approach presented in [PS22] a polynomial approximation of the geometry of a smooth closed embedded hypersurface M is considered, where a local interpolation polynomial of a closest point projection map π for each element T ∈ T 1 h will be constructed, with T 1 h denoting a conforming triangulation which interpolates the smooth closed hypersurface M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' The mapping π is a bijective mapping from the piecewise flat surface to the smooth surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' The purpose of this paper is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' First, we consider the problem of numerical integration of a function over a discrete triangulated surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' In order to do that, we will assume that our smooth closed hypersurface is composed of non-overlapping regions Vi whose union is denoted with M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='e, M = �n i=1 Vi where each Vi is a smooth closed surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Via a proper map ψi : σ −→ Vi ⊆ M (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='1) we provide a smooth parametrization of the surface Vi in R3 such that ∂sψi(s, t) × ∂tψi(s, t) ̸= 0 at all points, where σ is a reference simplex defined as σ := {(s, t) , 0 ≤ s ≤ 1, 0 ≤ t ≤ 1 − s} in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Then, the surface integration problem becomes: � M fdS = n � i=1 � Vi fdS = N � i=1 � 1 0 � 1−s 0 f � ψi(s, t) � gi(s, t)dsdt (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='2) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='02996v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='NA] 8 Jan 2023 where gi(s, t) = � det � ∂sψT i (s, t)∂tψi(s, t) � with ∂sψi(s, t), ∂tψi(s, t) denoting the Jacobian of ψi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' In this paper, we approximate the function f and each parametrization ψi by polynomials of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Thus, we approximate the integration by evaluating N � i=1 � 1 0 � 1−s 0 Qf,n � ψi(s, t) � ˜gi(s, t)dsdt , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='3) where ˜gi(s, t) = � det � ∂sQT ψi,n(s, t)∂tQψi,n(s, t) � , with Qψi,n denoting a n−th order polynomial approx- imating the map ψi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Second, this study was also motivated by [RWJG12], where a stabilized least squares approximation, a blending procedure based on linear shape functions, and high-degree quadrature rules are combined to produce a method for integration over discrete surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' This method has an accuracy order of O � hk + h5� ([RWJG12], Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Based on their numerical experiments, they observe that even- degree polynomials exhibit a higher convergence rate than predicted by their theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' The aim of this paper is to publicize this phenomenon and expand our understanding of it with the aid of new numerical experiments (section 4) and a new Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='2, which also provides a detailed analysis of errors for integration over discrete triangulated surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Among the methods that can benefit from such an error analysis are the finite element method, the surface finite element method, the boundary integral method, and the finite volume methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' The article is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Section 2 gives a mathematical description of the surface parametrization introduced in [PS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' In Section 3, we analyze the error using this approach and justify why even-degree polynomials exhibit higher convergence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' In section 4 we conclude with some numerical examples to illustrate our findings and propose an alternative approach that overcomes some of the disadvantages of the actual method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Codes for reproducing the results of this manuscript are summarized and available in the repository https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='com/zavala92/code_paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 2 Polynomial Approximation for Closed Surfaces Let M be a smooth connected, orientable closed hypersurface, with smooth hypersurface we mean a C∞ topological manifold that is second-countable, Hausdorff, and locally Euclidean of dimension 2, which is embedded in an ambient space of dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' According to the Jordan–Brouwer decomposition theorem [Lim88], M divides R3 into an interior and an exterior domain and we denote by d the signed distance function to M oriented in such a way that d > 0 in the exterior, d < 0 in the interior of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Let us denote the outward unit normal of M with n(x) = ∇d(x), where ∇ is the standard gradient in R3 (see [Dem09] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Let M ⊂ R3, the disjoint union of tangent spaces to M is the tangent bundle of M: TM := ⨿x∈MTxM = {(x, y) | x ∈ M and y ∈ TxM}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' The normal space of M in R3 at x is the orthogonal complement of the tangent space TxM, namely NxM = � TxR3�⊥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' The normal bundle is a smooth embedded submanifold of R3 × R3 of dimension 3, NM := ⨿x∈MNxM = {(x, z) | x ∈ M and z ∈ NxM}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Let δ : M −→ R+ be a positive, continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Consider the following open tubular neighborhood of the normal bundle: Nδ = {(x, yx) | ∥yx∥ < δ (x)}, where yx is normal vector attached at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' The map F : NM −→ R3 (x, y) �→ x + y is smooth and there exists a δ such that the restriction F|Nδ becomes a diffeomorphism onto its image [Lee13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Consequently, 2 Nδ = F (Nδ) is a 3−dimensional, open, smooth, embedded submanifold of R3 that forms a tubular neighborhood of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Since a global parameterization of the whole surface is computationally difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' We will assume that we have a polyhedral surfaces Mh in Euclidean three-space which is defined to be a compact subset Mh ⊆ R3 and is homeomorphic to a smooth connected, orientable two-dimensional closed hypersurface M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' This discrete surface is composed of finitely many triangles such that each edge is contained in a certain (affine) line and each face is contained in a certain (affine) plane: Mh = n� i=1 Ti, T 1 h = n� i=1 {Ti}, where each triangle Ti is parameterized over a reference and T 1 h is a collection of flat triangles with mesh size h = maxTi∈T 1 h diam � Ti � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' The collection T 1 h is called a conforming triangulation if for any Ti, Tj ∈ T 1 h with Ti ̸= Tj the intersection Ti ∩ Tj is either empty or a proper k-sub-simplex of Ti (k < 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Let assume that Mh is contained in the tubular neighborhood Nδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Under these conditions, we define a unique nonlinear closest point projection map: π : Nδ ⊇ Mh −→ M ⊂ R3 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='1) of the form π (x) = x − d (x) n (x) which assigns to every x ∈ Mh the closest point on M, so that (x − π(x)) ⊥ Tπ(x)M, ∀x ∈ Mh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' We assume that Nδ has the additional property that for each point x ∈ Mh ⊆ Nδ there is a unique point π(x) on M that minimizes the distance from M to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' In other words, the computation of the closest point projection (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='1) is a local minimizer problem [BV21] in the sense: π : Mh ∋ x �→ argmin y∈M ∥y − x∥ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='2) where the nonlinear projection maps a point x ∈ Mh to the point on M that minimizes the distance to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' The accuracy of standard surface integration methods is limited to only first or second order due to piecewise linear approximations of the surface geometry and integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' To obtain high order accuracy, we must construct a high order approximation both to the geometry of the surface and to the integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' By relying on [PS22] we give now the construction of a surface Mk h which is locally parametrised over the reference simplex σ by polynomials of degree k and interpolates the smooth surface M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' As we stated earlier we first consider a piecewise flat representation Mh of the smooth surface having triangular faces with vertices lying on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' To simplify the notation, the subscripts from the transformation map (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='1) and the triangulation of Mh are dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' For any regular simplex T ∈ T 1 h with vertices q1, q2, q3 ∈ R3 we define the following map ξ : σ −→ T, qi = ξ ( ˆpi) , 1 ≤ i ≤ 3, with ξ (s, t) = q1 + � q3 − q1 � s + � q2 − q1 � t to be the affine linear parametrization which maps each vertex ˆpi of σ to the vertex qi of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' We de- note the images of the non-vertex nodes of σ on the simplex T by ¯qi = ξ ( ˆpi) , 3 < i ≤ N (2, k), where N (2, k) is the dimension of the vector space P2,k (σ) of bivariate polynomials of degree k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Let Lk 1 (ˆp) , Lk 2 (ˆp) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' , Lk N(2,k) (ˆp) be the local Lagrange basis functions of degree k on σ corresponding to the nodal points ˆp1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' ˆpN(2,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Set pi := π (¯qi) = � π ◦ ξ � ( ˆpi) = ψ ( ˆpi) , where ψ := π ◦ ξ, 3 < i ≤ N (2, n) 3 and define Qψ,k to be a k−th order polynomial interpolation of the mapping ψ: Ik : C0� σ, R � −→ Pk (σ) , ψ �→ Qψ,k, Qψ,k (ˆp) := N(2,n) � i=1 piLk i (ˆp) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='3) Figure 1: Construction of the second order approximation of the smooth surface M2 h (blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' A simplex of a ‘base’ triangulation Mh (green line) is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' The interpolation nodes, here the center ¯qi of an edge, are projected (grey line) onto the smooth surface M (red line) via the projection π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' The projected nodal points π (¯qi) and the vertices of Mh are then interpolated, giving the second order approximation of the smooth surface M2 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Now, Qψ,k : σ → ˜T k := Qψ,k � σ � defines a polynomial mapping Qψ,k by interpolating the points pi ∈ M with the Lagrange-polynomials Lk 1 (ˆp) , Lk 2 (ˆp) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' , Lk N(2,k) (ˆp) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Thus, for every simplex T ∈ T 1 h we compute the projection π(¯qi) and define an isoparametric simplex ˜T k by applying Lagrange interpolation of order k to the coordinates of the projected equidistant nodes (see Fig (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Furthermore, if the base-triangulation T 1 h is fine enough, then the map Qψ,k is a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' By differentiating the interpolation polynomial of the map ψ, we obtain: ∂sQψ,k (ˆp) := N(2,n) � i=1 pi∂sLk i (ˆp) , ∂tQψ,k (ˆp) := N(2,n) � i=1 pi∂tLk i (ˆp) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='4) The Jacobian of the transformation is calculated using the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Given that Qψ,k is a diffeo- morphism ∀ T ∈ T 1 h , then by union of non-overlapping mapped elements: Mk h := � T∈T 1 h Qψ,k (T) = � T∈T 1 h , ˆp∈σ N(2,n) � i=1 π � ξ ( ˆpi) � Lk i (ˆp) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='5) Thus, we have obtained a k−th order discrete approximation Mk h of the smooth surface M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' An illus- tration of this process for a torus and a sphere is presented in Fig (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 3 Accuracy of Integration In this section, we will prove our main theorem about the accuracy of integration when replacing M by its k−th order polygonal approximation Mk h and show that symmetric triangles of the mesh combined with even-degree polynomials improve global errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' This motivates the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 4 pi =π(qi) M Mh Mn qi(a) Mh (b) M4 h (c) Mh (d) M3 h Figure 2: Lagrange parametrization for a torus and a sphere using equidistant nodes on vertices and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Given a collection of flat triangles T 1 h , a pair of triangles Tx1,x2,x3, Tx1,x4,x5 ∈ T 1 h as in Fig (3) are called symmetric, if they satisfy the following property (x3 − x1) = − (x5 − x1) , (x2 − x1) = − (x4 − x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='1) Figure 3: A pair of symmetric triangles Next, we state one of our main results, which explains why even degree polynomials combined with symmetric triangles perform better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Let M be a smooth closed embedded hypersurface and f ∈ Ck+2 (M, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Consider a piecewise linear triangulation Mh with mesh size h of the smooth surface having vertices lie on M and let Mk h be the k−th order approximation of the smooth surface constructed using local fittings and assume π ∈ Ck+3 � Mh, R3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Assume that the triangulation mesh is composed of symmetric triangles, then ����� � M fdS − � Mk h Qf,kdS ����� L∞ ≤ � Chk+2, k ≡ 0 (mod 2) Chk+1, k ≡ 1 (mod 2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='2) where Qf,k : Mk h → R is a k− th order polynomial approximating the integrand f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 5 X4 X3 x1 X5 X2As we approach the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='2, we need three lemmata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Let T be a triangle in T 1 h and assume that ψ ∈ Ck+1 (T) and k ≥ 0, then we have ∥ψ − Qψ,k∥L∞(T) ≤ chk+1 l,p,q≥0 max l+p+q=k+1 max (x,y,z)∈T ���∂k+1ψ (x, y, z) ∂xl∂yp∂zq ��� (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='3) with h = diam(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' The constant c depends on k, but it is independent of both ψ and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' In order to prove this lemma, we will use Taylor’s formula in several variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Let’s denote with α = (α1, α2), where |α| = |α1| + |α2|, h := (h1, h2) = (s − 0, t − 0) and ∂α1 = ∂s, ∂α2 = ∂t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' As a first step, let’s consider the k− order interpolation of the map ψ Qψ,k (ˆp) := N(2,n) � i=1 piLk i (ˆp) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Using Taylor’s formula for ψ around the point (0, 0) we obtain: ψ (ˆp) = � |α|=k �∂α α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' ψ (0) hα� + 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' � 1 0 (1 − µ)k dk+1ψ (sµ, tµ) dk+1µ dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='4) Let us denote H (ˆp) = 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' � 1 0 (1 − µ)k dk+1ψ(sµ,tµ) dk+1µ dµ, by interpolating each term in the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='4) using a polynomial of order k and using the fact that the interpolation of the first term is exact because it is a polynomial of degree ≤ k,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='e � Ik � |α|=k � ∂α α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' ψ (0) hα� = � |α|=k � ∂α α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' ψ (0) hα�� , we obtain Qψ,k (ˆp) = � |α|=k �∂α α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' ψ (0) hα� + N(2,n) � i=1 H (ˆpi) Lk i (ˆp) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='5) Subtracting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='5) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='4), we have ψ (ˆp) − Qψ,k (ˆp) = H (ˆp) − N(2,n) � i=1 H (ˆpi) Lk i (ˆp) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='6) The right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='6) can be written H (ˆp) − N(2,n) � i=1 H (ˆpi) Lk i (ˆp) = 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' � 1 0 (1 − µ)kE (µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' s, t) dµ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='7) where E (µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' s, t) has the following form E (µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' s, t) := dk+1ψ (sµ, tµ) dk+1µ − N(2,n) � i=1 Lk i (ˆp) dk+1ψ (siµ, tiµ) dk+1µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='8) It is important to note that the left-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='6) is affected by the behavior of the term E (µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' s, t), which itself is influenced by dk+1ψ(sµ,tµ) dk+1µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' For k = 0, we have max 0≤µ≤1, ˆp∈σ ���� dψ (sµ, tµ) dµ ���� ≤ ch max ����� ∂ψ ∂x ���� L∞ , ���� ∂ψ ∂y ���� L∞ , ���� ∂ψ ∂z ���� L∞ � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='9) 6 where ���� ∂ψ ∂x ���� L∞ := max (x,y,z)∈T ���� ∂ψ ∂x ���� and analogously for ∂ψ ∂y , ∂ψ ∂z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Following the same line of reasoning with the higher-order derivatives of ψ, we obtain max 0≤µ≤1, ˆp∈σ ���� dk+1ψ (sµ, tµ) dk+1µ ���� ≤ chk+1 max ����� ∂ψ ∂x ���� L∞ , ���� ∂ψ ∂y ���� L∞ , ���� ∂ψ ∂z ���� L∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='10) Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='7), yields H (ˆp) − N(2,n) � i=1 H (ˆpi) Lk i (ˆp) = O � hk+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='11) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='6) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='11), yields Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Next, we recall the following Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='4 ([Chi93]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Let k be an even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Let ψ (ˆp) ∈ Pk+1 be a polynomial of degree k + 1 on σ, and let Qψ,k (ˆp) be its interpolant of degree k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Then, for each ˆp ∈ σ � σ ∂s (ψ (ˆp) − Qψ,k (ˆp)) dsdt = 0, � σ ∂t (ψ (ˆp) − Qψ,k (ˆp)) dsdt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='12) Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Let M be a smooth closed hypersurface and consider a piecewise linear triangulation Mh with mesh size h of the smooth surface having vertices lie in M and let Mk h be the k−th order approximation of the smooth surface constructed using local fittings and assume π ∈ Ck+3 � Mh, R3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Assume that the triangulation mesh is composed of symmetric triangles, then ����� � M dS − � Mk h dS ����� L∞ ≤ � Chk+1, k ≡ 1 (mod 2) Chk+2, k ≡ 0 (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='13) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' First, let us write every term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='13) over a reference simplex � M dS = n � i=1 � σ gidsdt, � Mk h dS = n � i=1 � σ ˜gidsdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='14) In the same manner, as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='3, we obtain ψ (ˆp) − Qψ,k (ˆp) = Rk+1 + Rk+2 + O � hk+3� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='15) where Rk+1 := � |α|=k+1 �∂α α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' ψ (0) hα − N(2,n) � i=1 hα ∂α α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' ψ (0) Lk i (ˆp) � Rk+2 := � |α|=k+2 �∂α α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' ψ (0) hα − N(2,n) � i=1 hα ∂α α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' ψ (0) Lk i (ˆp) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Analogously, an expansion can be given for the errors in the partial derivatives of ∂α1ψ (ˆp) , ∂α2ψ (ˆp), amounting to apply the derivative on (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Therefore, we have ∂α1ψ (ˆp) − ∂α1Qψ,k (ˆp) = ∂α1Rk+1 + ∂α1Rk+2 + O � hk+2� (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='16) 7 ∂α2ψ (ˆp) − ∂α2Qψ,k (ˆp) = ∂α2Rk+1 + ∂α2Rk+2 + O � hk+2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='17) Using Taylor’s theorem and expanding about (s = 0, t = 0), we obtain ∥˜gi − gi∥L∞ := ���� � det(∂sQT ψ,n (ˆp) ∂tQψ,n (ˆp)) − � det(∂sψT (ˆp) ∂tψ (ˆp)) ���� L∞ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='18) = E � hk+2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (xk − xj) , (xℓ − xj) � + E � hk+3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (xk − xj) , (xℓ − xj) � + O � hk+4� , where E � hk+2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (xk − xj) , (xℓ − xj) � and E � hk+3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (xk − xj) , (xℓ − xj) � , describes the collection of terms with order k+2, k+3 in h, whose coefficients are combinations of the vertices of triangles with appropriate indices j, ℓ, and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' For example for a triangle Tx1,x2,x3 the coefficients are combination of (x3 − x1) and (x2 − x1) both for E (k + 2) and E (k + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' From above computation we see that ∥˜gi −gi∥L∞ is at least of order O � hk+2� , confirming also the result obtained in [RWJG12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' However, this result can be improved under certain conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=', an approximation of order O � hk+4� can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Thus, integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='18) over a reference simplex, we have � σ ��� ˜gi − gi ��� L∞ dsdt = � σ E (k + 2) dsdt + � σ E (k + 3) dsdt + O � hk+4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='19) At this point we notice that if k is even, then � σ E (k + 2) dsdt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' This is due to the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='4 because E (k + 2) represents errors when integrating a polynomial of order k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Thus, we have � σ ��� ˜gi − gi ��� L∞ dsdt = � σ E (k + 3) dsdt + O � hk+4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='20) This shows that the left hand side of the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='20) is at least of order O � hk+3� for each triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' At this point using that the number of triangles in the surface mesh is inversely proportional to the average area of triangle n = O � h−2� , yields the first part of inequality(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' While if k is odd, yields � σ E (k + 2) dsdt ̸= 0 then, the left hand side of the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='19) is at least of order O � hk+2� for every T ∈ T 1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' It should be noted that this theorem can be generalized and proved for d− dimensional simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Assuming that k is even and writing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='20) with respect to a pair of symmetric triangles, we obtain Tx1,x2,x3 and Tx1,x4,x5 � σ ��� ˜gi − gi ��� L∞ dsdt ���� Tx1,x2,x3 = � σ E (k + 3) dsdt + O � hk+4� (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='21) � σ ��� ˜gi − gi ��� L∞ dsdt ���� Tx1,x4,x5 = � σ E (k + 3) dsdt + O � hk+4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='22) Now, if k is even and triangles are pairwise symmetric then the error contributed is � σ ��� ˜gi − gi ��� L∞ dsdt = O � hk+4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='23) 8 This is due to the fact that the left hand side integrand of the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='21) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='22), are the collection of terms which are of order k + 3 in h, where the coefficients are the combination of the vertices of triangles, thus each integrand is an odd function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' In general for a triangle Tx1,x2,x3 we may write: E � hk+3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' − (x3 − x1) , − (x2 − x1) � = −E � hk+3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (x3 − x1) , (x2 − x1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='24) In the same manner for Tx1,x4,x5 E � hk+3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' − (x4 − x1) , − (x5 − x1) � = −E � hk+3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (x4 − x1) , (x5 − x1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='25) But using the property (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='1), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='24) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='25), we obtain E � hk+3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (x3 − x1) , (x2 − x1) � + E � hk+3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (x4 − x1) , (x5 − x1) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Again, using the fact that the number of triangles in the surface mesh is inversely proportional to the average area of triangle n = O � h−2� and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='23), yields the second part of inequality(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' We can now prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='2 Poof of theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' As in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='5, expanding the function f using Taylor’s formula around the point (0, 0), yields f (ψ (ˆp)) − Qf,k (ψ (ˆp)) = Rk+1 + O � hk+2� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='26) where Rk+1 = O � hk+1� and it has the form Rk+1 := � |α|=k+1 �∂α α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' f (0) hα − N(2,n) � i=1 hα ∂α α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' f (0) Lk i (ˆp) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Integrate both side of the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='26), we have � σ (f (ψ (ˆp)) − Qf,k (ψ (ˆp))) gidsdt = � σ Rk+1gidsdt + O � hk+4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='27) Using ∥gi∥L∞ = O � h2� , the integral term on the right hand side is at least of order O � hk+3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' If k is even due to the presence of symmetric triangles the first term on the right-hand side integral is 0, so we can compute the left-hand integral with an accuracy of order O � hk+4� , while for k odd, we have the following � σ ∥(f (ψ (ˆp)) − Qf,k (ψ (ˆp))) gi∥L∞ dsdt ≤ Chk+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='28) Assume that Mk h is composed of triangles ˜T k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='e Mk h = �n i=1 ˜T k i and the smooth closed surface M = �n i=1 Ti where n is the number of triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Then � M fdS = n � i=1 � Ti fdS, � Mk h Qf,kdS = n � i=1 � ˜T k i Qf,kdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='29) Let us rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='29) over a reference simplex where the quadrature rules are defined � M fdS = n � i=1 � σ f (ψ (ˆp)) gidsdt, � Mk h Qf,kdS = n � i=1 � σ Qf,k (ψ (ˆp)) ˜gidsdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='30) 9 By making use of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='5 and the fact that the number of triangles in the surface mesh is inversely proportional to the average area of triangle n = O � h−2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' we have ����� � M fdS − � Mk h Qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='kdS ����� L∞ ≤ ����� n � i=1 � σ � f (ψ (ˆp)) gi − Qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='k (ψ (ˆp)) ˜gi � dsdt ����� L∞ ≤ n � i=1 ���� � σ (f (ψ (ˆp)) − Qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='k (ψ (ˆp))) gi + Qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='k (ψ (ˆp)) (gi − ˜gi) dsdt ���� L∞ ≤ n � i=1 �� σ ∥(f (ψ (ˆp)) − Qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='k (ψ (ˆp))) gi∥L∞ dsdt + � σ ∥Qf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='k (ψ (ˆp)) (gi − ˜gi)∥L∞ dsdt � ≤ � Chk+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' k ≡ 0 (mod 2) Chk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' k ≡ 1 (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' For the last inequality, we have used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='27) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 4 Results and Discussion In order to validate our findings, we developed tests employing the Gauss-Bonnet theorem [PPCT01], [Spi99] for a series of classic smooth closed surfaces given by the following equations: 1) Ellipsoid x2 a2 + y2 b2 + z2 c2 = 1, a, b, c ∈ R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 2) Torus (x2 + y2 + z2 + R2 − r2)2 − 4R2(x2 + y2) = 0, 0 < r < R ∈ R 3) Sphere x2 + y2 + z2 = R2, R ∈ R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' where the analytic expression for the Gaussian Curvature are 1) Ellipsoid KGauss = 1 (abc)2 � x2 a4 + y2 b4 + z2 c4 �2 , a, b, c ∈ R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 2) Torus KGauss = cos v r(R+r cos v), where we used toric coordinates: (x, y, z) = � (R + r cos θ) cos ϕ, (R + r cos θ) sin ϕ, r sin θ � , ϕ, θ ∈ [0, 2π) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 3) Sphere KGauss = 1 R2 , r ∈ R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' In all experiments, we applied a grade-12 Gaussian quadrature rule on each triangle, with 32 quadrature points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 4 and Fig 5 we show the relative errors under mesh refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' In order to calculate the relative error, we integrate the Gaussian curvature on the manifold and compare it with the predictions of the Gauss-Bonnet Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' The convergence rates shown on the plots confirm our theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' However, as can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 6, the approach presented in [PS22] has some limitations, it runs into Runge’s phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' This is due to the fact that the Lebesgue constant Λ for the set of equidistant points grows exponentially [MS92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' To mitigate this effect, we propose an alternative approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' It is well known that under a proper map the operations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=', interpolation, numerical differentiation, and quadrature) on a triangular element can be performed on the reference square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Below we will state and prove a theorem that we hope will pave the way for developing another powerful numerical integration method for closed surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 10 10 1 h 10 10 10 8 10 6 10 4 10 2 error fdS k hfkdS , k = 2 fdS k hfkdS , k = 4 h4 h6 (a) Even order 10 1 h 10 11 10 9 10 7 10 5 10 3 error fdS k hfkdS , k = 3 fdS k hfkdS , k = 5 h4 h6 (b) Odd order Figure 4: Relative errors by integrating the Gaussian curvature over the torus with radii R = 2, r = 1 with the ideal convergence lines hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 10 1 h 10 10 10 8 10 6 10 4 10 2 error fdS k hfkdS , k = 2 fdS k hfkdS , k = 4 h4 h6 (a) Even order 10 1 h 10 10 10 8 10 6 10 4 10 2 error fdS k hfkdS , k = 3 fdS k hfkdS , k = 5 h4 h6 (b) Odd order Figure 5: Relative errors by integrating the Gaussian curvature over the unit sphere with the ideal convergence lines hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Let f ∈ Ck � [−1, 1]2� , we have the estimate ∥f (x, y) − Qf,n (x, y)∥L∞([−1,1]2) ≤ ˜C ln2 (n) n−k � ω � ∂k xf (x, y) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 1 n, 0 � + ω � ∂k yf (x, y) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 0, 1 n �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='1) Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Based on Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='1 in future work we plan to incorporate recent advances in mul- tivariate interpolation [HGM+20], providing a stable approach suppressing Runge’s phenomenon by interpolating the map ψ in proper chosen, transformed Chebyshev-Lobatto nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' In order to prove the theorem we recall the modulus of continuity from [TIM63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' For δ1, δ2 the modulus of continuity ω (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' δ1, δ2) is defined as: ω (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' δ1, δ2) = sup |x1−x2|≤δ1, x1,x2∈[−1,1] sup |y1−y2|≤δ2, y1,y2∈[−1,1] |f (x1, x2) − f (y1, y2) |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='2) The function ω (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' δ1, δ2) is semi-additive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='e, ω (δ1 + δ2, λ1 + λ2) ≤ ω (δ1 + λ1) + ω (δ2 + λ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 11 (a) Visualization of Gaussian Curvature for the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (b) Visualization of Gaussian Curvature for the ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 3 4 5 6 7 8 9 10 Degree of Polynomial 10 11 10 10 10 9 10 8 10 7 10 6 10 5 10 4 error (c) torus with radii R = 2, r = 1 3 4 5 6 7 8 9 10 Degree of Polynomial 10 12 10 10 10 8 10 6 error (d) ellipsoid with a = b = 1, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Figure 6: Relative errors by integrating the Gaussian curvature over the torus and the ellipsoid using N∆ = 2528, N∆ = 6152 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Poof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' We consider the interpolation operator I[−1,1]2 : C0� [−1, 1]2, R � → P2,n with P2,n being the space of bivariate polynomials of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Let us denote with Q∗ f,n (x, y) the best polynomial approximation of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' The identity theorem for polynomials yields Q∗ f,n (x, y) = I[−1,1]2 � Q∗ f,n (x, y) � , by making use of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='4 in [HGM+20], we have ∥f (x, y) − Qf,n (x, y)∥L∞([−1,1]2) ≤ ��Q∗ f,n (x, y) − f (x, y) �� L∞([−1,1]2) + ��Q∗ f,n (x, y) − Qf,n (x, y) �� L∞([−1,1]2) ≤ ��Q∗ f,n (x, y) − f (x, y) �� L∞([−1,1]2) + ��I[−1,1]2 � Q∗ f,n (x, y) − f (x, y) ��� L∞([−1,1]2) ≤ � 1 + CΛ ln2 (n) � ��f (x, y) − Q∗ f,n (x, y) �� L∞([−1,1]2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='3) At this point the multivariate version of Jackson’s inequality and semi-additivity of the modulus [TIM63], gives ��f (x, y) − Q∗ f,n (x, y) �� L∞([−1,1]2) ≤ C � 2kn−kω � ∂k xf (x, y) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 2 n, 0 � + 2kn−kω � ∂k yf (x, y) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 0, 2 n �� ≤ C2k+1n−k � ω � ∂k xf (x, y) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 1 n, 0 � + ω � ∂k yf (x, y) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 0, 1 n �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='4) 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='3e-01 21 K_Gauss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='0e+002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='8e+00 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='5 _Gauss 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='5 K 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='0e+00Combining inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='4) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='3), we obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Additionally, it is easy to see that, for f ∈ Ck(T), we have ∥f (x, y) − Qf,n (x, y)∥L∞(T) ≤ c ∥f (x, y) − Qf,n (x, y)∥L∞([−1,1]2) = O � n−k� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='5) In light of the multivariate extension of Jackson’s theorem [BBL02], we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Let f ∈ Ck � [−1, 1]2� , we have the estimate ∥∂xf (x, y) − ∂xQf,n (x, y)∥L∞([−1,1]2) ≤ C (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' n) ln2 (n) n−(k−1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='6) and similarly ∥∂yf (x, y) − ∂yQf,n (x, y)∥L∞([−1,1]2) ≤ C (f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' n) ln2 (n) n−(k−1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='7) where C is a suitable constant (with n), dependent on f(x, y) and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' If f ∈ Ck � [−1, 1]2� satisfy the Dini–Lipschitz criterion in the sense that lim n−→∞ ln2 (n) ω � f (x, y) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 1 n, 1 n � = 0 then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content='1) guarantees uniform convergence lim n−→∞ ∥f (x, y) − Qf,n (x, y)∥L∞([−1,1]2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' It is anticipated that with this initiative, we can develop an efficient numerical integration method for closed surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' This will be discussed in a forthcoming work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Acknowledgement We deeply acknowledge Paul Breiding and Michael Hecht for many inspiring comments and helpful suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' The research of Gentian Zavalani was partially funded by the Center of Advanced Systems Understanding (CASUS) which is financed by Germany’s Federal Ministry of Education and Research (BMBF) and by the Saxon Ministry for Science, Culture, and Tourism (SMWK) with tax funds on the basis of the budget approved by the Saxon State Parliament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' The research of Elima Shehu was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Projektnummer 445466444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' References [BBL02] Thomas Bagby, Len Bos, and Norman Levenberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Multivariate simultaneous approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Constructive approximation, 18(4):569–577, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' [BV21] Paul Breiding and Nick Vannieuwenhoven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' The condition number of riemannian approxima- tion problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' SIAM Journal on Optimization, 31(1):1049–1077, jan 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' [Chi93] David Da-Kwun Chien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Piecewise polynomial collocation for integral equations with a smooth kernel on surfaces in three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' The Journal of Integral Equations and Ap- plications, 5:315–44, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 13 [Dem09] Alan Demlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' SIAM Journal on Numerical Analysis, 47(2):805–827, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' [Geo98] Kurt Georg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Approximation of integrals for boundary element methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' SIAM Journal on Scientific and Statistical Computing, 12, 08 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' [HGM+20] Michael Hecht, Krzysztof Gonciarz, Jannik Michelfeit, Vladimir Sivkin, and Ivo F Sbalzarini.' metadata={'source': 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Geom- etry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Springer undergraduate mathematics series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Springer, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' [PS22] Simon Praetorius and Florian Stenger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Dune-curvedgrid - a dune module for surface parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Archive of Numerical Software, page Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 1 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 1 (2022), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' [RWJG12] Navamita Ray, Duo Wang, Xiangmin Jiao, and James Glimm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' High-order numerical in- tegration over discrete surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' SIAM Journal on Numerical Analysis, 50(6):3061–3083, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' [Spi99] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Spivak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' A Comprehensive Introduction to Differential Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Number v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 1 in A Comprehensive Introduction to Differential Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Publish or Perish, Incorporated, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' [TIM63] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' TIMAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Theory of Approximation of Functions of a Real Variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' International Series of Monographs on Pure and Applied Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Pergamon, 1963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' [YQ09] Pinghai Yang and Xiaoping Qian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' A general, accurate procedure for calculating molecular interaction force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' Journal of Colloid and Interface Science, 337(2):594–605, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} +page_content=' 14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQfNAN9/content/2301.02996v1.pdf'} diff --git a/jtFRT4oBgHgl3EQfWjfo/vector_store/index.faiss b/jtFRT4oBgHgl3EQfWjfo/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..772796489665871754673275fa00c3ba4df7f684 --- /dev/null +++ b/jtFRT4oBgHgl3EQfWjfo/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4c7e3a7e6c2875591754680221cb2f962047dcd92c7b1b38b981d1b91961f8d3 +size 2097197 diff --git 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School of Chemistry and Molecular Biosciences, The University of Queensland, +Brisbane, Queensland, Australia. 4Department of Medicine (Hematology), Stanford University, Stanford, CA, USA. 5Department of Genetics, Stanford +University, Stanford, CA, USA. 6PolyBio Research Foundation, Kenmore, WA, USA. 7Department of Bioengineering, Stanford University, Stanford, CA, USA. +ChEM-H Institute, Stanford University, Stanford, CA, USA. +10Chan Zuckerberg Biohub, San Francisco, CA, USA. ✉e-mail: snayfach@lbl.gov; nckyrpides@lbl.gov +T +he gut microbiome is a complex microbial ecosystem with +important roles in human health and development1. Although +often overlooked, viruses are estimated to be abundant in the +microbiome2,3 and have been associated with human disease4–6. In +particular, bacteriophages (viruses that infect bacteria) constitute +the majority of viral particles3,7,8 and can impact microbial ecosystem +processes through phage predation9, lysogeny10 and horizontal gene +transfer11. Despite their ubiquity, our knowledge of viral genomic +diversity in the microbiome is limited, with most viral sequences +failing to match existing genome databases8. A comprehensive data- +base of viral genomes from the microbiome is a prerequisite for +assembly-free quantification of viruses, predicting host–virus inter- +actions12, comparative genomics and genome mining (for example, +anti-CRISPR genes13). +Traditionally, there have been two main approaches for sequenc- +ing viral genomes from the microbiome: viral metagenomic +sequencing and bulk metagenomic sequencing. Viral metagenom- +ics involves using size filtration to select for virus-like particles, +followed by viral DNA extraction, (often) whole-genome amplifica- +tion, shotgun sequencing and metagenomic assembly14–17. Although +size filtration is used to enrich extracellular viruses, it will not +remove all cellular organisms18 and can exclude some large viruses19. +Whole-genome amplification is often necessary due to low sample +biomass but can skew viral abundances and over-amplify small cir- +cular single-stranded DNA (ssDNA) viruses19–21. +whole-genome amplification. However, with bulk metagenomic +sequencing, it is more challenging to assemble low-abundance +viruses because the majority of reads derive from cellular organ- +isms24. Additionally, DNA-extraction protocols may not be opti- +mized for viruses16 and some viral sequences may originate from +degraded prophages in bacterial chromosomes10,25. +To date, numerous studies have used viral metagenomic sequenc- +ing to identify phage genomes from human stool samples across a +wide variety of phenotypes4–6. To integrate these disparate data sets, +Soto-Perez et al.26 formed the Human Virome Database (HuVirDB) +from 1,831 public samples (including skin, stool, lung and blood) +and Gregory et al.27 formed the Gut Virome Database (GVD) from +2,697 public samples. In contrast to these viral metagenomic stud- +ies, Paez-Espino et al22. formed the IMG/VR database by identifying +viruses from bulk metagenomes, including 490 stool samples from +the Human Microbiome Project28. Since this publication, the num- +ber of publicly available bulk metagenomes has rapidly grown, as +evidenced by recent, large-scale data mining efforts29–31. +To expand these existing resources and provide a complemen- +tary view of the gut virome, we performed large-scale identification +of viral genomes from 11,810 bulk metagenomes from human stool +samples derived from 61 previously published studies. We used +these data to form the Metagenomic Gut Virus (MGV) catalogue, +which contains 189,680 viral draft genomes estimated to be >50% +complete and representing 54,118 candidate viral species. These +Metagenomic compendium of 189,680 DNA +An alternative approach is to generate bulk metagenomes, with- +genomes vastly expand the known diversity of DNA viruses from +out size filtration or whole-genome amplification, followed by +the gut microbiome and improve knowledge of host–virus con- +computational separation of viral and cellular sequences22,23. This +nections. We expect the MGV catalogue will be a useful commu- +approach captures sequences of both extracellular and intracel- +nity resource for interrogating the role of the gut virome in human +lular viruses, including integrated prophages, and is not biased by +health and disease. +viruses from the human gut microbiome +Stephen Nayfach   1,2 ✉, David Páez-Espino   1,2, Lee Call1,2, Soo Jen Low3, Hila Sberro4,5, +Natalia N. Ivanova   1,2, Amy D. Proal6, Michael A. Fischbach   7,8,9,10, Ami S. Bhatt   4,5, +Philip Hugenholtz   3 and Nikos C. Kyrpides   1,2 ✉ +Bacteriophages have important roles in the ecology of the human gut microbiome but are under-represented in reference data- +bases. To address this problem, we assembled the Metagenomic Gut Virus catalogue that comprises 189,680 viral genomes +from 11,810 publicly available human stool metagenomes. Over 75% of genomes represent double-stranded DNA phages that +infect members of the Bacteroidia and Clostridia classes. Based on sequence clustering we identified 54,118 candidate viral spe- +cies, 92% of which were not found in existing databases. The Metagenomic Gut Virus catalogue improves detection of viruses +in stool metagenomes and accounts for nearly 40% of CRISPR spacers found in human gut Bacteria and Archaea. We also pro- +duced a catalogue of 459,375 viral protein clusters to explore the functional potential of the gut virome. This revealed tens of +thousands of diversity-generating retroelements, which use error-prone reverse transcription to mutate target genes and may +be involved in the molecular arms race between phages and their bacterial hosts. +NAturE MICroBIoLogy | VOL 6 | JULy 2021 | 960–970 | www.nature.com/naturemicrobiology +960 +ResouRce +https://doi.org/10.1038/s41564-021-00928-6 + +ResouRce +NATuRE MICROBIOlOGy +results +A genomic catalogue of DNA viruses from the gut microbiome. +We developed a viral detection pipeline for the current study using a +combination of well-established methods and signatures, including +VirFinder32, viral protein families from the Earth’s Virome Study23, +and the propensity for viral genes to lie on the same strand33 and +be functionally unannotated8 (Fig. 1a,b). Based on in silico bench- +marking, our pipeline was able to sensitively identify genome frag- +ments of diverse human-associated viruses and phages, including +crAss-like phages34 and megaphages35, with high specificity and per- +formed favourably compared with existing methods (Supplementary +Tables 1–2 and Methods). For genome fragments of 1, 10 and 100 kb +our pipeline achieved true-positive rates (TPR) of 41%, 74% and +96% at false-positive rates (FPR) of only 0.43%, 0.38% and 0.18%. +We then applied our pipeline to bulk metagenomes from 11,810 +distinct human gut samples that were assembled in previous stud- +ies29,31,36 to broadly capture lytic and lysogenic DNA viruses (Fig. 1a +and Supplementary Table 3). The analysed data sets span 61 stud- +ies across 24 countries and include individuals with a wide range +of ages, lifestyles and disease states (Supplementary Table 4). This +revealed 3.5 million unique, single-contig viral genomes longer than +1 kb. Based on an analysis of metagenomes found in all three stud- +ies, we found that choice of assembler (that is, MEGAHIT versus +metaSPAdes) had little effect on the quality or identity of recovered +viruses (Extended Data Fig. 1). Viral genomes were largely derived +from individuals in Europe (46%), China (23%) and the USA (13%) +reflecting the amount of metagenomic data from these sources +(45%, 24% and 11% of total assembly length, respectively). +The completeness of metagenome-assembled viruses can vary +widely, ranging from short fragments to complete or near-complete +genomes. To assess genome completeness, we applied CheckV37, +revealing 189,680 genomes that were at least 50% complete (Fig. 1c), +including 26,030 complete genomes identified on the basis of direct +terminal repeats (n = 19,704), host–provirus boundaries (n = 5,123) +and inverted terminal repeats (n = 1,203). To improve genome qual- +ity, we removed flanking host regions from these sequences (Fig. +1a); confirming that viral genomes were free of host contamination, +we identified only one full-length 16S rRNA gene (flanking an inte- +grated prophage) among all 189,680 viruses compared with 83,050 +16S rRNA genes in the full set of metagenomic contigs used for +viral discovery (Methods). We focused all subsequent analysis on +the 189,680 genomes with >50% completeness to avoid limitations +associated with small genome fragments38 and to be consistent with +quality standards applied to microbial genomes39. +Because there was no separation of viral-like particles prior +to sequencing, we anticipated many viruses were derived from +Viral taxonomy +Africa +Asia +Europe +North America +Oceania +South America +Geography +0 +20 +40 +60 +Gene strand switch +rate +0 +0.4 +0.8 +VirFinder score +Yes +0 +40 +80 +Percentage viral +protein families +0 +40 +80 +Percentage non-viral +Pfams +No +Predicted virus +Yes +No +Predicted virus +d +a +b +c +Unknown completeness +<50% complete +50–90% complete +>90% complete +Complete +Viral contig length (kb) +1 +10 +100 +1,000 +(n = 671,842) +(n = 2,620,162) +(n = 110,430) +(n = 53,220) +(n = 26,030) +CheckV genome completeness level +for 3.5 million viral contigs +1. Assembled, whole human gut metagenomes +(no enrichment for virus-like particles) +Pasolli et al. 2019 +7,768 metaSPAdes assemblies +HGM dataset (Nayfach et al. 2019) +3,771 MEGAHIT assemblies +MGnify (Mitchell et al. 2019) +6,732 metaSPAdes assemblies +2. Extract viral-informative features (panel b) +Viral +(3.5 million contigs) +3. Identify viral contigs (>1 kb) +Non-viral +(373.8 million contigs) +4. Estimate genome completeness (panel c) +≥50% complete +(n = 189,680 contigs) +<50% complete +(n = 3,292,004 contigs) +5. Trim flanking host regions from prophages +• Genome clustering +• Host prediction +• Taxonomic annotation +• Functional annotation +• Phylogenetic analysis +6. Annotation and analysis +• Viral protein families (Paez-Espino et al. 2016) +• VirFinder score (Ren et al. 2017) +• Non-viral Pfams (El-Gebali et al. 2019) +• Gene strand switch rate (Roux et al. 2015) +Host +(673 × 106 bp) +Virus +(8,497 × 106 bp) +Viral-informative features for +metagenomic contigs +Summary statistics for 189,680 viral +contigs with >50% completeness +Clostridia +Bacteroidota +Actinobacteriota +Bacilli +Negativicutes +Proteobacteria +Other +Unknown +Host prediction +crAssphage +Other +Unknown +No +Yes +Integrated prophage +Complete +>90% +complete +50–90% +complete +Genome completeness +Podoviridae +Siphoviridae +Myoviridae +Microviridae +Fig. 1 | thousands of high-quality viral genomes recovered from human gut metagenomes. a, Overview of viral discovery effort and formation of the MGV +catalogue. b, Genomic signatures of predicted viral and non-viral metagenomic contigs longer than 20 kb. Displayed data is for 1,000 randomly selected +contigs from each category. c, Distribution of estimated genome completeness and classification of MGVs into quality tiers (complete, n = 26,030; >90% +complete, n = 53,220; 50–90% complete, n = 110,430; <50% complete, n = 2,620,162; completeness not determined, n = 671,842). d, Metadata and +annotations for 189,680 genomes with >50% completeness. For box plots, the middle line denotes the median, the box denotes the interquartile range and +the whiskers denote 1.5× the interquartile range. +NAturE MICroBIoLogy | VOL 6 | JULy 2021 | 960–970 | www.nature.com/naturemicrobiology +961 + +ResouRce +NATuRE MICROBIOlOGy +bacterial chromosomes. However, only 24% of viral genomes +had evidence of host integration (Fig. 1d) and only 10% where +the flanking host region was >5 kb. Furthermore, the major- +ity of non-integrated viruses were classified as virulent based on +BACPHLIP40 (65% of 140,689) which is a computational tool that +predicts bacteriophage lifestyle from conserved protein domains. +Likewise, BACPHLIP classified 58% of the 26,030 complete +genomes as virulent, indicating that this result is not due to incom- +plete genome assembly because integrase genes often occur at the +ends of prophage genomes41. Together, these results demonstrate +that it is not uncommon to recover the genome sequences of lytic +viruses from unfiltered stool metagenomes. +Host prediction and taxonomic annotation. Predicting the cellular +hosts of viruses is important for understanding phage predation and +an essential first step towards utilizing host–virus interactions to +design innovative phage therapies42. Towards this goal, we leveraged +the Unified Human Gastrointestinal Genome (UHGG) database +of 286,997 genomes of Bacteria and Archaea from the gut micro- +biome43, which represents 4,644 prokaryotic species (Fig. 2). First, +we extracted 1,846,441 CRISPR spacers from the UHGG genomes, +and looked for near-exact matches to the 189,680 viral genomes, +resulting in host–virus connections that covered 81% of viruses +(n = 153,892). Interestingly, just 21% of viruses were connected to +a host when using spacers extracted from the 4,644 species-level +representatives, indicating considerable CRISPR diversity between +bacterial strains and active community infection. Although most +viruses were targeted by a spacer, CRISPR arrays were found in only +28% (n = 79,734) of UHGG genomes and in <1% of many prevalent +species including Alistipes putredinis, Bacteroides cellulosilyticus and +Bifidobacterium breve, confirming the limited distribution of this +anti-viral defence system44. To expand the host–virus network, we +performed whole-genome alignment between the 189,680 viruses +and 286,997 hosts and identified connections based on near-exact +genomic matches (≥96% identity over ≥1 kb), resulting in connec- +tions that covered 96% of host genomes and 90% of viral genomes. +As expected, the majority of viruses were connected to Firmicutes +(predominantly Clostridia) and Bacteroidia, which are the two +dominant phyla of bacteria in the gut microbiome (Fig. 1d). These +results show that host–virus interactions can be systematically elu- +cidated through extensive assembly of both viral and microbial +genomes from the same environment. +Next, we assigned viruses to families from the ICTV data- +base based on alignments to genomes from NCBI GenBank and +crAss-like viruses from recent studies34,45,46 (Fig. 1d). Only 56.6% +of viruses could be annotated at the family level, confirming a +large knowledge gap in the taxonomy of human gut viruses8. To +increase sensitivity, we used taxonomically informative profile +hidden Markov models (HMMs) from the VOG database (http:// +vogdb.org), revealing most unannotated viruses to be members of +the Caudovirales order. Among annotated sequences were 9,395 +genomes of putative crAss-like viruses (5% of total). Overall, only +274 (199) +415 (58) +896 (256) +1,053 (1,308) +1,188 (139) +1,493 (880) +2,547 (78) +3,085 (2,389) +4,043 (323) +4,785 (140) +7,588 (634) +11,089 (1,689) +16,511 (1,018) +18,082 (6,492) +69,411 (3,617) +84,590 (9,304) +90,083 (17,211) +93,699 (19,813) +172,132 (13,189) +301,781 (62,710) +961,366 (145,390) +Number of CRISPR spacers +per host class +Percentage of spacers +matching MGV catalogue +Siphoviridae +Myoviridae +crAss-like +Podoviridae +Microviridae +Other +Unknown +Taxonomy of +connected viruses +Percentage of host +genomes with spacer hit +Spacer hit +No spacer hit +No spacers +Percentage of host genomes +with spacer or 1 kb hit +Spacer hit +Spacer hit and ≥1 kb hit +≥1 kb hit +No viral linkage +Viral family +Viral linkage method +Spacer linkages +Elusimicrobia +Peptococcia +Spirochaetia +Vampirovibrionia +Thermoplasmata +Campylobacteria +Synergistia +Alphaproteobacteria +Lentisphaeria +Brachyspirae +Fusobacteriia +Desulfovibrionia +Methanobacteria +Coriobacteriia +Verrucomicrobiae +Actinobacteria +Bacilli +Gammaproteobacteria +Negativicutes +Bacteroidia +Clostridia +8.8% +1% +7.8% +30.5% +4% +0.5% +9.8% +7.6% +8.9% +3.2% +21.4% +36.1% +17.9% +26.4% +38.5% +52.4% +31.8% +10.3% +41.5% +47.8% +44.5% +(Number of host genomes +from UHGG catalogue) +a +b +c +e +d +Fig. 2 | Viral connections to human gut Bacteria and Archaea. a, Bar plots indicating the number of CRISPR spacers across 286,997 human gut Bacteria +and Archaea, with the number of genomes indicated in parentheses. Each row indicates one host class containing at least 20 genomes and 100 spacers. +The majority of CRISPR spacers are derived from Clostridia and Bacteroidia, reflecting their abundance in the human gut. b, Percentage of CRISPR spacers +matching viral genomes with a maximum of one mismatch. c, Host genomes containing a CRISPR-spacer array, and those with a CRISPR-spacer array match +to a viral genome. d, Genomes linked to a virus using a combination of approaches as indicated. e, Distribution of known viral families that are associated +with each host class. Each host class is infected by a distinct repertoire of viral families. +NAturE MICroBIoLogy | VOL 6 | JULy 2021 | 960–970 | www.nature.com/naturemicrobiology +962 + +ResouRce +NATuRE MICROBIOlOGy +0.51% (n = 48) of the putative crAss phages displayed clear evi- +dence of lysogeny (that is flanked by host region and contained an +integrase), which was more than 17× lower than other viruses in +the data set. Consistent with this, 56% of high-quality crAssphage +genomes (n = 5,439) could be circularized compared with 24% of +the other high-quality genomes (n = 36,872). crAss-like genomes +contained several other unusual features, including low GC con- +tent (mean = 32%), usage of an alternative genetic code and a pre- +dominance of hypothetical proteins. For example, TAG or TGA stop +codons were recoded to amino acids in 27% of crAss-like phages +versus 0.5% of other viruses. Likewise, only 12% of crAssphage pro- +teins had significant hits to Pfam, KEGG or TIGRFAM versus 28% +of proteins from other viruses. This large-scale analysis supports +previous findings that some crAss-like viruses have an obligate lytic +lifestyle46 and reveals several unusual features that further establish +crAssphage as an outlier among human gut viruses47. +Vastly expanded viral genomic diversity. To quantify the diversity +of genomes in the MGV catalogue, we first identified species-level +viral operational taxonomic units (vOTUs) using the MIUViG rec- +ommended criteria of 95% average nucleotide identity (ANI) over +85% of the length of the shorter sequence38. Small adjustments +to these parameters did impact the number of identified vOTUs, +suggesting a continuum of viral diversity beyond the species-level +boundary (Supplementary Table 5). Overall, we identified 54,118 +vOTUs, of which 8,086 included members from at least two samples +(Fig. 3a). The largest vOTUs were predicted to infect some of the most +prevalent species in the gut microbiome, including Bacteroides uni- +formis, Faecalibacterium prausnitzii and Agathobacter rectalis (for- +merly Eubacterium rectale). To identify higher-ranking viral clades, +we clustered genomes into approximately genus- and family-level +groups on the basis of pairwise average amino acid identity (AAI) +and gene sharing (Methods), revealing 5,800 genus-level vOTUs +and 1,434 family-level vOTUs (Fig. 3a). Accumulation curves of +vOTUs appeared to be approaching an asymptote at the family and +genus ranks but not yet for species (Fig. 3b). +Other recent studies have also compiled databases of DNA +viruses from the gut microbiome22,26,27. To identify vOTUs unique +to the MGV catalogue, we clustered the 189,680 genomes from +our study together with medium- and high-quality viral genomes +from three other genome catalogues (Fig. 3a): the HuVirDB (9,626 +genomes derived from 1,543 viral metagenomes), GVD v.1.0 +(4,494 genomes derived from 471 viral metagenomes and 98 whole +metagenomes) and IMG/VR v.2.0 (6,895 genomes derived from 490 +whole metagenomes). Note that during the review of this manu- +script, the IMG/VR and GVD were updated to new versions which +were not analysed here. To enable comparability between all studies, +CheckV was run on all viral data sets and genome fragments with +<50% completeness were excluded. +Strikingly, we found that 50,048 of the 54,118 species-level +vOTUs from the MGV catalogue (92%), comprising 100,398 of the +189,680 genomes (53%), did not cluster with any genome from the +other databases (Fig. 3a). In contrast, the three reference databases +combined represented 10,391 species-level vOTUs, nearly half of +which were also found in the MGV. The MGV and IMG/VR data- +bases, which were both derived from whole metagenomes, shared +the greatest number of vOTUs and contained a relatively high pro- +portion of lysogenic phages from the order Caudovirales, whereas +the HuVirDB and GVD data sets, which were largely derived from +viral metagenomes, were enriched in small circular ssDNA viruses +from the Microviridae, Anelloviridae and CRESS families. +Next, we compared the four genome catalogues based on their +ability to recruit sequencing reads from a geographically diverse set +of whole metagenomes and viral metagenomes (Fig. 3c). To prevent +self matches we discarded alignments between sequencing reads +and viral genomes derived from the same original study. Overall, +MGV genomes recruited 8.6% of whole-metagenome reads, which +was 4.0-fold higher than any other database, and 40.1% of virome +reads, which was comparable with the HuVirDB at 42.3%. We also +compared the recruitment of CRISPR spacers to each viral database +as a way of quantifying host–virus connections (Fig. 3c). Overall, +37.5% of the 1.8 M spacers from UHGG genomes matched a genome +from the MGV catalogue, which was 3.25-fold higher than any +other database. The number of matched spacers and metagenomic +reads did not change considerably when using a viral database of +only species-level representatives (Fig. 3c). Together, these results +show that the MGV catalogue has substantially increased known +viral diversity, improved detection of viral reads in whole metage- +nomes and expanded coverage of host–virus connections. +Phylogenomics of intestinal Caudovirales. Caudovirales comprise +an expansive order of tailed double-stranded DNA (dsDNA) phages +found in numerous environments48 and were highly represented in +the stool metagenomes we analysed. To explore the evolution of this +group in the gut microbiome, we constructed a species-level phylo- +genetic tree based on a concatenated alignment of 77 protein-coding +marker genes (Fig. 4a)49. After removing genomes with insufficient +data (fewer than three markers or <5% representation in align- +ment), the final tree contained 25,528 species-level viral genomes +derived from the four databases of uncultivated gut viruses (MGV, +IMG/VR, HuVirDB and GVD). +Based on cumulative branch length, the MGV catalogue cov- +ered 95.7% of the total phylogenetic diversity (PD) and contained +genomes representing all major lineages across the tree (Fig. +4b). Compared with the three other databases combined, MGV +genomes resulted in a 287% increase in PD that was evenly distrib- +uted across viral and host taxonomic groups. Clostridia phages were +by far the most diverse group (41.8% of PD) because of the large +number and broad phylogenetic distribution of these vOTUs. In +contrast, Bacteroidota phages represented only 11.1% of PD with +most vOTUs falling into four primary clusters (Fig. 4a) including +one dominated by crAss-like phages (2.17% of PD). Overall, there +was poor correspondence between the classical viral families based +on tail morphology and genome-based phylogeny (for example, +nearly all lineages contained Siphoviridae annotated genomes) +which further highlights the need for a phylogeny driven taxonomy +of Caudovirales49 and other viral groups, analogous to the GTDB +taxonomy developed for Bacteria and Archaea50. +Notably, several lineages contained jumbo phages with genomes +exceeding 200 kb (518 genomes from 245 species-level vOTUs). As +with other analyses, we carefully removed flanking host regions +as well as assembly artefacts resulting in the same genome being +repeated multiple times (Methods). The largest genome was a +553,716-bp near-complete linear genome closely related to a +Prevotella phage Lak-A1 (ref. 35; 94.5% AAI over 87.1% of genes). +As with crAss-like phages, jumbo phages were rarely integrated +into a host (n = 13) although they sometimes contained integrases +(n = 121). To characterize the diversity of these viruses in greater +detail, we constructed a separate tree based on the large terminase +subunit (TerL). Compared with a recently published collection of +jumbo phages from diverse environments51, MGVs resulted in a +large expansion of phylogenetic diversity and coverage of most lin- +eages (Extended Data Fig. 2). +Interestingly, jumbo phages and other Caudovirales appeared +to have little to no preference in biogeographic distribution, as +most clades were found in all continents. We hypothesized that +region-specific phylotypes might be apparent over shorter evolu- +tionary timescales, as observed for human gut bacteria52. Towards +this goal, we used single-nucleotide variants to construct strain-level +phylogenies for 146 prevalent vOTUs with more than 100 members +(Methods). Strikingly, we observed discrete subspecies that were +highly enriched in specific geographic regions for many vOTUs +NAturE MICroBIoLogy | VOL 6 | JULy 2021 | 960–970 | www.nature.com/naturemicrobiology +963 + +ResouRce +NATuRE MICROBIOlOGy +(Extended Data Fig. 3). For example, one crAss-like subspecies pre- +dicted to infect Parabacteroides was prevalent among samples from +Asia, but rare or absent from Europe and North America. More +work is needed to understand the evolutionary drivers and genomic +adaptations underlying these phylogenetic patterns. +Functional capacity of the gut virome. Although the functional +potential of human gut bacteria and archaea has been extensively +studied43,53,54, that of intestinal phages is less well understood. To +explore this, we identified 11,837,198 protein-coding genes with +at least 20 amino acids (98.4% with start and stop codons) across +the 189,680 viral genomes from our study and compared these with +HMM databases, including KEGG55, TIGRFAM56, Pfam57, VOGDB +(http://vogdb.org/) and the Earth’s Virome database23. Overall, 45% +of viral genes did not have significant matches to any database and +75% were not assigned any biological function (Fig. 5a,b), indicat- +ing that remarkably little is known about the functional potential of +human gut viruses. +To identify the most common functions among intesti- +nal phages, we clustered the 11.8 million viral genes at 30% AAI +using MMseqs2 (ref. 58) into 459,375 de novo viral protein clusters +(Fig. 5c) including 61% with at least two members (Fig. 5d). An +accumulation curve displayed no plateau, indicating that gut phages +have a large reservoir of functional diversity that is not fully cap- +tured by this study (Fig. 5e). Clostridia phages contained the most +functional diversity with 173,187 protein clusters, reflecting the +large phylogenetic diversity of these phages. Several of the largest +protein clusters had no predicted function, including the fourth +largest with 8,319 genes, and are therefore good candidates for +experimental characterization in the future (Fig. 5f). Other large +clusters were annotated with typical viral functions, including cap- +sid formation, packaging, lysis, lysogeny, replication and transcrip- +tional regulation (Fig. 5f). +Although it is outside the scope of this article to enumerate all +viral functions and auxiliary metabolic genes, we explored two par- +ticularly unusual findings. Based on HMM searches against Pfam, +we uncovered 11,496 putative viral beta-lactamases (PF12706), +including the majority of sequences in a single protein clus- +ter with 5,832 members (Fig. 5f). Beta-lactamases are enzymes +that enable resistance to beta-lactam antibiotics such as penicil- +lins, cephalosporins and cephamycins, and pose a major global +health problem59. To validate this result, we performed homology +a +Species-level vOTUs (60,439) +MGV +(54,118) +IMG/VR +(3,132) +HuVirDB +(5,111) +GVD v.1.0 +(4,450) +15 +17 +34 +41 +169 +360 +0 +1 +552 +125 +75 +28 +0 +9 +84 +66 +154 +203 +207 +508 +505 +0 +7 +3,269 +481 +224 +127 +0 +47 +479 +2,630 +1,314 +1,672 +532 +241 +170 +0 +22 +50,048 +870 +161 +221 3 +680 +1,875 +Genus-level vOTUs (6,277) +MGV +(5,800) +IMG/VR +(1,389) +HuVirDB +(1,567) +GVD v.1.0 +(2,082) +Family-level vOTUs (1,510) +MGV +(1,434) +IMG/VR +(557) +HuVirDB +(630) +GVD v.1.0 +(781) +0 +50,000 +150,000 +0 +20,000 +50,000 +Species vOTUs +0 +0 +2,000 +4,000 +6,000 +0 +0 +400 +800 +1,400 +Number of genomes (MGV data set) +100,000 +50,000 +150,000 +Genus vOTUs +100,000 +50,000 +150,000 +Family vOTUs +100,000 +Number of genomes (MGV data set) +Number of genomes (MGV data set) +b +Percentage of reads mapped +to viral genomes +c +0 +2 +4 +6 +8 +10 +0 +10 +20 +30 +40 +50 +GVD +HuVirDB +IMG/VR +MGV +Viral data set +All viral +genomes +One genome +per vOTU +GenBank +Bulk metagenomic reads +Viral metagenomic reads +CRISPR spacers from +UHGG genomes +GVD +HuVirDB +IMG/VR +MGV +All data sets +GenBank +GVD +HuVirDB +IMG/VR +MGV +GenBank +0 +10 +20 +30 +40 +Viral data set +Viral data set +Percentage of reads mapped +to viral genomes +Percentage of spacers mapped +to viral genomes +All data sets +All data sets +Fig. 3 | genome clustering and comparison with existing databases. The 189,680 genomes from the MGV catalogue were compared with human gut virus +genomes >50% complete from three databases: IMG/VR (n = 6,895), HuVirDB (n = 9,626) and GVD (n = 4,494). a, Viral genomes were clustered into +vOTUs at approximately species, genus and family levels. b, Accumulation curves for vOTUs from the MGV catalogue. c, Percentage of reads from 1,257 +unfiltered stool metagenomes, percentage of reads from 585 viral stool metagenomes and percentage of CRISPR spacers from 286,997 UHGG genomes +mapped to viral genomes from various databases. +NAturE MICroBIoLogy | VOL 6 | JULy 2021 | 960–970 | www.nature.com/naturemicrobiology +964 + +ResouRce +NATuRE MICROBIOlOGy +searches against curated databases of antimicrobial resistance genes +using Resfams60, the NCBI AMRFinder61 and the Resistance Gene +Identifier (RGI)62. These tools revealed a combined total of only +88 resistance genes (63 using Resfams, 56 using AMRFinder and +30 using RGI), indicating low similarity between the 11,496 puta- +tive viral beta-lactamases and validated resistance genes (Extended +Data Fig. 4). Although functional metagenomic assays may uncover +bona fide viral beta-lactamases in the gut microbiome, these results +appear to support the conclusion that phages rarely encode antibi- +otic resistance genes63. +Another interesting finding was a large number of phage reverse +transcriptases (RTs) (Fig. 5f and Supplementary Table 6). Overall, +the RT domain (PF00078) was the third most common functional +annotation, next to only the helix–turn–helix DNA-binding domain +(PF01381) and phage integrase family (PF00589). RTs are known +to occur in retroviruses64, RNA-targeting CRISPR–Cas systems65 +and diversity-generating retroelements (DGRs)66. DGRs utilize +error-prone reverse transcription to generate random mutations +in the transcript of a template region (TR), which is then inserted +back into the genome at a variable region (VR), thereby generat- +ing population-level hyper-variability in a specific gene. Since the +DGR system was first characterized in a Bordetella bacteriophage66, +it has been found in human microbiomes67 and in several human +gut phages68,69. +To determine whether the viral RTs were part of the DGR sys- +tem, we used the tool DGRscan67 to identify TR–VR pairs across +79,250 high-quality viral genomes with >90% estimated complete- +ness. Confirming our hypothesis, the great majority of genomes +with an RT also contained a TR–VR (85.7% of 25,620) compared +with a small minority of those without an RT (6.5% of 53,630) +(Fig. 5g). DGRs were remarkably common in certain Caudovirales +families (for example, 84% of 6,616 Myoviridae) and among lysog- +enized viruses (50.1% of 18,187), whereas they were rare or com- +pletely absent from other Caudovirales families, ssDNA viruses +and eukaryotic viruses (Fig. 5h). Although the vast majority of +DGR gene targets were not functionally annotated, we observed +highly significant enrichment within several Pfam domains +(Supplementary Table 7) including an immunoglobulin-like +domain that was 5.9-fold more common among DGR-targeted +genes and is believed to play a role in phage interactions with car- +bohydrates on the cell surface of bacteria70. Together, these results +reveal DGRs to be more common in intestinal phages than pre- +viously thought and may point towards viral proteins involved in +molecular phage–host interactions. +Tree +scale: 1.0 +4. Newly identified +5. Lysogeny rate +6. Genome size +65.1% +Current study and previous studies +Previous +studies +only +Current +study +only +30.6% +Percentage of Caudovirales +phylogenetic diversity (PD) +3. Host taxonomy +Actinobacteriota +Proteobacteria +Bacteroidota +Firmicutes +---Bacilli +---Clostridia +---Negativicutes +Other +crAss-like +Myoviridae +Podoviridae +Siphoviridae +Other +2. Viral taxonomy +1. Viral contigs (log10) +1,000 +0 +6. Genome size +<100 kb +100–200 kb +>200 kb +5. Lysogeny rate +(Percentage of +prophages/OTU) +4. Newly identified +Yes +No +0 +>50 +25 +3. Host taxonomy +1. Viral contigs from current study +2. Viral taxonomy +a +b +Newly identified Caudovirales lineage +Previously known Caudovirales lineage +Percentage + of PD +Total PD +Viral taxonomy +4.3% +c +d +10 +100 +crAssphage +Podoviridae +Myoviridae +Unknown +Siphoviridae +0.31% +2.17% +4.22% +13.88% +47.61% +61.03% +Other +Other +Negativicutes +Proteobacteria +Actinobacteriota +Bacilli +Bacteroidota +Unknown +Clostridia +4.8% +5.34% +6.92% +5.81% +10.24% +11.13% +40.9% +41.79% +Percentage + of PD +Total PD +Host taxonomy +Fig. 4 | Phylogenomics of intestinal Caudovirales. A phylogenetic tree was constructed from 25,528 species-level genomes derived from the MGV and +other databases (IMG/VR, HuVirDB and GVD). a, Phylogeny of intestinal Caudovirales. Tree was plotted using iToL74 and to improve visualization only +one genome per genus-level vOTU is displayed. Branch colour indicates whether a lineage is represented by a previously published study (black) or is +unique to the MGV catalogue (green). Outer rings display metadata for each vOTU. b, PD was calculated by taking the sum of branch lengths represented +by species-level viral genomes. c,d, MGVs from the current study result in a large gain in PD, which is consistent across (c) viral families and (d) viruses +infecting different host groups. +NAturE MICroBIoLogy | VOL 6 | JULy 2021 | 960–970 | www.nature.com/naturemicrobiology +965 + +ResouRce +NATuRE MICROBIOlOGy +Discussion +In this study, we performed large-scale data mining of publicly +available metagenomes to identify 189,680 draft-quality viral +genomes representing an estimated 54,118 species-, 5,800 genus- +and 1,434 family-level vOTUs. This large resource contains exten- +sive viral genomic diversity not found in other databases, improves +detection of viral reads in microbiomes and represents numerous +diverse and previously uncharacterized viral groups. Through a +combination of approaches, we were able to predict host–virus link- +ages that cover the majority of viral and prokaryotic diversity in +the gut microbiome. These host–virus linkages may be important +in the future for understanding disease processes, designing phage +therapies or understanding host–virus co-evolutionary dynamics. +Despite large-scale annotation efforts, we were only able to assign +preliminary biological functions to 25% of viral genes, indicating +that more work and new methods are needed to predict protein +function in viral genomes, such as deep learning71 and functional +metagenomic assays72. Although the current study focused exclu- +sively on DNA viruses, future studies could use metatranscrip- +tomics data to study RNA viruses or gene expression patterns. +While this manuscript was in review, Camarillo-Guerrero +et al.73 published the Gut Phage Database (GPD), a collection of +∼142,000 non-redundant viral genomes (>10 kb) identified from +28,060 human gut metagenomes and 2,898 gut bacterial genomes. +After applying CheckV, we found the GPD represents 79,889 viral +contigs with >50% completeness that form 46,480 species-level +vOTUs, which is 14% less than the 54,118 vOTUs from the MGV +(Extended Data Fig. 5). Differences between viral catalogues are +4 +8 +12 +Phage portal protein (PF04860) +Phage antirepressor protein (PF03374) +ssDNA binding protein (PF05772) +HTH domain of resolvase (PF02796) +tRNA-splicing ligase RtcB (K14415) +HNH endonuclease (PF01844) +Reverse transcriptase (PF00078) +Phage terminase (PF03354) +DNA repair protein (K03546) +DNA methylase (PF02384) +Restriction endonuclease-like domain (PF08774) +Phage tail tube protein (PF09393) +Baseplate J-like protein (PF04865) +DNA methylase (PF01555) +Recombinase, RecT family (K07455) +DNA polymerase (K21237) +Thymidylate synthase (K03465) +Reverse transcriptase (PF00078) +LysM domain (PF01476) +Peptidoglycan amidohydrolase (PF01520) +Prohead serine protease (PF04586) +SNF2 family helicase (PF00176) +Terminase small subunit (PF03592) +Phage connector (PF05352) +Phage tail protein (PF05489) +Phage integrase (PF00589) +Putative beta-lactamase domain (PF12706) +Phage portal protein (PF05136) +Phage tail sheath protein (PF04984) +ssDNA binding protein (PF00436) +Phage terminase large subunit +ssDNA binding protein (PF00436) +Phage tail tube protein (PF04985) +Phage terminase large subunit +ssDNA binding protein (PF00436) +ATPase family (PF00004) +Reverse transcriptase (PF00078) +Reverse transcriptase (PF00078) +Exoribonuclease (K03698) +Phage portal protein (PF05133) +Phage portal protein (PF04860) +HTH domain (PF13936) +Phage terminase (PF03354) +Terminase large subunit (PF05876) +Peptidoglycan amidohydrolase (PF01510) +Reverse transcriptase (PF00078) +a +f +g +h +b +c +d +e +Percentage of 11,837,198 genes +hitting database +Earth’s Virome VPFs +VOGDB +Pfam-A +KEGG +TIGRFAM +10.5% +4.6% +Unknown +function +No hit +45% +Known +function +25% +30% +11,837,198 viral genes +459,375 protein clusters +14% +11% +No hit +75% +Unknown +function +Known +function +Number of genes (×106) +Number of genes (×103) +Number of +protein clusters (×105) +0 +4 +8 +12 +0 +2 +4 +Structural +Packaging and assembly +Lysis +Replication +DNA binding/regulation +Other +Unknown function +Functional category +0 +Largest 75 viral protein clusters +34.5% +27.5% +27.3% +Reverse +transcriptase +(PF00078) +TR–VR +(DGRscan) +n = 21,945 +(27.7%) +n = 3,675 +(4.6%) +n = 3,487 +(4.4%) +DGR absent: lack of RT and TR–VR +n = 50,143 (63.3%) +DGR +present +High-quality viral genomes +from current study +n = 79,250 +Myoviridae (6,616) +Siphoviridae (23,364) +crAssphage (4,543) +Papillomaviridae (75) +Podoviridae (4,151) +Microviridae (2,029) +Inoviridae (107) +Cress DNA (68) +Herelleviridae (16) +Adenoviridae (11) +Retrovirales (10) +Anelloviridae (5) +Fraction of high-quality genomes +with DGRs +0 +0.2 +0.4 +0.6 +0.8 +1.0 +Viral taxonomy +Yes (18,187) +No (53,901) +dsDNA (69,803) +ssDNA (2,215) +Baltimore classification +Predicted prophage +459,375 protein clusters +2–10 +genes +(41%) +1 gene +(39%) +10–100 +genes +>100 +genes +16% +4% +Fig. 5 | Functional landscape of intestinal phages. a, Protein-coding viral genes were identified across all MGVs and compared with profile HMMs from +five databases. b, Forty-five per cent of genes fail to match any HMM, 30% match an HMM of unknown function and 25% match an HMM of known +function. c, The 11,837,198 genes were clustered at 30% AAI using MMseqs2 into 459,375 protein clusters. d, Size distribution of protein clusters. e, An +accumulation curve of protein clusters has not reached an asymptote. f, Functional annotations for the largest 75 protein clusters. Reverse transcriptases +are highlighted in red. g, Prediction of DGRs based on the combination of the reverse transcriptase gene (PF00078) and TR–VR pair identified using +DGRscan. A large fraction of MGVs contain the DGR system. h, DGR prevalence across different categories of viruses. DGRs are most common in +lysogenic, dsDNA viruses from the Myoviridae family. +NAturE MICroBIoLogy | VOL 6 | JULy 2021 | 960–970 | www.nature.com/naturemicrobiology +966 + +ResouRce +NATuRE MICROBIOlOGy +due to several factors, including data sets used for metagenome +mining, methods for viral identification and criteria for sequence +inclusion. For example, the MGV had greatly improved coverage of +Microviridae which were excluded from the GPD due to their short +length (mean = 4.9 kb). Combined, the MGV and GPD represented +75,187 species-level vOTUs, indicating that the two catalogues con- +tain complementary viral diversity. In the future, these and other +large-scale viral genome catalogues could be integrated to create +a unified and standardized community resource, as recently per- +formed for human gut microbial genome catalogues43. +Methods +Development of viral detection pipeline. We used a combination of four viral +signatures to identify viral metagenomic contigs: (1) the presence of viral protein +families, (2) the absence of microbial protein families, (3) the presence of viral +nucleotide signatures, and (4) multiple adjacent genes on the same strand. For +the presence of viral protein families, we used HMMs for 23,841 viral protein +families from the IMG/VR database23 (downloaded 1 June 2019) after excluding +1,440 commonly found in microbial genomes or plasmids. For the absence of +microbial protein families, we used HMMs for 16,260 protein families from the +Pfam-A database57 (release 31) after excluding 452 commonly found in viruses. +Proteins from metagenomic contigs were searched against HMMs from IMG/VR +and Pfam-A using hmmsearch within the HMMER package v.3.1b2 (options: −Z +1, e-value: <1 × 10−10)75 and were classified as either viral or microbial based on +the database containing the top hit. For the presence of viral nucleotide signatures, +we applied the tool VirFinder v.1.1 (ref. 32) to metagenomic contigs, which scores +sequences using a combination of k-mer frequencies and machine learning. For +multiple adjacent genes on the same strand, we quantified the strand switch rate by +dividing the number of strand switches by the number of genes on each contig. +Benchmarking viral detection pipeline. We evaluated our viral detection +pipeline on mock data sets we created that contained genome fragments from +human-associated viruses and bacteria. Each mock data set contained genome +fragments from six diverse categories of viruses: (1) crAss-like phages from +the human gut45, (2) Lak-phages from human and mammalian microbiomes35, +(3) bacteriophages assembled from human gut viromes76, (4) phages with +CRISPR-spacer matches to gut isolated microbial genomes, (5) isolate dsDNA +human viruses and (6) isolate ssDNA human viruses. Non-viral genome fragments +were derived from: (1) gut isolated microbial genomes and (2) plasmids genomes. +We generated 2,000 genomic fragments from randomly sampled genomes within +each of the eight categories at each of seven different fragments lengths (1, 2, 5, +10, 20, 50 and 100 kb). The TPR (percentage of viral contigs classified as viral) +and FPR (percentage of non-viral contigs classified as viral) were calculated +for over 77,000 combinations of cut-off values for the four viral signatures. We +selected up to five different combinations of cut-offs that resulted in the highest +classification score for each fragment length, where the classification score was +based on a weighted combination of the TPR and FPR (score = TPR − 50 × FPR; +Supplementary Table 3). We assigned a very high negative weight to the FPR to +avoid many false positives in the metagenomes which are expected to contain +mostly non-viral sequences. We compared the performance of our method with +VirSorter v.1.0.5 (ref. 33) and to VirFinder v.1.1 (ref. 32) using the same benchmark +data set (Supplementary Table 2). VirFinder was run using default options and +we applied p-value thresholds of 0.05, 0.01 and 0.001 for classifying genome +fragments as viral. VirSorter was run with and without the ‘-virome’ option, and +we used VirSorter categories 1 and 2 to classify a fragment as viral (excluding low +confidence predictions and integrated prophages). We also evaluated VirSorter +when including predicted prophages (categories 4 and 5). +Application of pipeline to identify human gut viruses from whole +metagenomes. To perform a comprehensive search for human gut viruses, we +downloaded 18,271 publicly available metagenomic assemblies from human +stool samples totalling 2.25 ×1012 bases and corresponding to 11,810 unique +biological samples (Supplementary Table 1). Assemblies were obtained from two +recent studies29,31 and the MGnify database (accessed on 16 April 2019)36. We +excluded assemblies from environments other than human gut and those that +could not be assigned to an accession number from the NCBI SRA database. +Metadata were obtained from previous studies and the NCBI BioSample database77 +(Supplementary Table 2). We applied our viral detection pipeline method to +identify 4,436,008 contigs longer than 1 kb across the 18,271 metagenomic +assemblies (Supplementary Table 1), which were de-replicated to 3,481,684 +sequences at 100% ANI over 100% the length of the shorter sequence. +Gene calling and identifying viruses with alternative genetic codes. Prodigal v.2.6.3 +(ref. 78) was used to identify protein-coding genes in the 3,481,684 viral genomes +using the flag ‘-p meta’ optimized for metagenomes. Additionally, we ran a custom +pipeline to identify viruses using an alternative genetic code. Specifically, Prodigal was +run using the standard code (11), and three alternative genetic codes: TGA recoded +(code 4 or 25), TAG recoded (code 15) and TAA recoded (code 90), as previously +described by Ivanova et al.79. To reduce false positives this procedure was only run on +viral contigs longer than 10 kb with GC content <50%. For each viral contig, Prodigal +outputs a GFF file that includes a coding potential score for every predicted gene. +To evaluate the genetic codes, we took the sum of coding potential scores per contig. +An alternative genetic code was predicted if it’s total coding potential score was the +greatest and at least 10% greater than the standard genetic code. +Viral reference genomes used for comparison. Viral genomes from the MGV +were compared against four reference databases: IMG/VR v.2.0 (ref. 22), GVD v.1.0 +(ref. 27), HuVirDB v.1.0 (ref. 26) and NCBI GenBank. For IMG/VR, we extracted +28,697 viral contigs which were identified from 490 whole metagenomes from +human stool samples using the Earth’s Virome Pipeline23. For GVD, we used +all 13,203 viral contigs, which were identified from 471 viral metagenomes and +98 whole metagenomes using a combination of tools including VirSorter and +VirFinder and previously clustered into viral populations. An updated version of +the GVD was released while the paper was under review but was not analysed here. +For the HuVirDB, we extracted 929,886 contigs longer than 1 kb from 1,543 viral +metagenomes from human stool samples. Because no viral prediction was previously +applied, we ran the viral prediction pipeline developed for the current manuscript. +For NCBI GenBank (downloaded 1 June 2019), we extracted 28,996 complete viral +genomes after removing those labelled as incomplete, contaminated, or chimeric. +Quality control of viral genomes. We applied CheckV v.0.7.0 (database v.0.6)37 +to all viral sequences to identify closed genomes, estimate genome completeness +and remove flanking host regions on assembled proviruses. Putative complete +genomes were predicted based on direct terminal repeats (minimum 20 bp), +inverted terminal repeats (minimum 20 bp) or provirus integration sites (host +region predicted on both ends of viral contig), and were additionally required to +display >90% estimated completeness based on comparison with CheckV reference +genomes. A small number of sequences were removed that contained large repeats +spanning >30% of the contig length. We selected all genomes with >50% estimated +completeness for further analysis, resulting in 189,680 viral contigs from the MGV +catalogue, 6,895 contigs from IMG/VR, 4,494 from GVD, 9,626 from HuVirDB +and 28,996 from GenBank. We estimated the amount of non-viral DNA from +cellular organisms among MGV sequences by searching for 16S and 18S rRNA +genes using Barrnap v.0.9-dev (https://github.com/tseemann/barrnap) with models +for Bacteria, Archaea and Eukaryotes. Alignments were required to cover ≥70% of +the 16S or 18S rRNA gene and display an e-value <1 × 10−5. This same procedure +was applied to the 18,271 metagenomic assemblies used for viral discovery to +estimate the background levels of 16S and 18S rRNA genes. +Taxonomic annotation. Viral genomes were annotated based on amino acid +alignments to a database of proteins derived from complete NCBI GenBank +genomes and crAss-like genomes. Annotations were performed using the +Baltimore classification (DNA, dsDNA, ssDNA, ssDNA-RT, dsRNA, RNA, ssRNA, +ssRNA-RT) as well the ICTV taxonomy at the order, family and genus ranks. +DIAMOND v.0.9.32 (options: –query-cover 50–subject-cover 50–e-value 1e-5– +max-target-seqs 1000)80 was used to align viral proteins to the reference database. +The taxonomy of the top database hit was then transferred to each protein at each +taxonomic rank (Baltimore, order, family, genus). In cases where the taxonomy +of the top hit was missing, we used the next hit if its bit-score was within 25% +of the top hit. For each viral genome, we aggregated annotations across proteins +after weighting by bit-scores. Each viral genome was then annotated at the lowest +taxonomic rank having >70% agreement across annotated proteins. At the family +rank, we required genomes to have a minimum of two annotated proteins with +>30% AAI to the database. At the genus rank, we required genomes to have a +minimum of three annotated proteins with >40% average AAI to the database. +As validation, we applied our pipeline to taxonomically annotated genomes from +NCBI GenBank after removing closely related genes from the database. Our +pipeline achieved average TPRs of 90.0%, 98.7%, 92.2% and 73.5% at precision +values of 95.6%, 99.9%, 99.3% and 96.5% for taxonomic ranks of Baltimore, order, +family and genus, respectively. +Host prediction. We used a combination of CRISPR-spacer matches and ≥1 kb +genome sequence matches to associate viral genomes to Bacterial and Archaeal +genomes from the UHGG collection43. The UHGG contains 286,997 genomes, +representing 4,644 species of Bacteria and Archaea from the human gut that are +taxonomically annotated using GTDB-tk v.0.3.1 (GTDB release 89)81. Many of +the UHGG genomes are metagenome-assembled genomes, which sometimes +contain erroneously binned sequences, including those from viruses. To address +this, we conservatively identified and removed 2,043,531 contigs from UHGG +genomes where the host region comprised <50% of the contig length. We then +compared the remaining UHGG contigs with viral genomes and identified ≥1 kb +genome sequence matches with ≥96% DNA identity using blastn from the blast+ +package v.2.9.0 (ref. 82). Next, we identified 1,846,441 spacers from 145,053 +CRISPR arrays from 79,735 UHGG genomes using a combination of CRT83 and +PILER-CR84 with default parameters. Redundant CRISPR arrays predicted by both +tools were merged based on genomic coordinates. Spacers were searched against +NAturE MICroBIoLogy | VOL 6 | JULy 2021 | 960–970 | www.nature.com/naturemicrobiology +967 + +ResouRce +NATuRE MICROBIOlOGy +viral genomes using blastn from the blast+ package v.2.9.0 (options: -dust = no +-word-size = 18), allowing a maximum of one mismatch or gap over ≥95% of +the spacer length. For each viral genome, we then aggregated connections to +UHGG genomes and identified the lowest host taxonomic rank resulting in >70% +agreement across connections. +Clustering viral genomes into vOTUs. All viral genomes with >50% completeness +were clustered into species-level vOTUs on the basis of 95% ANI and 85% +alignment fraction (AF) of the shorter sequence, as recommended by Roux et al.38. +ANI and AF were estimated between all genome pairs using a custom script from +the CheckV repository. The script performs all-versus-all local alignments using +blastn from the blast+ package v.2.9.0 (options: perc_identity = 90 max_target_ +seqs = 10000). ANI is computed as the length-weighted average DNA identity +across local alignments between each genome pair. AF is computed by merging +alignment coordinates between each genome pair and dividing by the length of +each genome. This approach gave consistent results compared to MUMMer4 +(ref. 85), while running in a small fraction of the time. Clustering was performed +using a greedy, centroid-based algorithm in which: (1) genomes were sorted by +length, (2) the longest genome was designated as the centroid of a new cluster, +(3) all genomes within 95% ANI and 85% AF were assigned to that cluster, and +steps 2 and 3 were repeated until all genomes had been assigned to a cluster. +To identify genus- and family-level vOTUs, we clustered viral genomes using +a combination of gene sharing and AAI. For computational efficiency, only the +longest genome per species-level vOTU was included. Blastp from the DIAMOND +package v.0.9.25.126 was used with options ‘-e-value 1 × 10−5–max-target-seqs +10,000’ to align all viral proteins. For each pair of genomes, we identified shared +genes (e-value <1 × 10−5), computed their AAI, and computed the percentage of +genes shared. Edges between genomes were filtered based on their minimum AAI +and gene sharing. Clustering was performed with MCL v.14-137 using different +values for the inflation factor parameter. We then selected the filtering thresholds +and MCL inflation factor that resulted in the highest agreement with genus- and +family-level annotations from NCBI RefSeq, respectively. At the family level, we +filtered connections between genomes with <20% AAI or <10% genes shared and +used an inflation factor of 1.2. At the genus level, we filtered connections between +genomes with <50% AAI or <20% gene sharing and used an inflation factor of +2.0. We benchmarked our approach on taxonomically annotated genomes from +NCBI, showing that viral clusters displayed high taxonomic homogeneity (that is +the percentage of genomes from each cluster assigned to the same taxon; genus +rank = 95.1%, family rank = 93.7%), though sometimes split known taxa into +multiple clusters (that is percentage of genomes from each taxon assigned to the +same cluster: genus rank = 92.6%, family rank = 74.5%). +Metagenomic read recruitment. Read mapping was performed to viral genomes +databases to assess their coverage of viruses in microbiomes. First, we downloaded +short reads from human gut viromes analysed by the HuVirDB plus short reads +from three recent gut virome studies14,86,87. Short reads from whole metagenomes +were downloaded for 1,257 stool samples from various countries (representing +up to 50 samples per country). To ensure that viromes were mostly free of +cellular contamination, we ran the viromeQC tool88 and retained viromes with +an enrichment score >10, as recommended by the authors. For computational +efficiency, we only analysed the first 1,000,000 sequencing reads from each data +set. For quality control, we discarded reads that were either too short (<70 bp), +contained ambiguous base calls, had low base quality scores (mean quality score +<30) or mapped to the human genome (build hg19). +Next, we used Bowtie v.2.3.2 (ref. 89) to construct genome indexes for read +mapping. Five indexes were created using all genomes from each of the four gut +human virus databases (MGV, IMG/VR, HuVirDB, GVD), plus NCBI GenBank. +Five additional indexes were created using only a single genome per species-level +vOTU. Next, we used Bowtie 2 (options ‘–very-sensitive -k 20’) to align sequencing +reads to each of the 10 genome indexes. Alignments between sequencing reads +and viral genomes derived from the same SRA study were discarded to prevent +overestimation of mapping rates. Additionally, alignments with mapping identity +<95% (for example, edit distance >5 for 100-bp read) were discarded. After these +filtering steps we quantified the percentage of high-quality, non-human reads that +mapped to each database. +Phylogenetic analyses. We constructed a phylogeny of Caudovirales genomes +using the method described by Low et al.49. First, we identified the set of 77 +Caudovirales markers in the representative genomes of 60,439 species-level vOTUs. +HMMs for the 77 markers were searched against the protein sequences and the top +hits individually aligned to the profile HMMs using HMMER v.3.1b2. Individual +marker alignments were then trimmed to retain positions with less than 50% gaps +using trimAl v.1.4 (ref. 90) and concatenated, filling in gaps for missing markers +where necessary. Only genomes containing at least three markers and having data +at >5% of alignment columns were retained. This resulted in a multiple sequence +alignment of 28,780 genomes with 22,711 alignment columns. We then inferred +a concatenated protein phylogeny from the multiple sequence alignment using +FastTree v.2.1.9 (ref. 91) under the WAG + G model with the additional flags ‘-mlacc +2’ and ‘-slownni’. The tree was then midpoint-rooted and visualized using iToL74. +In addition, we constructed core-genome single-nucleotide polymorphism +(SNP) phylogenies of individual species-level vOTUs with at least 100 genomes. +SNPs were identified by aligning all genomes to the longest genome in the cluster +using nucmer from the MUMmer4 package v.4.0.0beta2 (ref. 85) with default +options. SNPs were identified at genomic positions covered by ≥50% of genomes +and we retained all genomes with data at ≥50% of positions. FastTree v.2.1.9 was +used to construct phylogenetic trees using default options. +Functional annotation and protein clustering. Some 11,837,198 protein-coding +genes were identified from the 189,680 MGVs using Prodigal and genes were +annotated based on HMM searches against protein family databases: KEGG55, +TIGRFAM56, Pfam-A57, VOGDB (http://vogdb.org) and the Earth’s Virome viral +protein families database23. All searches were performed using the hmmsearch +utility in the HMMER package v.3.1b2 (ref. 75) with default parameters. Each gene +was annotated by each database according to its top scoring alignment with a +bit-score ≥50, except for Pfam and TIGRFAM where trusted cut-offs were used. +Antibiotic resistance genes were identified using three tools: (1) the Resistance +Gene Identifier v.5.1.0 (ref. 62) using option ‘–low_quality’ with gene-specific +bit-score thresholds, (2) the NCBI AMRFinder tool v.3.8.4 (ref. 61) using default +options and (3) the Resfams database60 using hmmsearch with HMM-specific +bit-score thresholds. DGRs were identified using the tool DGRscan67 with default +options. All proteins were clustered at 30% AAI and 70% alignment coverage using +MMseqs2 v.10.6d92c58. +Reporting Summary. Further information on research design is available in the +Nature Research Reporting Summary linked to this article. +Data availability +Access to the full catalogue of viral genomes, protein clusters, diversity-generating +retroelements and CRISPR spacers is provided without restrictions at https:// +portal.nersc.gov/MGV. Any requests for further data should be directed to the +corresponding authors. +Code availability +Supporting code, including our viral detection pipeline, is provided at https:// +github.com/snayfach/MGV. +Received: 9 March 2021; Accepted: 25 May 2021; +Published online: 24 June 2021 +references + 1. Lynch, S. V. & Pedersen, O. The human intestinal microbiome in health and +disease. N. Engl. J. Med. 375, 2369–2379 (2016). + 2. Ogilvie, L. A. et al. Genome signature-based dissection of human gut +metagenomes to extract subliminal viral sequences. Nat. Commun. 4, +2420 (2013). + 3. Reyes, A. et al. 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The human gut virome is highly diverse, stable, and +individual specific. Cell Host Microbe 26, 527–541 (2019). + 88. Zolfo, M. et al. Detecting contamination in viromes using ViromeQC. +Nat. Biotechnol. 37, 1408–1412 (2019). + 89. Pongor, L. S., Vera, R. & Ligeti, B. Fast and sensitive alignment of microbial +whole genome sequencing reads to large sequence datasets on a desktop PC: +application to metagenomic datasets and pathogen identification. PLoS ONE +9, e103441 (2014). + 90. Capella-Gutierrez, S., Silla-Martinez, J. M. & Gabaldon, T. trimAl: a tool for +automated alignment trimming in large-scale phylogenetic analyses. +Bioinformatics 25, 1972–1973 (2009). + 91. Price, M. N., Dehal, P. S. & Arkin, A. P. A. FastTree 2 – approximately +maximum-likelihood trees for large alignments. PLoS ONE 5, e9490 (2010). +Acknowledgements +We thank S. Roux for analysis of jumbo phages. The work was conducted in the +Environmental Genomics and Systems Biology Division at the E.O. Lawrence Berkeley +National Laboratory. Funding was provided by the Chan Zuckerberg Biohub, the +Autoimmunity Research Foundation (FP00010476), the Australian Research Council +Laureate Fellowship (FL150100038) and the National Insititutes of Health (R01AI148623 +and P30 CA124435). +Author contributions +S.N., D.P.-E. and N.C.K. conceived of the project. S.N. performed experiments, analysed +data and wrote the manuscript. S.J.L. constructed the Caudovirales phylogeny. D.P.-E., +H.S. and L.C. contributed to analysis of protein families. N.N.I. identified phages using +alternative genetic codes. A.D.P. and M.A.F contributed funding. N.C.K. supervised the +project. All authors reviewed and approved the manuscript. +Competing interests +P.H. is a co-founder of Microba Life Sciences, which is a microbial genomics company +developing microbiome-based diagnostics and therapeutics and offers metagenomic gut +microbiome reports. All other authors declare no competing interests. +Additional information +Extended data is available for this paper at https://doi.org/10.1038/s41564-021-00928-6. +Supplementary information The online version contains supplementary material +available at https://doi.org/10.1038/s41564-021-00928-6. +Correspondence and requests for materials should be addressed to S.N. or N.C.K. +Reprints and permissions information is available at www.nature.com/reprints. +Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in +published maps and institutional affiliations. +Open Access This article is licensed under a Creative Commons +Attribution 4.0 International License, which permits use, sharing, adap- +tation, distribution and reproduction in any medium or format, as long +as you give appropriate credit to the original author(s) and the source, provide a link to +the Creative Commons license, and indicate if changes were made. The images or other +third party material in this article are included in the article’s Creative Commons license, +unless indicated otherwise in a credit line to the material. If material is not included in +the article’s Creative Commons license and your intended use is not permitted by statu- +tory regulation or exceeds the permitted use, you will need to obtain permission directly +from the copyright holder. To view a copy of this license, visit http://creativecommons. +org/licenses/by/4.0/. +© The Author(s) 2021 +NAturE MICroBIoLogy | VOL 6 | JULy 2021 | 960–970 | www.nature.com/naturemicrobiology +970 + +ResouRce +NATuRE MICROBIOlOGy +ResouRce +NATuRE MICROBIOlOGy +Extended Data Fig. 1 | Impact of assembly methods on viral recovery from gut metagenomes. The MGV catalogue was formed using metagenomic viral +contigs identified from three studies that performed large-scale assembly of human stool metagenomes. The CIBIO and MGnify studies used MetaSPAdes +for metagenomic assembly while the JGI study used MEGAHIT. To explore the effect of assembler on virus identification, we compared viral contigs +identified from a common set of 752 stool samples which were assembled by all three studies and were each represented by a single SRA run accession. a, +The number of vOTUs represented by viral contigs (>50% completeness) from each of the three studies. A similar number of vOTUs were identified from +metagenomic contigs assembled by each study. b, The number of viral contigs at different quality levels identified from each of the three studies. A greater +number of complete and high-quality viral genomes are recovered from the MEGAHIT assemblies. +NAturE MICroBIoLogy | www.nature.com/naturemicrobiology + +Dataset +Citation +Complete +High-quality Medium-guality +Low-quality +MGnify +Mitchelletal.2019 +363 +616 +1,778 +63,192 +CIBIO +Pasollietal.2019 +389 +592 +1,755 +62,198 +JGI +Nayfachetal.2019 +497 +676 +1,651 +60,885ResouRce +NATuRE MICROBIOlOGy +ResouRce +NATuRE MICROBIOlOGy +Extended Data Fig. 2 | Diversity of jumbo phages identified in the MgV dataset. The tree includes MGV sequences alongside a reference set of +metagenome-assembled jumbo phages published by Al-Shayeb et al.51. Branches leading to MGV sequences, or clades composed exclusively of MGV +sequences, are highlighted in red. Nodes with support < 50% were collapsed, and nodes with support ≥ 80% are indicated with a grey circle on the +corresponding branch. Outer rings indicate the genome quality and continent of origin for MGV sequences. When sequences from different continents +were 100% identical and only 1 sequence was included in the tree, the different continents of origin are indicated with stacked coloured squares. For box +plots, the middle line denotes the median, the box denotes the interquartile range (IQR), and the whiskers denote 1.5× the IQR. +NAturE MICroBIoLogy | www.nature.com/naturemicrobiology + +ResouRce +NATuRE MICROBIOlOGy +ResouRce +NATuRE MICROBIOlOGy +Extended Data Fig. 3 | Strain level phylogeography of prevalent human gut phages. Core-genome SNP phylogenies were constructed for individual +species-level vOTUs with at least 100 genomes. The figure shows three distinct vOTUs displaying a strong signature of phylogeography. For each tree, viral +genomes are displayed as tips with colours indicating the geographic origin of the metagenomic sample. +NAturE MICroBIoLogy | www.nature.com/naturemicrobiology + +ResouRce +NATuRE MICROBIOlOGy +ResouRce +NATuRE MICROBIOlOGy +Extended Data Fig. 4 | Antibiotic resistance genes identified from 11.8 million viral proteins. a-b, Viral genes with putative beta-lactamase domains +identified based on hits to the Pfam and KEGG databases, respectively. c-e, Resistance genes (including beta-lactamases) identified using Resfinder, +AMRfinder, or the Resistance Gene Identifier (RGI), respectively. f, Overlap of resistance genes identified by Resfinder, AMRfinder, and RGI. Most viral +proteins identified with putative beta-lactamase domains are not confirmed as antibiotic resistance genes. +NAturE MICroBIoLogy | www.nature.com/naturemicrobiology + +ResouRce +NATuRE MICROBIOlOGy +ResouRce +NATuRE MICROBIOlOGy +Extended Data Fig. 5 | Comparison of viral contigs from the MgV and gPD catalogues. a, The number of viral contigs with at least 50% completeness +from the MGV and GPD catalogues. The GPD catalogue contains 142,809 viral contigs when including those with <50% completeness. Contigs from +each catalogue where clustered at 95% ANI over 85% the length of the shorter sequence to form species-level vOTUs. b, MGV and GPD catalogues +were clustered together using the longest contig from each vOTU. c, The histograms show the similarity between contigs from the MGV (n = 54,118) and +GPD (n = 46,480) catalogues. d, Similarity to the GPD catalogue for MGV contigs from different viral families: Siphoviridae (n = 22,513), Podoviridae +(n = 5,075), Myoviridae (n = 2,560), crAss-like (n = 948), Caudovirales other (n = 19,633), Microviridae (n = 2,133), CRESS DNA (n = 115), other (n = 1,141). +NAturE MICroBIoLogy | www.nature.com/naturemicrobiology + +1 +nature research | reporting summary +October 2018 +Corresponding author(s): +Stephen Nayfach and Nikos Kyrpides +Last updated by author(s): April 15, 2021 +Reporting Summary +Nature Research wishes to improve the reproducibility of the work that we publish. This form provides structure for consistency and transparency +in reporting. For further information on Nature Research policies, see Authors & Referees and the Editorial Policy Checklist. +Statistics +For all statistical analyses, confirm that the following items are present in the figure legend, table legend, main text, or Methods section. +n/a Confirmed +The exact sample size (n) for each experimental group/condition, given as a discrete number and unit of measurement +A statement on whether measurements were taken from distinct samples or whether the same sample was measured repeatedly +The statistical test(s) used AND whether they are one- or two-sided +Only common tests should be described solely by name; describe more complex techniques in the Methods section. +A description of all covariates tested +A description of any assumptions or corrections, such as tests of normality and adjustment for multiple comparisons +A full description of the statistical parameters including central tendency (e.g. means) or other basic estimates (e.g. regression coefficient) +AND variation (e.g. standard deviation) or associated estimates of uncertainty (e.g. confidence intervals) +For null hypothesis testing, the test statistic (e.g. F, t, r) with confidence intervals, effect sizes, degrees of freedom and P value noted +Give P values as exact values whenever suitable. +For Bayesian analysis, information on the choice of priors and Markov chain Monte Carlo settings +For hierarchical and complex designs, identification of the appropriate level for tests and full reporting of outcomes +Estimates of effect sizes (e.g. Cohen's d, Pearson's r), indicating how they were calculated +Our web collection on statistics for biologists contains articles on many of the points above. +Software and code +Policy information about availability of computer code +Data collection +Prodigal v2.6.3, HMMER v3.1b2, VirFinder v1.1, DIAMOND v0.9.25.126, Barrnap v0.9-dev, blast+ v.2.9.0, Bowtie v2.3.2, MCL v14-137, +FAMSA v1.2.5, trimAL v1.4, FastTree v2.1.9, iTOL, MUMmer4 v4.0.0beta2, CRT, PILER-CR, AMRFinder v3.8.4, Resistance Gene Identifier +v.5.1.0, MMseqs2 v10.6d92c, PSI-BLAST, CheckV v0.7.0, viromeQC, DGRscan +Data analysis +See software listed above +For manuscripts utilizing custom algorithms or software that are central to the research but not yet described in published literature, software must be made available to editors/reviewers. +We strongly encourage code deposition in a community repository (e.g. GitHub). See the Nature Research guidelines for submitting code & software for further information. +Data +Policy information about availability of data +All manuscripts must include a data availability statement. This statement should provide the following information, where applicable: +- Accession codes, unique identifiers, or web links for publicly available datasets +- A list of figures that have associated raw data +- A description of any restrictions on data availability +Access to the full dataset of viral genomes, protein clusters, diversity generating retroelements, and CRISPR spacers is provided without restrictions at https:// +portal.nersc.gov/MGV. Any requests for further data should be directed to the corresponding authors. + +natureresearch2 +nature research | reporting summary +October 2018 +Field-specific reporting +Please select the one below that is the best fit for your research. If you are not sure, read the appropriate sections before making your selection. +Life sciences +Behavioural & social sciences + Ecological, evolutionary & environmental sciences +For a reference copy of the document with all sections, see nature.com/documents/nr-reporting-summary-flat.pdf +Life sciences study design +All studies must disclose on these points even when the disclosure is negative. +Sample size +We used 18,271 assembled human gut metagenomes for 11,810 samples. These represent all available datasets from the gut microbiome +with SRA accession codes at the time we started our project. +Data exclusions +We excluded datasets that were not from a human stool sample or where SRA accession codes could not be determined. +Replication +Not applicable to our study. All the datasets analyzed were publicly available, and therefore we did not generate any additional data for +replication. +Randomization +Not applicable to our study. Any conditions of the samples (e.g. geographic location or disease state) were already determined before our +study began since the datasets were publicly available. +Blinding +Not applicable to our study for the same reason given above. +Reporting for specific materials, systems and methods +We require information from authors about some types of materials, experimental systems and methods used in many studies. Here, indicate whether each material, +system or method listed is relevant to your study. If you are not sure if a list item applies to your research, read the appropriate section before selecting a response. +Materials & experimental systems +n/a Involved in the study +Antibodies +Eukaryotic cell lines +Palaeontology +Animals and other organisms +Human research participants +Clinical data +Methods +n/a Involved in the study +ChIP-seq +Flow cytometry +MRI-based neuroimaging + diff --git a/kb_49/content/tmp_files/load_file.txt b/kb_49/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..51a204f96514f24e4031849df1c76908f7d8731c --- /dev/null +++ b/kb_49/content/tmp_files/load_file.txt @@ -0,0 +1,1608 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf,len=1607 +page_content='8Department of Microbiology and Immunology, Stanford University, Stanford, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 9 1Environmental Genomics and Systems Biology Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 2U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Department of Energy Joint Genome Institute, Berkeley, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 3Australian Centre for Ecogenomics, School of Chemistry and Molecular Biosciences, The University of Queensland, Brisbane, Queensland, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 4Department of Medicine (Hematology), Stanford University, Stanford, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 5Department of Genetics, Stanford University, Stanford, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 6PolyBio Research Foundation, Kenmore, WA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 7Department of Bioengineering, Stanford University, Stanford, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' ChEM-H Institute, Stanford University, Stanford, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 10Chan Zuckerberg Biohub, San Francisco, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' ✉e-mail: snayfach@lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='gov; nckyrpides@lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='gov T he gut microbiome is a complex microbial ecosystem with important roles in human health and development1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Although often overlooked, viruses are estimated to be abundant in the microbiome2,3 and have been associated with human disease4–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' In particular, bacteriophages (viruses that infect bacteria) constitute the majority of viral particles3,7,8 and can impact microbial ecosystem processes through phage predation9, lysogeny10 and horizontal gene transfer11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Despite their ubiquity, our knowledge of viral genomic diversity in the microbiome is limited, with most viral sequences failing to match existing genome databases8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' A comprehensive data- base of viral genomes from the microbiome is a prerequisite for assembly-free quantification of viruses, predicting host–virus inter- actions12, comparative genomics and genome mining (for example, anti-CRISPR genes13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Traditionally, there have been two main approaches for sequenc- ing viral genomes from the microbiome: viral metagenomic sequencing and bulk metagenomic sequencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Viral metagenom- ics involves using size filtration to select for virus-like particles, followed by viral DNA extraction, (often) whole-genome amplifica- tion, shotgun sequencing and metagenomic assembly14–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Although size filtration is used to enrich extracellular viruses, it will not remove all cellular organisms18 and can exclude some large viruses19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Whole-genome amplification is often necessary due to low sample biomass but can skew viral abundances and over-amplify small cir- cular single-stranded DNA (ssDNA) viruses19–21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' whole-genome amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' However, with bulk metagenomic sequencing, it is more challenging to assemble low-abundance viruses because the majority of reads derive from cellular organ- isms24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Additionally, DNA-extraction protocols may not be opti- mized for viruses16 and some viral sequences may originate from degraded prophages in bacterial chromosomes10,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To date, numerous studies have used viral metagenomic sequenc- ing to identify phage genomes from human stool samples across a wide variety of phenotypes4–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To integrate these disparate data sets, Soto-Perez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='26 formed the Human Virome Database (HuVirDB) from 1,831 public samples (including skin, stool, lung and blood) and Gregory et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='27 formed the Gut Virome Database (GVD) from 2,697 public samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' In contrast to these viral metagenomic stud- ies, Paez-Espino et al22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' formed the IMG/VR database by identifying viruses from bulk metagenomes, including 490 stool samples from the Human Microbiome Project28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Since this publication, the num- ber of publicly available bulk metagenomes has rapidly grown, as evidenced by recent, large-scale data mining efforts29–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To expand these existing resources and provide a complemen- tary view of the gut virome, we performed large-scale identification of viral genomes from 11,810 bulk metagenomes from human stool samples derived from 61 previously published studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We used these data to form the Metagenomic Gut Virus (MGV) catalogue, which contains 189,680 viral draft genomes estimated to be >50% complete and representing 54,118 candidate viral species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' These Metagenomic compendium of 189,680 DNA An alternative approach is to generate bulk metagenomes, with- genomes vastly expand the known diversity of DNA viruses from out size filtration or whole-genome amplification, followed by the gut microbiome and improve knowledge of host–virus con- computational separation of viral and cellular sequences22,23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' This nections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We expect the MGV catalogue will be a useful commu- approach captures sequences of both extracellular and intracel- nity resource for interrogating the role of the gut virome in human lular viruses, including integrated prophages, and is not biased by health and disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' viruses from the human gut microbiome Stephen Nayfach 1,2 ✉, David Páez-Espino 1,2, Lee Call1,2, Soo Jen Low3, Hila Sberro4,5, Natalia N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Ivanova 1,2, Amy D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Proal6, Michael A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Fischbach 7,8,9,10, Ami S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Bhatt 4,5, Philip Hugenholtz 3 and Nikos C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Kyrpides 1,2 ✉ Bacteriophages have important roles in the ecology of the human gut microbiome but are under-represented in reference data- bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To address this problem, we assembled the Metagenomic Gut Virus catalogue that comprises 189,680 viral genomes from 11,810 publicly available human stool metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Over 75% of genomes represent double-stranded DNA phages that infect members of the Bacteroidia and Clostridia classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Based on sequence clustering we identified 54,118 candidate viral spe- cies, 92% of which were not found in existing databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The Metagenomic Gut Virus catalogue improves detection of viruses in stool metagenomes and accounts for nearly 40% of CRISPR spacers found in human gut Bacteria and Archaea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We also pro- duced a catalogue of 459,375 viral protein clusters to explore the functional potential of the gut virome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' This revealed tens of thousands of diversity-generating retroelements, which use error-prone reverse transcription to mutate target genes and may be involved in the molecular arms race between phages and their bacterial hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' NAturE MICroBIoLogy | VOL 6 | JULy 2021 | 960–970 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='com/naturemicrobiology 960 ResouRce https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1038/s41564-021-00928-6 ResouRce NATuRE MICROBIOlOGy results A genomic catalogue of DNA viruses from the gut microbiome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We developed a viral detection pipeline for the current study using a combination of well-established methods and signatures, including VirFinder32, viral protein families from the Earth’s Virome Study23, and the propensity for viral genes to lie on the same strand33 and be functionally unannotated8 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 1a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Based on in silico bench- marking, our pipeline was able to sensitively identify genome frag- ments of diverse human-associated viruses and phages, including crAss-like phages34 and megaphages35, with high specificity and per- formed favourably compared with existing methods (Supplementary Tables 1–2 and Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For genome fragments of 1, 10 and 100 kb our pipeline achieved true-positive rates (TPR) of 41%, 74% and 96% at false-positive rates (FPR) of only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='43%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='38% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='18%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We then applied our pipeline to bulk metagenomes from 11,810 distinct human gut samples that were assembled in previous stud- ies29,31,36 to broadly capture lytic and lysogenic DNA viruses (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 1a and Supplementary Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The analysed data sets span 61 stud- ies across 24 countries and include individuals with a wide range of ages, lifestyles and disease states (Supplementary Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' This revealed 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5 million unique, single-contig viral genomes longer than 1 kb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Based on an analysis of metagenomes found in all three stud- ies, we found that choice of assembler (that is, MEGAHIT versus metaSPAdes) had little effect on the quality or identity of recovered viruses (Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Viral genomes were largely derived from individuals in Europe (46%), China (23%) and the USA (13%) reflecting the amount of metagenomic data from these sources (45%, 24% and 11% of total assembly length, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The completeness of metagenome-assembled viruses can vary widely, ranging from short fragments to complete or near-complete genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To assess genome completeness, we applied CheckV37, revealing 189,680 genomes that were at least 50% complete (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 1c), including 26,030 complete genomes identified on the basis of direct terminal repeats (n = 19,704), host–provirus boundaries (n = 5,123) and inverted terminal repeats (n = 1,203).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To improve genome qual- ity, we removed flanking host regions from these sequences (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 1a); confirming that viral genomes were free of host contamination, we identified only one full-length 16S rRNA gene (flanking an inte- grated prophage) among all 189,680 viruses compared with 83,050 16S rRNA genes in the full set of metagenomic contigs used for viral discovery (Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We focused all subsequent analysis on the 189,680 genomes with >50% completeness to avoid limitations associated with small genome fragments38 and to be consistent with quality standards applied to microbial genomes39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Because there was no separation of viral-like particles prior to sequencing, we anticipated many viruses were derived from Viral taxonomy Africa Asia Europe North America Oceania South America Geography 0 20 40 60 Gene strand switch rate 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='8 VirFinder score Yes 0 40 80 Percentage viral protein families 0 40 80 Percentage non-viral Pfams No Predicted virus Yes No Predicted virus d a b c Unknown completeness <50% complete 50–90% complete >90% complete Complete Viral contig length (kb) 1 10 100 1,000 (n = 671,842) (n = 2,620,162) (n = 110,430) (n = 53,220) (n = 26,030) CheckV genome completeness level for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5 million viral contigs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Assembled, whole human gut metagenomes (no enrichment for virus-like particles) Pasolli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 2019 7,768 metaSPAdes assemblies HGM dataset (Nayfach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 2019) 3,771 MEGAHIT assemblies MGnify (Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 2019) 6,732 metaSPAdes assemblies 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Extract viral-informative features (panel b) Viral (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5 million contigs) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Identify viral contigs (>1 kb) Non-viral (373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='8 million contigs) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Estimate genome completeness (panel c) ≥50% complete (n = 189,680 contigs) <50% complete (n = 3,292,004 contigs) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Trim flanking host regions from prophages Genome clustering Host prediction Taxonomic annotation Functional annotation Phylogenetic analysis 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Annotation and analysis Viral protein families (Paez-Espino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 2016) VirFinder score (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 2017) Non-viral Pfams (El-Gebali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 2019) Gene strand switch rate (Roux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 2015) Host (673 × 106 bp) Virus (8,497 × 106 bp) Viral-informative features for metagenomic contigs Summary statistics for 189,680 viral contigs with >50% completeness Clostridia Bacteroidota Actinobacteriota Bacilli Negativicutes Proteobacteria Other Unknown Host prediction crAssphage Other Unknown No Yes Integrated prophage Complete >90% complete 50–90% complete Genome completeness Podoviridae Siphoviridae Myoviridae Microviridae Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 1 | thousands of high-quality viral genomes recovered from human gut metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' a, Overview of viral discovery effort and formation of the MGV catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' b, Genomic signatures of predicted viral and non-viral metagenomic contigs longer than 20 kb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Displayed data is for 1,000 randomly selected contigs from each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' c, Distribution of estimated genome completeness and classification of MGVs into quality tiers (complete, n = 26,030; >90% complete, n = 53,220; 50–90% complete, n = 110,430; <50% complete, n = 2,620,162; completeness not determined, n = 671,842).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' d, Metadata and annotations for 189,680 genomes with >50% completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For box plots, the middle line denotes the median, the box denotes the interquartile range and the whiskers denote 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5× the interquartile range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' NAturE MICroBIoLogy | VOL 6 | JULy 2021 | 960–970 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='com/naturemicrobiology 961 ResouRce NATuRE MICROBIOlOGy bacterial chromosomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' However, only 24% of viral genomes had evidence of host integration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 1d) and only 10% where the flanking host region was >5 kb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Furthermore, the major- ity of non-integrated viruses were classified as virulent based on BACPHLIP40 (65% of 140,689) which is a computational tool that predicts bacteriophage lifestyle from conserved protein domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Likewise, BACPHLIP classified 58% of the 26,030 complete genomes as virulent, indicating that this result is not due to incom- plete genome assembly because integrase genes often occur at the ends of prophage genomes41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Together, these results demonstrate that it is not uncommon to recover the genome sequences of lytic viruses from unfiltered stool metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Host prediction and taxonomic annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Predicting the cellular hosts of viruses is important for understanding phage predation and an essential first step towards utilizing host–virus interactions to design innovative phage therapies42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Towards this goal, we leveraged the Unified Human Gastrointestinal Genome (UHGG) database of 286,997 genomes of Bacteria and Archaea from the gut micro- biome43, which represents 4,644 prokaryotic species (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' First, we extracted 1,846,441 CRISPR spacers from the UHGG genomes, and looked for near-exact matches to the 189,680 viral genomes, resulting in host–virus connections that covered 81% of viruses (n = 153,892).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Interestingly, just 21% of viruses were connected to a host when using spacers extracted from the 4,644 species-level representatives, indicating considerable CRISPR diversity between bacterial strains and active community infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Although most viruses were targeted by a spacer, CRISPR arrays were found in only 28% (n = 79,734) of UHGG genomes and in <1% of many prevalent species including Alistipes putredinis, Bacteroides cellulosilyticus and Bifidobacterium breve, confirming the limited distribution of this anti-viral defence system44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To expand the host–virus network, we performed whole-genome alignment between the 189,680 viruses and 286,997 hosts and identified connections based on near-exact genomic matches (≥96% identity over ≥1 kb), resulting in connec- tions that covered 96% of host genomes and 90% of viral genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' As expected, the majority of viruses were connected to Firmicutes (predominantly Clostridia) and Bacteroidia, which are the two dominant phyla of bacteria in the gut microbiome (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' These results show that host–virus interactions can be systematically elu- cidated through extensive assembly of both viral and microbial genomes from the same environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Next, we assigned viruses to families from the ICTV data- base based on alignments to genomes from NCBI GenBank and crAss-like viruses from recent studies34,45,46 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Only 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='6% of viruses could be annotated at the family level, confirming a large knowledge gap in the taxonomy of human gut viruses8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To increase sensitivity, we used taxonomically informative profile hidden Markov models (HMMs) from the VOG database (http:// vogdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='org), revealing most unannotated viruses to be members of the Caudovirales order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Among annotated sequences were 9,395 genomes of putative crAss-like viruses (5% of total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Overall, only 274 (199) 415 (58) 896 (256) 1,053 (1,308) 1,188 (139) 1,493 (880) 2,547 (78) 3,085 (2,389) 4,043 (323) 4,785 (140) 7,588 (634) 11,089 (1,689) 16,511 (1,018) 18,082 (6,492) 69,411 (3,617) 84,590 (9,304) 90,083 (17,211) 93,699 (19,813) 172,132 (13,189) 301,781 (62,710) 961,366 (145,390) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Number of CRISPR spacers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='per host class ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Percentage of spacers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='matching MGV catalogue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Siphoviridae ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Myoviridae ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='crAss-like ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Podoviridae ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Microviridae ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Other ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Unknown ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Taxonomy of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='connected viruses ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Percentage of host ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='genomes with spacer hit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Spacer hit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='No spacer hit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='No spacers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Percentage of host genomes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='with spacer or 1 kb hit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Spacer hit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Spacer hit and ≥1 kb hit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='≥1 kb hit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='No viral linkage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Viral family ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Viral linkage method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Spacer linkages ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Elusimicrobia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Peptococcia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Spirochaetia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Vampirovibrionia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Thermoplasmata ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Campylobacteria ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Synergistia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Alphaproteobacteria ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Lentisphaeria ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Brachyspirae ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Fusobacteriia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Desulfovibrionia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Methanobacteria ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Coriobacteriia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Verrucomicrobiae ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Actinobacteria ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Bacilli ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Gammaproteobacteria ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Negativicutes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Bacteroidia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Clostridia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='8% 1% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='8% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5% 4% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5% 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='8% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='6% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='9% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='4% 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='9% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='4% 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5% 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='4% 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='8% 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='3% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5% 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='8% 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5% (Number of host genomes from UHGG catalogue) a b c e d Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 2 | Viral connections to human gut Bacteria and Archaea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' a, Bar plots indicating the number of CRISPR spacers across 286,997 human gut Bacteria and Archaea, with the number of genomes indicated in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Each row indicates one host class containing at least 20 genomes and 100 spacers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The majority of CRISPR spacers are derived from Clostridia and Bacteroidia, reflecting their abundance in the human gut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' b, Percentage of CRISPR spacers matching viral genomes with a maximum of one mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' c, Host genomes containing a CRISPR-spacer array, and those with a CRISPR-spacer array match to a viral genome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' d, Genomes linked to a virus using a combination of approaches as indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' e, Distribution of known viral families that are associated with each host class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Each host class is infected by a distinct repertoire of viral families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' NAturE MICroBIoLogy | VOL 6 | JULy 2021 | 960–970 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='com/naturemicrobiology 962 ResouRce NATuRE MICROBIOlOGy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='51% (n = 48) of the putative crAss phages displayed clear evi- dence of lysogeny (that is flanked by host region and contained an integrase), which was more than 17× lower than other viruses in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Consistent with this, 56% of high-quality crAssphage genomes (n = 5,439) could be circularized compared with 24% of the other high-quality genomes (n = 36,872).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' crAss-like genomes contained several other unusual features, including low GC con- tent (mean = 32%), usage of an alternative genetic code and a pre- dominance of hypothetical proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For example, TAG or TGA stop codons were recoded to amino acids in 27% of crAss-like phages versus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5% of other viruses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Likewise, only 12% of crAssphage pro- teins had significant hits to Pfam, KEGG or TIGRFAM versus 28% of proteins from other viruses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' This large-scale analysis supports previous findings that some crAss-like viruses have an obligate lytic lifestyle46 and reveals several unusual features that further establish crAssphage as an outlier among human gut viruses47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Vastly expanded viral genomic diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To quantify the diversity of genomes in the MGV catalogue, we first identified species-level viral operational taxonomic units (vOTUs) using the MIUViG rec- ommended criteria of 95% average nucleotide identity (ANI) over 85% of the length of the shorter sequence38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Small adjustments to these parameters did impact the number of identified vOTUs, suggesting a continuum of viral diversity beyond the species-level boundary (Supplementary Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Overall, we identified 54,118 vOTUs, of which 8,086 included members from at least two samples (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The largest vOTUs were predicted to infect some of the most prevalent species in the gut microbiome, including Bacteroides uni- formis, Faecalibacterium prausnitzii and Agathobacter rectalis (for- merly Eubacterium rectale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To identify higher-ranking viral clades, we clustered genomes into approximately genus- and family-level groups on the basis of pairwise average amino acid identity (AAI) and gene sharing (Methods), revealing 5,800 genus-level vOTUs and 1,434 family-level vOTUs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Accumulation curves of vOTUs appeared to be approaching an asymptote at the family and genus ranks but not yet for species (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Other recent studies have also compiled databases of DNA viruses from the gut microbiome22,26,27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To identify vOTUs unique to the MGV catalogue, we clustered the 189,680 genomes from our study together with medium- and high-quality viral genomes from three other genome catalogues (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 3a): the HuVirDB (9,626 genomes derived from 1,543 viral metagenomes), GVD v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0 (4,494 genomes derived from 471 viral metagenomes and 98 whole metagenomes) and IMG/VR v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0 (6,895 genomes derived from 490 whole metagenomes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Note that during the review of this manu- script, the IMG/VR and GVD were updated to new versions which were not analysed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To enable comparability between all studies, CheckV was run on all viral data sets and genome fragments with <50% completeness were excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Strikingly, we found that 50,048 of the 54,118 species-level vOTUs from the MGV catalogue (92%), comprising 100,398 of the 189,680 genomes (53%), did not cluster with any genome from the other databases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' In contrast, the three reference databases combined represented 10,391 species-level vOTUs, nearly half of which were also found in the MGV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The MGV and IMG/VR data- bases, which were both derived from whole metagenomes, shared the greatest number of vOTUs and contained a relatively high pro- portion of lysogenic phages from the order Caudovirales, whereas the HuVirDB and GVD data sets, which were largely derived from viral metagenomes, were enriched in small circular ssDNA viruses from the Microviridae, Anelloviridae and CRESS families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Next, we compared the four genome catalogues based on their ability to recruit sequencing reads from a geographically diverse set of whole metagenomes and viral metagenomes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To prevent self matches we discarded alignments between sequencing reads and viral genomes derived from the same original study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Overall, MGV genomes recruited 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='6% of whole-metagenome reads, which was 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0-fold higher than any other database, and 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1% of virome reads, which was comparable with the HuVirDB at 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We also compared the recruitment of CRISPR spacers to each viral database as a way of quantifying host–virus connections (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Overall, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5% of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='8 M spacers from UHGG genomes matched a genome from the MGV catalogue, which was 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='25-fold higher than any other database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The number of matched spacers and metagenomic reads did not change considerably when using a viral database of only species-level representatives (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Together, these results show that the MGV catalogue has substantially increased known viral diversity, improved detection of viral reads in whole metage- nomes and expanded coverage of host–virus connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Phylogenomics of intestinal Caudovirales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Caudovirales comprise an expansive order of tailed double-stranded DNA (dsDNA) phages found in numerous environments48 and were highly represented in the stool metagenomes we analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To explore the evolution of this group in the gut microbiome, we constructed a species-level phylo- genetic tree based on a concatenated alignment of 77 protein-coding marker genes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 4a)49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' After removing genomes with insufficient data (fewer than three markers or <5% representation in align- ment), the final tree contained 25,528 species-level viral genomes derived from the four databases of uncultivated gut viruses (MGV, IMG/VR, HuVirDB and GVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Based on cumulative branch length, the MGV catalogue cov- ered 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='7% of the total phylogenetic diversity (PD) and contained genomes representing all major lineages across the tree (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Compared with the three other databases combined, MGV genomes resulted in a 287% increase in PD that was evenly distrib- uted across viral and host taxonomic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Clostridia phages were by far the most diverse group (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='8% of PD) because of the large number and broad phylogenetic distribution of these vOTUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' In contrast, Bacteroidota phages represented only 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1% of PD with most vOTUs falling into four primary clusters (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 4a) including one dominated by crAss-like phages (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='17% of PD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Overall, there was poor correspondence between the classical viral families based on tail morphology and genome-based phylogeny (for example, nearly all lineages contained Siphoviridae annotated genomes) which further highlights the need for a phylogeny driven taxonomy of Caudovirales49 and other viral groups, analogous to the GTDB taxonomy developed for Bacteria and Archaea50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Notably, several lineages contained jumbo phages with genomes exceeding 200 kb (518 genomes from 245 species-level vOTUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' As with other analyses, we carefully removed flanking host regions as well as assembly artefacts resulting in the same genome being repeated multiple times (Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The largest genome was a 553,716-bp near-complete linear genome closely related to a Prevotella phage Lak-A1 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 35; 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5% AAI over 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1% of genes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' As with crAss-like phages, jumbo phages were rarely integrated into a host (n = 13) although they sometimes contained integrases (n = 121).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To characterize the diversity of these viruses in greater detail, we constructed a separate tree based on the large terminase subunit (TerL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Compared with a recently published collection of jumbo phages from diverse environments51, MGVs resulted in a large expansion of phylogenetic diversity and coverage of most lin- eages (Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Interestingly, jumbo phages and other Caudovirales appeared to have little to no preference in biogeographic distribution, as most clades were found in all continents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We hypothesized that region-specific phylotypes might be apparent over shorter evolu- tionary timescales, as observed for human gut bacteria52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Towards this goal, we used single-nucleotide variants to construct strain-level phylogenies for 146 prevalent vOTUs with more than 100 members (Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Strikingly, we observed discrete subspecies that were highly enriched in specific geographic regions for many vOTUs NAturE MICroBIoLogy | VOL 6 | JULy 2021 | 960–970 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='com/naturemicrobiology 963 ResouRce NATuRE MICROBIOlOGy (Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For example, one crAss-like subspecies pre- dicted to infect Parabacteroides was prevalent among samples from Asia, but rare or absent from Europe and North America.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' More work is needed to understand the evolutionary drivers and genomic adaptations underlying these phylogenetic patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Functional capacity of the gut virome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Although the functional potential of human gut bacteria and archaea has been extensively studied43,53,54, that of intestinal phages is less well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To explore this, we identified 11,837,198 protein-coding genes with at least 20 amino acids (98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='4% with start and stop codons) across the 189,680 viral genomes from our study and compared these with HMM databases, including KEGG55, TIGRFAM56, Pfam57, VOGDB (http://vogdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='org/) and the Earth’s Virome database23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Overall, 45% of viral genes did not have significant matches to any database and 75% were not assigned any biological function (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 5a,b), indicat- ing that remarkably little is known about the functional potential of human gut viruses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To identify the most common functions among intesti- nal phages, we clustered the 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='8 million viral genes at 30% AAI using MMseqs2 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 58) into 459,375 de novo viral protein clusters (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 5c) including 61% with at least two members (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' An accumulation curve displayed no plateau, indicating that gut phages have a large reservoir of functional diversity that is not fully cap- tured by this study (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 5e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Clostridia phages contained the most functional diversity with 173,187 protein clusters, reflecting the large phylogenetic diversity of these phages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Several of the largest protein clusters had no predicted function, including the fourth largest with 8,319 genes, and are therefore good candidates for experimental characterization in the future (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 5f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Other large clusters were annotated with typical viral functions, including cap- sid formation, packaging, lysis, lysogeny, replication and transcrip- tional regulation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 5f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Although it is outside the scope of this article to enumerate all viral functions and auxiliary metabolic genes, we explored two par- ticularly unusual findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Based on HMM searches against Pfam, we uncovered 11,496 putative viral beta-lactamases (PF12706), including the majority of sequences in a single protein clus- ter with 5,832 members (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 5f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Beta-lactamases are enzymes that enable resistance to beta-lactam antibiotics such as penicil- lins, cephalosporins and cephamycins, and pose a major global health problem59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To validate this result, we performed homology a Species-level vOTUs (60,439) MGV (54,118) IMG/VR (3,132) HuVirDB (5,111) GVD v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0 (4,450) 15 17 34 41 169 360 0 1 552 125 75 28 0 9 84 66 154 203 207 508 505 0 7 3,269 481 224 127 0 47 479 2,630 1,314 1,672 532 241 170 0 22 50,048 870 161 221 3 680 1,875 Genus-level vOTUs (6,277) MGV (5,800) IMG/VR (1,389) HuVirDB (1,567) GVD v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0 (2,082) Family-level vOTUs (1,510) MGV (1,434) IMG/VR (557) HuVirDB (630) GVD v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0 (781) 0 50,000 150,000 0 20,000 50,000 Species vOTUs 0 0 2,000 4,000 6,000 0 0 400 800 1,400 Number of genomes (MGV data set) 100,000 50,000 150,000 Genus vOTUs 100,000 50,000 150,000 Family vOTUs 100,000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Number of genomes (MGV data set) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Number of genomes (MGV data set) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Percentage of reads mapped ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='to viral genomes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='GVD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='HuVirDB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='IMG/VR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='MGV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Viral data set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='All ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='viral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='genomes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='One genome ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='per vOTU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='GenBank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Bulk metagenomic reads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Viral metagenomic reads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='CRISPR spacers from ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='UHGG genomes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='GVD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='HuVirDB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='IMG/VR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='MGV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='All data sets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='GenBank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='GVD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='HuVirDB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='IMG/VR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='MGV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='GenBank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Viral data set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Viral data set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Percentage of reads mapped ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='to viral genomes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Percentage of spacers mapped ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='to viral genomes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='All data sets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='All data sets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 3 | genome clustering and comparison with existing databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The 189,680 genomes from the MGV catalogue were compared with human gut virus genomes >50% complete from three databases: IMG/VR (n = 6,895), HuVirDB (n = 9,626) and GVD (n = 4,494).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' a, Viral genomes were clustered into vOTUs at approximately species, genus and family levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' b, Accumulation curves for vOTUs from the MGV catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' c, Percentage of reads from 1,257 unfiltered stool metagenomes, percentage of reads from 585 viral stool metagenomes and percentage of CRISPR spacers from 286,997 UHGG genomes mapped to viral genomes from various databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' NAturE MICroBIoLogy | VOL 6 | JULy 2021 | 960–970 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='com/naturemicrobiology 964 ResouRce NATuRE MICROBIOlOGy searches against curated databases of antimicrobial resistance genes using Resfams60, the NCBI AMRFinder61 and the Resistance Gene Identifier (RGI)62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' These tools revealed a combined total of only 88 resistance genes (63 using Resfams, 56 using AMRFinder and 30 using RGI), indicating low similarity between the 11,496 puta- tive viral beta-lactamases and validated resistance genes (Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Although functional metagenomic assays may uncover bona fide viral beta-lactamases in the gut microbiome, these results appear to support the conclusion that phages rarely encode antibi- otic resistance genes63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Another interesting finding was a large number of phage reverse transcriptases (RTs) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 5f and Supplementary Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Overall, the RT domain (PF00078) was the third most common functional annotation, next to only the helix–turn–helix DNA-binding domain (PF01381) and phage integrase family (PF00589).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' RTs are known to occur in retroviruses64, RNA-targeting CRISPR–Cas systems65 and diversity-generating retroelements (DGRs)66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' DGRs utilize error-prone reverse transcription to generate random mutations in the transcript of a template region (TR), which is then inserted back into the genome at a variable region (VR), thereby generat- ing population-level hyper-variability in a specific gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Since the DGR system was first characterized in a Bordetella bacteriophage66, it has been found in human microbiomes67 and in several human gut phages68,69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To determine whether the viral RTs were part of the DGR sys- tem, we used the tool DGRscan67 to identify TR–VR pairs across 79,250 high-quality viral genomes with >90% estimated complete- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Confirming our hypothesis, the great majority of genomes with an RT also contained a TR–VR (85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='7% of 25,620) compared with a small minority of those without an RT (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5% of 53,630) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 5g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' DGRs were remarkably common in certain Caudovirales families (for example, 84% of 6,616 Myoviridae) and among lysog- enized viruses (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1% of 18,187), whereas they were rare or com- pletely absent from other Caudovirales families, ssDNA viruses and eukaryotic viruses (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 5h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Although the vast majority of DGR gene targets were not functionally annotated, we observed highly significant enrichment within several Pfam domains (Supplementary Table 7) including an immunoglobulin-like domain that was 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='9-fold more common among DGR-targeted genes and is believed to play a role in phage interactions with car- bohydrates on the cell surface of bacteria70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Together, these results reveal DGRs to be more common in intestinal phages than pre- viously thought and may point towards viral proteins involved in molecular phage–host interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Tree scale: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Newly identified 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Lysogeny rate 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Genome size 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1% Current study and previous studies Previous studies only Current study only 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='6% Percentage of Caudovirales phylogenetic diversity (PD) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Host taxonomy Actinobacteriota Proteobacteria Bacteroidota Firmicutes ---Bacilli ---Clostridia ---Negativicutes Other crAss-like Myoviridae Podoviridae Siphoviridae Other 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Viral taxonomy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Viral contigs (log10) 1,000 0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Genome size <100 kb 100–200 kb >200 kb 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Lysogeny rate (Percentage of prophages/OTU) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Newly identified Yes No 0 >50 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Host taxonomy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Viral contigs from current study 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Viral taxonomy a b Newly identified Caudovirales lineage Previously known Caudovirales lineage Percentage of PD Total PD Viral taxonomy 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='3% c d 10 100 crAssphage Podoviridae Myoviridae Unknown Siphoviridae 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='31% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='17% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='22% 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='88% 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='61% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='03% Other Other Negativicutes Proteobacteria Actinobacteriota Bacilli Bacteroidota Unknown Clostridia 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='8% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='34% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='92% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='81% 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='24% 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='13% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='9% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='79% Percentage of PD Total PD Host taxonomy Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 4 | Phylogenomics of intestinal Caudovirales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' A phylogenetic tree was constructed from 25,528 species-level genomes derived from the MGV and other databases (IMG/VR, HuVirDB and GVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' a, Phylogeny of intestinal Caudovirales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Tree was plotted using iToL74 and to improve visualization only one genome per genus-level vOTU is displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Branch colour indicates whether a lineage is represented by a previously published study (black) or is unique to the MGV catalogue (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Outer rings display metadata for each vOTU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' b, PD was calculated by taking the sum of branch lengths represented by species-level viral genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' c,d, MGVs from the current study result in a large gain in PD, which is consistent across (c) viral families and (d) viruses infecting different host groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' NAturE MICroBIoLogy | VOL 6 | JULy 2021 | 960–970 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='com/naturemicrobiology 965 ResouRce NATuRE MICROBIOlOGy Discussion In this study, we performed large-scale data mining of publicly available metagenomes to identify 189,680 draft-quality viral genomes representing an estimated 54,118 species-, 5,800 genus- and 1,434 family-level vOTUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' This large resource contains exten- sive viral genomic diversity not found in other databases, improves detection of viral reads in microbiomes and represents numerous diverse and previously uncharacterized viral groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Through a combination of approaches, we were able to predict host–virus link- ages that cover the majority of viral and prokaryotic diversity in the gut microbiome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' These host–virus linkages may be important in the future for understanding disease processes, designing phage therapies or understanding host–virus co-evolutionary dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Despite large-scale annotation efforts, we were only able to assign preliminary biological functions to 25% of viral genes, indicating that more work and new methods are needed to predict protein function in viral genomes, such as deep learning71 and functional metagenomic assays72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Although the current study focused exclu- sively on DNA viruses, future studies could use metatranscrip- tomics data to study RNA viruses or gene expression patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' While this manuscript was in review, Camarillo-Guerrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='73 published the Gut Phage Database (GPD), a collection of ∼142,000 non-redundant viral genomes (>10 kb) identified from 28,060 human gut metagenomes and 2,898 gut bacterial genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' After applying CheckV, we found the GPD represents 79,889 viral contigs with >50% completeness that form 46,480 species-level vOTUs, which is 14% less than the 54,118 vOTUs from the MGV (Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Differences between viral catalogues are ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Phage portal protein (PF04860) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Phage antirepressor protein (PF03374) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='ssDNA binding protein (PF05772) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='HTH domain of resolvase (PF02796) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='tRNA-splicing ligase RtcB (K14415) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='HNH endonuclease (PF01844) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Reverse transcriptase (PF00078) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Phage terminase (PF03354) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='DNA repair protein (K03546) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='DNA methylase (PF02384) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Restriction endonuclease-like domain (PF08774) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Phage tail tube protein (PF09393) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Baseplate J-like protein (PF04865) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='DNA methylase (PF01555) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Recombinase, RecT family (K07455) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='DNA polymerase (K21237) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Thymidylate synthase (K03465) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Reverse transcriptase (PF00078) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='LysM domain (PF01476) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Peptidoglycan amidohydrolase (PF01520) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Prohead serine protease (PF04586) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='SNF2 family helicase (PF00176) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Terminase small subunit (PF03592) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Phage connector (PF05352) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Phage tail protein (PF05489) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Phage integrase (PF00589) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Putative beta-lactamase domain (PF12706) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Phage portal protein (PF05136) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Phage tail sheath protein (PF04984) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='ssDNA binding protein (PF00436) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Phage terminase large subunit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='ssDNA binding protein (PF00436) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Phage tail tube protein (PF04985) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Phage terminase large subunit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='ssDNA binding protein (PF00436) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='ATPase family (PF00004) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Reverse transcriptase (PF00078) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Reverse transcriptase (PF00078) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Exoribonuclease (K03698) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Phage portal protein (PF05133) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Phage portal protein (PF04860) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='HTH domain (PF13936) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Phage terminase (PF03354) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Terminase large subunit (PF05876) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Peptidoglycan amidohydrolase (PF01510) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Reverse transcriptase (PF00078) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='Percentage of 11,837,198 genes hitting database Earth’s Virome VPFs VOGDB Pfam-A KEGG TIGRFAM 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='6% Unknown function No hit 45% Known function 25% 30% 11,837,198 viral genes 459,375 protein clusters 14% 11% No hit 75% Unknown function Known function Number of genes (×106) Number of genes (×103) Number of protein clusters (×105) 0 4 8 12 0 2 4 Structural Packaging and assembly Lysis Replication DNA binding/regulation Other Unknown function Functional category 0 Largest 75 viral protein clusters 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='3% Reverse transcriptase (PF00078) TR–VR (DGRscan) n = 21,945 (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='7%) n = 3,675 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='6%) n = 3,487 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='4%) DGR absent: lack of RT and TR–VR n = 50,143 (63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='3%) DGR present High-quality viral genomes from current study n = 79,250 Myoviridae (6,616) Siphoviridae (23,364) crAssphage (4,543) Papillomaviridae (75) Podoviridae (4,151) Microviridae (2,029) Inoviridae (107) Cress DNA (68) Herelleviridae (16) Adenoviridae (11) Retrovirales (10) Anelloviridae (5) Fraction of high-quality genomes with DGRs 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0 Viral taxonomy Yes (18,187) No (53,901) dsDNA (69,803) ssDNA (2,215) Baltimore classification Predicted prophage 459,375 protein clusters 2–10 genes (41%) 1 gene (39%) 10–100 genes >100 genes 16% 4% Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 5 | Functional landscape of intestinal phages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' a, Protein-coding viral genes were identified across all MGVs and compared with profile HMMs from five databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' b, Forty-five per cent of genes fail to match any HMM, 30% match an HMM of unknown function and 25% match an HMM of known function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' c, The 11,837,198 genes were clustered at 30% AAI using MMseqs2 into 459,375 protein clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' d, Size distribution of protein clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' e, An accumulation curve of protein clusters has not reached an asymptote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' f, Functional annotations for the largest 75 protein clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Reverse transcriptases are highlighted in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' g, Prediction of DGRs based on the combination of the reverse transcriptase gene (PF00078) and TR–VR pair identified using DGRscan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' A large fraction of MGVs contain the DGR system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' h, DGR prevalence across different categories of viruses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' DGRs are most common in lysogenic, dsDNA viruses from the Myoviridae family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' NAturE MICroBIoLogy | VOL 6 | JULy 2021 | 960–970 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='com/naturemicrobiology 966 ResouRce NATuRE MICROBIOlOGy due to several factors, including data sets used for metagenome mining, methods for viral identification and criteria for sequence inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For example, the MGV had greatly improved coverage of Microviridae which were excluded from the GPD due to their short length (mean = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='9 kb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Combined, the MGV and GPD represented 75,187 species-level vOTUs, indicating that the two catalogues con- tain complementary viral diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' In the future, these and other large-scale viral genome catalogues could be integrated to create a unified and standardized community resource, as recently per- formed for human gut microbial genome catalogues43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Methods Development of viral detection pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We used a combination of four viral signatures to identify viral metagenomic contigs: (1) the presence of viral protein families, (2) the absence of microbial protein families, (3) the presence of viral nucleotide signatures, and (4) multiple adjacent genes on the same strand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For the presence of viral protein families, we used HMMs for 23,841 viral protein families from the IMG/VR database23 (downloaded 1 June 2019) after excluding 1,440 commonly found in microbial genomes or plasmids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For the absence of microbial protein families, we used HMMs for 16,260 protein families from the Pfam-A database57 (release 31) after excluding 452 commonly found in viruses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Proteins from metagenomic contigs were searched against HMMs from IMG/VR and Pfam-A using hmmsearch within the HMMER package v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1b2 (options: −Z 1, e-value: <1 × 10−10)75 and were classified as either viral or microbial based on the database containing the top hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For the presence of viral nucleotide signatures, we applied the tool VirFinder v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 32) to metagenomic contigs, which scores sequences using a combination of k-mer frequencies and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For multiple adjacent genes on the same strand, we quantified the strand switch rate by dividing the number of strand switches by the number of genes on each contig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Benchmarking viral detection pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We evaluated our viral detection pipeline on mock data sets we created that contained genome fragments from human-associated viruses and bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Each mock data set contained genome fragments from six diverse categories of viruses: (1) crAss-like phages from the human gut45, (2) Lak-phages from human and mammalian microbiomes35, (3) bacteriophages assembled from human gut viromes76, (4) phages with CRISPR-spacer matches to gut isolated microbial genomes, (5) isolate dsDNA human viruses and (6) isolate ssDNA human viruses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Non-viral genome fragments were derived from: (1) gut isolated microbial genomes and (2) plasmids genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We generated 2,000 genomic fragments from randomly sampled genomes within each of the eight categories at each of seven different fragments lengths (1, 2, 5, 10, 20, 50 and 100 kb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The TPR (percentage of viral contigs classified as viral) and FPR (percentage of non-viral contigs classified as viral) were calculated for over 77,000 combinations of cut-off values for the four viral signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We selected up to five different combinations of cut-offs that resulted in the highest classification score for each fragment length, where the classification score was based on a weighted combination of the TPR and FPR (score = TPR − 50 × FPR; Supplementary Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We assigned a very high negative weight to the FPR to avoid many false positives in the metagenomes which are expected to contain mostly non-viral sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We compared the performance of our method with VirSorter v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 33) and to VirFinder v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 32) using the same benchmark data set (Supplementary Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' VirFinder was run using default options and we applied p-value thresholds of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='001 for classifying genome fragments as viral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' VirSorter was run with and without the ‘-virome’ option, and we used VirSorter categories 1 and 2 to classify a fragment as viral (excluding low confidence predictions and integrated prophages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We also evaluated VirSorter when including predicted prophages (categories 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Application of pipeline to identify human gut viruses from whole metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To perform a comprehensive search for human gut viruses, we downloaded 18,271 publicly available metagenomic assemblies from human stool samples totalling 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='25 ×1012 bases and corresponding to 11,810 unique biological samples (Supplementary Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Assemblies were obtained from two recent studies29,31 and the MGnify database (accessed on 16 April 2019)36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We excluded assemblies from environments other than human gut and those that could not be assigned to an accession number from the NCBI SRA database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Metadata were obtained from previous studies and the NCBI BioSample database77 (Supplementary Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We applied our viral detection pipeline method to identify 4,436,008 contigs longer than 1 kb across the 18,271 metagenomic assemblies (Supplementary Table 1), which were de-replicated to 3,481,684 sequences at 100% ANI over 100% the length of the shorter sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Gene calling and identifying viruses with alternative genetic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Prodigal v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='3 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 78) was used to identify protein-coding genes in the 3,481,684 viral genomes using the flag ‘-p meta’ optimized for metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Additionally, we ran a custom pipeline to identify viruses using an alternative genetic code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Specifically, Prodigal was run using the standard code (11), and three alternative genetic codes: TGA recoded (code 4 or 25), TAG recoded (code 15) and TAA recoded (code 90), as previously described by Ivanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To reduce false positives this procedure was only run on viral contigs longer than 10 kb with GC content <50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For each viral contig, Prodigal outputs a GFF file that includes a coding potential score for every predicted gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To evaluate the genetic codes, we took the sum of coding potential scores per contig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' An alternative genetic code was predicted if it’s total coding potential score was the greatest and at least 10% greater than the standard genetic code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Viral reference genomes used for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Viral genomes from the MGV were compared against four reference databases: IMG/VR v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 22), GVD v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 27), HuVirDB v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 26) and NCBI GenBank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For IMG/VR, we extracted 28,697 viral contigs which were identified from 490 whole metagenomes from human stool samples using the Earth’s Virome Pipeline23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For GVD, we used all 13,203 viral contigs, which were identified from 471 viral metagenomes and 98 whole metagenomes using a combination of tools including VirSorter and VirFinder and previously clustered into viral populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' An updated version of the GVD was released while the paper was under review but was not analysed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For the HuVirDB, we extracted 929,886 contigs longer than 1 kb from 1,543 viral metagenomes from human stool samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Because no viral prediction was previously applied, we ran the viral prediction pipeline developed for the current manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For NCBI GenBank (downloaded 1 June 2019), we extracted 28,996 complete viral genomes after removing those labelled as incomplete, contaminated, or chimeric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Quality control of viral genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We applied CheckV v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0 (database v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='6)37 to all viral sequences to identify closed genomes, estimate genome completeness and remove flanking host regions on assembled proviruses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Putative complete genomes were predicted based on direct terminal repeats (minimum 20 bp), inverted terminal repeats (minimum 20 bp) or provirus integration sites (host region predicted on both ends of viral contig), and were additionally required to display >90% estimated completeness based on comparison with CheckV reference genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' A small number of sequences were removed that contained large repeats spanning >30% of the contig length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We selected all genomes with >50% estimated completeness for further analysis, resulting in 189,680 viral contigs from the MGV catalogue, 6,895 contigs from IMG/VR, 4,494 from GVD, 9,626 from HuVirDB and 28,996 from GenBank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We estimated the amount of non-viral DNA from cellular organisms among MGV sequences by searching for 16S and 18S rRNA genes using Barrnap v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='9-dev (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='com/tseemann/barrnap) with models for Bacteria, Archaea and Eukaryotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Alignments were required to cover ≥70% of the 16S or 18S rRNA gene and display an e-value <1 × 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' This same procedure was applied to the 18,271 metagenomic assemblies used for viral discovery to estimate the background levels of 16S and 18S rRNA genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Taxonomic annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Viral genomes were annotated based on amino acid alignments to a database of proteins derived from complete NCBI GenBank genomes and crAss-like genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Annotations were performed using the Baltimore classification (DNA, dsDNA, ssDNA, ssDNA-RT, dsRNA, RNA, ssRNA, ssRNA-RT) as well the ICTV taxonomy at the order, family and genus ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' DIAMOND v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='32 (options: –query-cover 50–subject-cover 50–e-value 1e-5– max-target-seqs 1000)80 was used to align viral proteins to the reference database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The taxonomy of the top database hit was then transferred to each protein at each taxonomic rank (Baltimore, order, family, genus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' In cases where the taxonomy of the top hit was missing, we used the next hit if its bit-score was within 25% of the top hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For each viral genome, we aggregated annotations across proteins after weighting by bit-scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Each viral genome was then annotated at the lowest taxonomic rank having >70% agreement across annotated proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' At the family rank, we required genomes to have a minimum of two annotated proteins with >30% AAI to the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' At the genus rank, we required genomes to have a minimum of three annotated proteins with >40% average AAI to the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' As validation, we applied our pipeline to taxonomically annotated genomes from NCBI GenBank after removing closely related genes from the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Our pipeline achieved average TPRs of 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0%, 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='7%, 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2% and 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5% at precision values of 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='6%, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='9%, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='3% and 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5% for taxonomic ranks of Baltimore, order, family and genus, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Host prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We used a combination of CRISPR-spacer matches and ≥1 kb genome sequence matches to associate viral genomes to Bacterial and Archaeal genomes from the UHGG collection43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The UHGG contains 286,997 genomes, representing 4,644 species of Bacteria and Archaea from the human gut that are taxonomically annotated using GTDB-tk v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1 (GTDB release 89)81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Many of the UHGG genomes are metagenome-assembled genomes, which sometimes contain erroneously binned sequences, including those from viruses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To address this, we conservatively identified and removed 2,043,531 contigs from UHGG genomes where the host region comprised <50% of the contig length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We then compared the remaining UHGG contigs with viral genomes and identified ≥1 kb genome sequence matches with ≥96% DNA identity using blastn from the blast+ package v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Next, we identified 1,846,441 spacers from 145,053 CRISPR arrays from 79,735 UHGG genomes using a combination of CRT83 and PILER-CR84 with default parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Redundant CRISPR arrays predicted by both tools were merged based on genomic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Spacers were searched against NAturE MICroBIoLogy | VOL 6 | JULy 2021 | 960–970 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='com/naturemicrobiology 967 ResouRce NATuRE MICROBIOlOGy viral genomes using blastn from the blast+ package v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0 (options: -dust = no word-size = 18), allowing a maximum of one mismatch or gap over ≥95% of the spacer length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For each viral genome, we then aggregated connections to UHGG genomes and identified the lowest host taxonomic rank resulting in >70% agreement across connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Clustering viral genomes into vOTUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' All viral genomes with >50% completeness were clustered into species-level vOTUs on the basis of 95% ANI and 85% alignment fraction (AF) of the shorter sequence, as recommended by Roux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' ANI and AF were estimated between all genome pairs using a custom script from the CheckV repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The script performs all-versus-all local alignments using blastn from the blast+ package v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0 (options: perc_identity = 90 max_target_ seqs = 10000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' ANI is computed as the length-weighted average DNA identity across local alignments between each genome pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' AF is computed by merging alignment coordinates between each genome pair and dividing by the length of each genome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' This approach gave consistent results compared to MUMMer4 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 85), while running in a small fraction of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Clustering was performed using a greedy, centroid-based algorithm in which: (1) genomes were sorted by length, (2) the longest genome was designated as the centroid of a new cluster, (3) all genomes within 95% ANI and 85% AF were assigned to that cluster, and steps 2 and 3 were repeated until all genomes had been assigned to a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To identify genus- and family-level vOTUs, we clustered viral genomes using a combination of gene sharing and AAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For computational efficiency, only the longest genome per species-level vOTU was included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Blastp from the DIAMOND package v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='126 was used with options ‘-e-value 1 × 10−5–max-target-seqs 10,000’ to align all viral proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For each pair of genomes, we identified shared genes (e-value <1 × 10−5), computed their AAI, and computed the percentage of genes shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Edges between genomes were filtered based on their minimum AAI and gene sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Clustering was performed with MCL v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='14-137 using different values for the inflation factor parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We then selected the filtering thresholds and MCL inflation factor that resulted in the highest agreement with genus- and family-level annotations from NCBI RefSeq, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' At the family level, we filtered connections between genomes with <20% AAI or <10% genes shared and used an inflation factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' At the genus level, we filtered connections between genomes with <50% AAI or <20% gene sharing and used an inflation factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We benchmarked our approach on taxonomically annotated genomes from NCBI, showing that viral clusters displayed high taxonomic homogeneity (that is the percentage of genomes from each cluster assigned to the same taxon; genus rank = 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1%, family rank = 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='7%), though sometimes split known taxa into multiple clusters (that is percentage of genomes from each taxon assigned to the same cluster: genus rank = 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='6%, family rank = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Metagenomic read recruitment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Read mapping was performed to viral genomes databases to assess their coverage of viruses in microbiomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' First, we downloaded short reads from human gut viromes analysed by the HuVirDB plus short reads from three recent gut virome studies14,86,87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Short reads from whole metagenomes were downloaded for 1,257 stool samples from various countries (representing up to 50 samples per country).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To ensure that viromes were mostly free of cellular contamination, we ran the viromeQC tool88 and retained viromes with an enrichment score >10, as recommended by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For computational efficiency, we only analysed the first 1,000,000 sequencing reads from each data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For quality control, we discarded reads that were either too short (<70 bp), contained ambiguous base calls, had low base quality scores (mean quality score <30) or mapped to the human genome (build hg19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Next, we used Bowtie v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 89) to construct genome indexes for read mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Five indexes were created using all genomes from each of the four gut human virus databases (MGV, IMG/VR, HuVirDB, GVD), plus NCBI GenBank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Five additional indexes were created using only a single genome per species-level vOTU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Next, we used Bowtie 2 (options ‘–very-sensitive -k 20’) to align sequencing reads to each of the 10 genome indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Alignments between sequencing reads and viral genomes derived from the same SRA study were discarded to prevent overestimation of mapping rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Additionally, alignments with mapping identity <95% (for example, edit distance >5 for 100-bp read) were discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' After these filtering steps we quantified the percentage of high-quality, non-human reads that mapped to each database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Phylogenetic analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We constructed a phylogeny of Caudovirales genomes using the method described by Low et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' First, we identified the set of 77 Caudovirales markers in the representative genomes of 60,439 species-level vOTUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' HMMs for the 77 markers were searched against the protein sequences and the top hits individually aligned to the profile HMMs using HMMER v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Individual marker alignments were then trimmed to retain positions with less than 50% gaps using trimAl v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='4 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 90) and concatenated, filling in gaps for missing markers where necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Only genomes containing at least three markers and having data at >5% of alignment columns were retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' This resulted in a multiple sequence alignment of 28,780 genomes with 22,711 alignment columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We then inferred a concatenated protein phylogeny from the multiple sequence alignment using FastTree v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='9 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 91) under the WAG + G model with the additional flags ‘-mlacc 2’ and ‘-slownni’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The tree was then midpoint-rooted and visualized using iToL74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' In addition, we constructed core-genome single-nucleotide polymorphism (SNP) phylogenies of individual species-level vOTUs with at least 100 genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' SNPs were identified by aligning all genomes to the longest genome in the cluster using nucmer from the MUMmer4 package v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0beta2 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 85) with default options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' SNPs were identified at genomic positions covered by ≥50% of genomes and we retained all genomes with data at ≥50% of positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' FastTree v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='9 was used to construct phylogenetic trees using default options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Functional annotation and protein clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Some 11,837,198 protein-coding genes were identified from the 189,680 MGVs using Prodigal and genes were annotated based on HMM searches against protein family databases: KEGG55, TIGRFAM56, Pfam-A57, VOGDB (http://vogdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='org) and the Earth’s Virome viral protein families database23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' All searches were performed using the hmmsearch utility in the HMMER package v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1b2 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 75) with default parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Each gene was annotated by each database according to its top scoring alignment with a bit-score ≥50, except for Pfam and TIGRFAM where trusted cut-offs were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Antibiotic resistance genes were identified using three tools: (1) the Resistance Gene Identifier v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 62) using option ‘–low_quality’ with gene-specific bit-score thresholds, (2) the NCBI AMRFinder tool v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='4 (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 61) using default options and (3) the Resfams database60 using hmmsearch with HMM-specific bit-score thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' DGRs were identified using the tool DGRscan67 with default options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' All proteins were clustered at 30% AAI and 70% alignment coverage using MMseqs2 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='6d92c58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Reporting Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Further information on research design is available in the Nature Research Reporting Summary linked to this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Data availability Access to the full catalogue of viral genomes, protein clusters, diversity-generating retroelements and CRISPR spacers is provided without restrictions at https:// portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='nersc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='gov/MGV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Any requests for further data should be directed to the corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Code availability Supporting code, including our viral detection pipeline, is provided at https:// github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='com/snayfach/MGV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Received: 9 March 2021; Accepted: 25 May 2021; Published online: 24 June 2021 references 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Lynch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' V.' metadata={'source': 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+page_content=' Acknowledgements We thank S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Roux for analysis of jumbo phages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The work was conducted in the Environmental Genomics and Systems Biology Division at the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Lawrence Berkeley National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Funding was provided by the Chan Zuckerberg Biohub, the Autoimmunity Research Foundation (FP00010476), the Australian Research Council Laureate Fellowship (FL150100038) and the National Insititutes of Health (R01AI148623 and P30 CA124435).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Author contributions S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='N., D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' conceived of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' performed experiments, analysed data and wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' constructed the Caudovirales phylogeny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='-E., H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' contributed to analysis of protein families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' identified phages using alternative genetic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='F contributed funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' supervised the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' All authors reviewed and approved the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Competing interests P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' is a co-founder of Microba Life Sciences, which is a microbial genomics company developing microbiome-based diagnostics and therapeutics and offers metagenomic gut microbiome reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' All other authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Additional information Extended data is available for this paper at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1038/s41564-021-00928-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Supplementary information The online version contains supplementary material available at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1038/s41564-021-00928-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Correspondence and requests for materials should be addressed to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' or N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Reprints and permissions information is available at www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='com/reprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Open Access This article is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0 International License, which permits use, sharing, adap- tation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' If material is not included in the article’s Creative Commons license and your intended use is not permitted by statu- tory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To view a copy of this license, visit http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' © The Author(s) 2021 NAturE MICroBIoLogy | VOL 6 | JULy 2021 | 960–970 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='com/naturemicrobiology 970 ResouRce NATuRE MICROBIOlOGy ResouRce NATuRE MICROBIOlOGy Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 1 | Impact of assembly methods on viral recovery from gut metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The MGV catalogue was formed using metagenomic viral contigs identified from three studies that performed large-scale assembly of human stool metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The CIBIO and MGnify studies used MetaSPAdes for metagenomic assembly while the JGI study used MEGAHIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' To explore the effect of assembler on virus identification, we compared viral contigs identified from a common set of 752 stool samples which were assembled by all three studies and were each represented by a single SRA run accession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' a, The number of vOTUs represented by viral contigs (>50% completeness) from each of the three studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' A similar number of vOTUs were identified from metagenomic contigs assembled by each study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' b, The number of viral contigs at different quality levels identified from each of the three studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' A greater number of complete and high-quality viral genomes are recovered from the MEGAHIT assemblies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' NAturE MICroBIoLogy | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='com/naturemicrobiology Dataset Citation Complete High-quality Medium-guality Low-quality MGnify Mitchelletal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2019 363 616 1,778 63,192 CIBIO Pasollietal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2019 389 592 1,755 62,198 JGI Nayfachetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2019 497 676 1,651 60,885ResouRce NATuRE MICROBIOlOGy ResouRce NATuRE MICROBIOlOGy Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 2 | Diversity of jumbo phages identified in the MgV dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The tree includes MGV sequences alongside a reference set of metagenome-assembled jumbo phages published by Al-Shayeb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Branches leading to MGV sequences, or clades composed exclusively of MGV sequences, are highlighted in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Nodes with support < 50% were collapsed, and nodes with support ≥ 80% are indicated with a grey circle on the corresponding branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Outer rings indicate the genome quality and continent of origin for MGV sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' When sequences from different continents were 100% identical and only 1 sequence was included in the tree, the different continents of origin are indicated with stacked coloured squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For box plots, the middle line denotes the median, the box denotes the interquartile range (IQR), and the whiskers denote 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5× the IQR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' NAturE MICroBIoLogy | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='com/naturemicrobiology ResouRce NATuRE MICROBIOlOGy ResouRce NATuRE MICROBIOlOGy Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 3 | Strain level phylogeography of prevalent human gut phages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Core-genome SNP phylogenies were constructed for individual species-level vOTUs with at least 100 genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The figure shows three distinct vOTUs displaying a strong signature of phylogeography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For each tree, viral genomes are displayed as tips with colours indicating the geographic origin of the metagenomic sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' NAturE MICroBIoLogy | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='com/naturemicrobiology ResouRce NATuRE MICROBIOlOGy ResouRce NATuRE MICROBIOlOGy Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 4 | Antibiotic resistance genes identified from 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='8 million viral proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' a-b, Viral genes with putative beta-lactamase domains identified based on hits to the Pfam and KEGG databases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' c-e, Resistance genes (including beta-lactamases) identified using Resfinder, AMRfinder, or the Resistance Gene Identifier (RGI), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' f, Overlap of resistance genes identified by Resfinder, AMRfinder, and RGI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Most viral proteins identified with putative beta-lactamase domains are not confirmed as antibiotic resistance genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' NAturE MICroBIoLogy | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='com/naturemicrobiology ResouRce NATuRE MICROBIOlOGy ResouRce NATuRE MICROBIOlOGy Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' 5 | Comparison of viral contigs from the MgV and gPD catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' a, The number of viral contigs with at least 50% completeness from the MGV and GPD catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' The GPD catalogue contains 142,809 viral contigs when including those with <50% completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Contigs from each catalogue where clustered at 95% ANI over 85% the length of the shorter sequence to form species-level vOTUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' b, MGV and GPD catalogues were clustered together using the longest contig from each vOTU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' c, The histograms show the similarity between contigs from the MGV (n = 54,118) and GPD (n = 46,480) catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' d, Similarity to the GPD catalogue for MGV contigs from different viral families: Siphoviridae (n = 22,513), Podoviridae (n = 5,075), Myoviridae (n = 2,560), crAss-like (n = 948), Caudovirales other (n = 19,633), Microviridae (n = 2,133), CRESS DNA (n = 115), other (n = 1,141).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' NAturE MICroBIoLogy | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='com/naturemicrobiology 1 nature research | reporting summary October 2018 Corresponding author(s): Stephen Nayfach and Nikos Kyrpides Last updated by author(s): April 15, 2021 Reporting Summary Nature Research wishes to improve the reproducibility of the work that we publish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' This form provides structure for consistency and transparency in reporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For further information on Nature Research policies, see Authors & Referees and the Editorial Policy Checklist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Statistics For all statistical analyses, confirm that the following items are present in the figure legend, table legend, main text, or Methods section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' n/a Confirmed The exact sample size (n) for each experimental group/condition, given as a discrete number and unit of measurement A statement on whether measurements were taken from distinct samples or whether the same sample was measured repeatedly The statistical test(s) used AND whether they are one- or two-sided Only common tests should be described solely by name; describe more complex techniques in the Methods section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' A description of all covariates tested A description of any assumptions or corrections, such as tests of normality and adjustment for multiple comparisons A full description of the statistical parameters including central tendency (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' means) or other basic estimates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' regression coefficient) AND variation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' standard deviation) or associated estimates of uncertainty (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' confidence intervals) For null hypothesis testing, the test statistic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' F, t, r) with confidence intervals, effect sizes, degrees of freedom and P value noted Give P values as exact values whenever suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' For Bayesian analysis, information on the choice of priors and Markov chain Monte Carlo settings For hierarchical and complex designs, identification of the appropriate level for tests and full reporting of outcomes Estimates of effect sizes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=" Cohen's d, Pearson's r), indicating how they were calculated Our web collection on statistics for biologists contains articles on many of the points above." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Software and code Policy information about availability of computer code Data collection Prodigal v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='3, HMMER v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1b2, VirFinder v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1, DIAMOND v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='126, Barrnap v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='9-dev, blast+ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0, Bowtie v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2, MCL v14-137, FAMSA v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5, trimAL v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='4, FastTree v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='9, iTOL, MUMmer4 v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0beta2, CRT, PILER-CR, AMRFinder v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='4, Resistance Gene Identifier v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0, MMseqs2 v10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='6d92c, PSI-BLAST, CheckV v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='0, viromeQC, DGRscan Data analysis See software listed above For manuscripts utilizing custom algorithms or software that are central to the research but not yet described in published literature, software must be made available to editors/reviewers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' We strongly encourage code deposition in a community repository (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' GitHub).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' See the Nature Research guidelines for submitting code & software for further information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Data Policy information about availability of data All manuscripts must include a data availability statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' This statement should provide the following information, where applicable: Accession codes, unique identifiers, or web links for publicly available datasets A list of figures that have associated raw data A description of any restrictions on data availability Access to the full dataset of viral genomes, protein clusters, diversity generating retroelements, and CRISPR spacers is provided without restrictions at https:// portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='nersc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='gov/MGV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Any requests for further data should be directed to the corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' natureresearch2 nature research | reporting summary October 2018 Field-specific reporting Please select the one below that is the best fit for your research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' If you are not sure, read the appropriate sections before making your selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Life sciences Behavioural & social sciences Ecological, evolutionary & environmental sciences For a reference copy of the document with all sections, see nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='com/documents/nr-reporting-summary-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='pdf Life sciences study design All studies must disclose on these points even when the disclosure is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Sample size We used 18,271 assembled human gut metagenomes for 11,810 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' These represent all available datasets from the gut microbiome with SRA accession codes at the time we started our project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Data exclusions We excluded datasets that were not from a human stool sample or where SRA accession codes could not be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Replication Not applicable to our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' All the datasets analyzed were publicly available, and therefore we did not generate any additional data for replication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Randomization Not applicable to our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Any conditions of the samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' geographic location or disease state) were already determined before our study began since the datasets were publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Blinding Not applicable to our study for the same reason given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Reporting for specific materials, systems and methods We require information from authors about some types of materials, experimental systems and methods used in many studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Here, indicate whether each material, system or method listed is relevant to your study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' If you are not sure if a list item applies to your research, read the appropriate section before selecting a response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} +page_content=' Materials & experimental systems n/a Involved in the study Antibodies Eukaryotic cell lines Palaeontology Animals and other organisms Human research participants Clinical data Methods n/a Involved in the study ChIP-seq Flow cytometry MRI-based neuroimaging' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_49/content/kb_49.pdf'} diff --git a/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf b/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..96efad59fee18ebabb4a7a9d90ae8ad85cc71692 --- /dev/null +++ b/l9FKT4oBgHgl3EQfDy1k/content/2301.11713v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:16339aa5bd3f01f97bfd14044580bbad567dc7d9624101e7d379d622d5c38c30 +size 181980 diff --git a/l9FKT4oBgHgl3EQfDy1k/vector_store/index.pkl b/l9FKT4oBgHgl3EQfDy1k/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..d856e652709b98fc747e893543215ac8ce62f84d 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a/mdE3T4oBgHgl3EQfKgkL/content/tmp_files/2301.04353v1.pdf.txt b/mdE3T4oBgHgl3EQfKgkL/content/tmp_files/2301.04353v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a7e3f48bcdbbcedc4bbbfdd2fecc21b70c6e8f23 --- /dev/null +++ b/mdE3T4oBgHgl3EQfKgkL/content/tmp_files/2301.04353v1.pdf.txt @@ -0,0 +1,569 @@ +K-space interpretation of image-scanning-microscopy +Tal I. Sommer,1, 2 Gil Weinberg,1 and Ori Katz1 +1)Department of Applied Physics, Hebrew University of Jerusalem, Jerusalem 9190401, +Israel +2)Alexander Grass Center for Bioengineering, Hebrew University of Jerusalem, Jerusalem 9190401, +Israel +(Dated: 12 January 2023) +In recent years, image-scanning microscopy (ISM, also termed pixel-reassignment microscopy) has emerged as +a technique that improves the resolution and signal-to-noise compared to confocal and widefield microscopy +by employing a detector array at the image plane of a confocal laser scanning microscope. Here, we present a +k-space analysis of coherent ISM, showing that ISM is equivalent to spotlight synthetic-aperture radar (SAR) +and analogous to oblique-illumination microscopy. This insight indicates that ISM can be performed with a +single detector placed in the k-space of the sample, which we numerically demonstrate. +Confocal imaging is a technique that improves axial- +sectioning and transverse resolution in optical microscopy +compared to conventional ’widefield’ imaging1,2. In wide- +field imaging, the entire field of view (FOV) is illumi- +nated, and all points in the sample are simultaneously +imaged to a detector (camera) plane. Confocal imaging +allows an improvement in the transverse and axial reso- +lution by illuminating the sample with a scanned focused +spot, and detecting the signal emerging from that focal +spot with a small ’confocal’ detector conjugated to the +illumination spot. The effective size of the confocal detec- +tor, which determines the spatial filtering and collection +efficiency, is set by a small ’confocal pinhole’ placed in +front of the detector. While the resolution improvement +in confocal-microscopy increases with decreasing pinhole +size3, a smaller pinhole lowers the detection efficiency, +resulting in a lower signal-to-noise ratio (SNR). +A recently-introduced method termed image-scanning +microscopy4–6 (ISM) or pixel-reassignment7 allows full +use of all signal photons in a confocal scanning system +without sacrificing transverse imaging resolution. More- +over, depending on the imaging point spread function +(PSF), ISM can also provide an improvement in imag- +ing resolution8,9. ISM achieves this feat by using an ar- +ray of detectors at the image plane instead of the con- +ventional single detector of confocal systems. The cen- +ter detector of the ISM array collects the same infor- +mation as would have been collected by a confocal pin- +hole. +However, the neighboring detectors collect light +that is otherwise rejected in a confocal system. To con- +struct the ISM image, at each illumination point, rin, +the signals from each detector position, rout, are reas- +signed to the midpoint between illumination and detec- +tion positions2,4: rISM = 1 +2(rin + rout). The reassigned +signals are summed over all scan positions, forming an +ISM image. ISM was first implemented in incoherent mi- +croscopy modalities4,10,11, and was recently adapted to +coherent imaging modalities12–15. +Here, we analyze coherent-ISM in the spatial Fourier +domain. +Utilizing the projection-slice theorem16 we +show that ISM is equivalent to spotlight synthetic- +aperture radar (SAR)17,18 (Fig. 1), a well-established +beam-scanning imaging technique, which utilizes a sin- +gle detector. As a direct result, we numerically demon- +strate that ISM can be performed with a single detector, +and leverage the k-space analysis to highlight the close +connection to oblique-illumination microscopy19. +We begin our analysis by representing the signals +collected in coherent-ISM using the reflection matrix +formalism14,20. In this formalism, the fields that are col- +lected at position rout at the detection plane when the +illumination is focused at rin in the object plane are given +by the matrix element R(rin, rout) (see coordinate nota- +tion in Fig. 1b). To simplify the mathematical deriva- +tions and without loss of generality, we consider below +one transverse dimension, whose coordinate is given by +x. +Following the pixel reassignment process (xISM += +1 +2(xin +xout)), the ISM image formation at a given imag- +ing depth, z, is given by: +IISM(x) = +� +xin +R(xin, xout = 2x − xin; z) +(1) +In this matrical representation, the ISM summation +can be interpreted as a sum over the anti-diagonal ele- +ments of the reflection matrix (Fig. 1a). +Such a line summation (Fig. 1a, left inset) is analo- +gous to a line projection performed in many tomographic +imaging techniques such as x-ray CT21. Leveraging the +Fourier-slice theorem16,22, the Fourier transform of this +line-projection is the main diagonal in the 2D Fourier +transform of R(xin, xout; z), scaled by a factor of half +(Fig. 1a, right inset): +F{IISM(x)} = ˜IISM(k) = ˜R(kin = k/2, kout = k/2; z) +(2) +where F{} is the spatial Fourier-transform, and kin +and kout are the k-vectors. For monochromatic illumi- +nation, this k-space reflection matrix23, ˜R(kin, kout), can +be interpreted as the measured reflected plane-wave with +a wavevector kout when illuminating the sample with a +plane-wave, kin. This k-space matrix is the 2D Fourier +transform of R: +arXiv:2301.04353v1 [physics.optics] 11 Jan 2023 + +2 +FIG. 1. +Coherent Image-Scanning-Microscopy (ISM), obtained by raster-scanning a focused illumination, and its k-space +interpretation as spotlight synthetic aperture radar (SAR), obtained by plane wave illumination. (a) ISM measurements can +be represented by a reflection matrix R(xin, xout) providing the measured signals at coordinates xout for an illumination +focused at xin (where the diagonal R(x, x) is the confocal image). For simplicity, a one-dimensional object and monochromatic +illumination are considered. An ISM image, IISM(x), is obtained by summing over the anti-diagonal elements of the reflection +matrix, IISM(x) = � +xin R(xin, xout = 2x − xin), i.e., for all points satisfying x = (xout + xin)/2 (white arrows). According to +the Fourier-slice theorem, this line summation (projection) is the Fourier-pair of the diagonal of the k-space reflection matrix +˜R(kin, kout) = F{R(xin, xout)}, where F{} is a 2D Fourier-transform. These k-space equivalent measurements, ˜R(kin, kout = +kin), represent the reflected amplitude of a plane-wave at kout obtained for a plane-wave illumination at kin = kout, for a +set of different illumination angles. These are analogous to spotlight SAR measurements. (b) Schematic of conventional ISM +measurement setup: focused illumination is raster-scanned across the object plane (x-space). At each illumination position, the +reflected fields at all positions around the illumination point are measured. (c) Schematic of a setup for k-space ISM (analogous +to spotlight SAR measurements): At each measurement position (green and purple beams), the object is illuminated by a tilted +plane wave, kin, and the plane-wave that is reflected in the same angle, kout = kin, is detected. +˜R(kin, kout; z) = +�� +R(xin, xout; z) e−ikinxin e−ikoutxout dxoutdxin +(3) +The Fourier-slice equivalence of Eq. 2 can be derived +by considering the rescaled diagonal of ˜R: +˜R(kin = k/2, kout = k/2; z) = +�� +R(xin, xout; z) e−i 1 +2 kxin e−i 1 +2 kxout dxoutdxin += +�� +R(xin, xout; z) e−ik xin+xout +2 +dxoutdxin +(4) +Performing inverse Fourier-transform yields: +� +˜R(kin = k/2, kout = k/2; z) eikx dk = +��� +R(xin, xout; z) e−ik xin+xout +2 +eikx dkdxoutdxin += +�� +R(xin, xout; z) δ +� +x − xin + xout +2 +� +dxoutdxin +(5) +Which is the same anti-diagonal summation as in ISM +image formation in Eq. 1. +This mathematical equivalence implies that the same +ISM image can be acquired in two different forms. One, +conventionally, utilizes a scanning array in the spatial +domain, detecting the reflected fields from a spatially- +focused scanning illumination beam (Fig. 1b). +Alter- +natively, the sample can be scanned by plane waves at + +X +X +X +2222 +LASER +Kin +SCAN3 +FIG. 2. +Numerical comparison of coherent monochromatic imaging via conventional ISM, plane-wave scanning ISM, and +confocal imaging. (a) In conventional ISM, a focused illumination beam raster scans the target object plane (x, y). A detector +array (CAM) that is conjugated to the object plane detects the reflected fields, and the ISM image is formed by reassignment +and summation of the detected signals. A confocal image is formed from the signal emerging only from the illumination point +((xin, yin) = (xout, yout)). (b) Monochromatic ISM can be performed with a single detector placed at the Fourier plane of the +object, by sequentially illuminating the object with tilted plane waves. (c-e) We display a comparison between the detected +amplitudes in confocal detection, an x-scanned ISM, and a k-scanned ISM. A vertical cross-section (depicted by a white dashed +line in the confocal image) is also displayed. This comparison shows the equivalence between x-scan ISM and k-scan ISM +results, which both result in a super-resolved image compared to the conventional confocal image. +different angles of incidence (given by kin), and a sin- +gle detector placed in the far field of the sample that +detects the reflected plane wave at the same angle, i.e., +the single Fourier component kout = kin. This second +approach, illustrated in Fig. 1c, is known as spotlight +synthetic aperture radar (SAR) imaging18,24. +The scanning plane wave illumination of k-space ISM +is closely related to oblique-illumination microscopy19. +In oblique-illumination microscopy (originally proposed +by Abbe25) the sample is illuminated by different an- +gled plane-waves, and a camera is measuring the result- +ing image. In the k-space, the plane-wave illumination +is manifested as a shift of the high frequencies angular- +spectrum information of the sample into the passband +of the system, allowing its detection26. In k-space ISM, +each illumination is identical to the plane wave illumina- +tion of oblique-illumination microscopy. The difference +from oblique illumination microscopy is that only a sin- +gle Fourier component is detected in each illumination +in k-space ISM (Eq. 2). +Similar to oblique illumina- +tion microscopy, k-space ISM measurements thus yield +extended k-space support. The differences between the +techniques are the use of a single detector in k-space ISM +rather than the detector-array of oblique-illumination mi- +croscopy, which requires a larger number of illuminations, +and the angular-spectrum reweighting that is required in +oblique illumination microscopy19,27. +To support and demonstrate our mathematical deriva- +tion, we numerically simulated the coherent imaging of a +test sample with conventional ISM (Fig. 2a), and k-space +ISM (Fig. 2b). While the two approaches differ in both +their illumination scheme (focused vs. plane-wave), and +detection scheme (array detection in the object plane vs. +a single detector in the k-space), the resulting images +are identical and present an improvement over confocal +imaging. +The simulated conventional ISM setup (Fig. 2a) con- +sists of a 4-f imaging system, where the scanning laser +and the detection array (CAM) are both imaged onto the +object plane. A focused illumination beam raster scans +the target object plane (x, y). At each illumination posi- +tion, (xin, yin), the reflected field across the image plane, +(xout, yout), are measured. The simulated k-space ISM +setup (Fig. 2b) is based on Fourier-transforming a point +illumination (LASER) and confocal detection, from the +detector plane (DET) to the object plane using a lens. +The point illumination is thus converted to a tilted plane- +wave in the object plane, and the detector measures a +single Fourier component, kout = kin. Observing Fig. 2b +reveals that k-space ISM is, in fact, confocal coherent +imaging performed in the k-space domain. See simula- +tion parameters in Supplementary Material, Section A. + +2222 +G +LASER +22 +ASER +BS +Confocal +k-scan ISM +X-scan ISM +100μm4 +In Fig. 3 we compare the two ISM processing ap- +proaches and confocal imaging using an experimental +ultrasound echography dataset. The dataset represents +measurements performed on an acoustic phantom with +multi-plane-wave transmission28. Data were acquired us- +ing a Verasonics P4–2v probe at a center frequency of +f0 = 4.8MHz, having 64 elements with a total aperture +size of D = 19.2mm. The imaged target is composed of +five pins (Fig. 3a, red X’s) having a diameter of 0.1mm +at a depth of 60mm. Data were post-processed with the +proper phases to perform coherent compounding for ei- +ther confocal, x-space ISM14 or k-space ISM (Fig. 3a,b,c, +respectively). These experimental results agree with the +analytic and numerical investigations (Fig. 3d). +See a +discussion of the data processing in Supplementary Ma- +terial, Section B. +FIG. 3. +Experimental comparison of confocal ultrasound +imaging (a), conventional ISM (b), and k-space ISM (c), of +five reflecting pins (marked by red X’s) in an acoustic phan- +tom. The same experimental data, acquired by plane wave +sonications, were processed offline to generate the results. (a- +c) presents the reconstructed intensity (absolute value square +of the reconstructed fields). The two approaches for ISM re- +sults in practically identical images, having the expected no- +ticeable resolution improvement over the confocal image. (d) +A cross-section comparison of (a-c) along the dotted line in +(a). +In conclusion, by leveraging the Fourier slice theorem, +we show that it is possible to perform coherent ISM with +a single detector located at the Fourier plane of the ob- +ject. This allows the same resolution improvement gained +in ISM over confocal imaging without requiring a detec- +tor array. This may be interesting for utilizing ISM in +new modalities where detector arrays are not accessible. +In addition, the k-space acquisition and simple (confocal- +like) analysis reduce the memory requirements and may +be important also for the reduction of computational bur- +den. +We note that a single detector approach will result in +a lower SNR than the detector-array approach in most +scenarios since only a fraction of the reflected field is cap- +tured. On the other hand, the plane-wave illumination +reduces the power density concentrated on the sample, +as compared to focused scanned illumination, and thus +may allow higher illumination power. +ACKNOWLEDGMENTS +This work has received funding from The Israel Sci- +ence Foundation (Grant no. +1361/18), European Re- +search Council (ERC) Horizon 2020 research and innova- +tion program (677909), and was supported by the Min- +istry of Science and Technology in Israel. +AUTHOR CONTRIBUTIONS +Tal Sommer and Gil Weinberg contributed equally to +this work. +Tal +Sommer: +Conceptualization +(equal); +Data +Curation (equal); Formal Analysis(equal); Methodol- +ogy (equal); Writing/Original Draft Preparation (lead); +Writing/Review +& +Editing +(equal). +Gil +Wein- +berg: Conceptualization (equal); Data Curation (equal); +Formal Analysis(equal); +Methodology (equal); +Writ- +ing/Review & Editing (equal). Ori Katz: Supervision +(lead); Funding Acquisition (lead); Writing/Review & +Editing (equal) +DATA AVAILABILITY +The data that support the findings of this study are +available from the corresponding author upon reasonable +request. +SUPPLEMENTARY MATERIAL +See supplementary material for the simulation param- +eters and a discussion of the data processing. +1J. Mertz, Introduction to optical microscopy (Cambridge Univer- +sity Press, 2019). +2C. J. Sheppard, M. Castello, G. Tortarolo, T. Deguchi, S. V. +Koho, G. Vicidomini, +and A. Diaspro, “Pixel reassignment in +image scanning microscopy: a re-evaluation,” JOSA A 37, 154– +162 (2020). + +Confocal +X--X-X +■ +Imm +x-scan ISM +k-scan ISM5 +3J. G. Fujimoto and D. Farkas, Biomedical optical imaging (Ox- +ford University Press, 2009). +4C. B. M¨uller and J. Enderlein, “Image scanning microscopy,” +Physical review letters 104, 198101 (2010). +5E. N. Ward and R. 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Oron, “Super- +resolved second harmonic generation imaging by coherent im- +age scanning microscopy,” Applied physics letters 120, 071111 +(2022). +16R. N. Bracewell, “Strip integration in radio astronomy,” Aus- +tralian Journal of Physics 9, 198–217 (1956). +17W. M. Brown, “Synthetic aperture radar,” IEEE Transactions +on Aerospace and Electronic Systems , 217–229 (1967). +18J. C. Kirk, “A discussion of digital processing in synthetic aper- +ture radar,” IEEE Transactions on aerospace and electronic sys- +tems , 326–337 (1975). +19S. Chowdhury, A.-H. Dhalla, and J. Izatt, “Structured oblique +illumination microscopy for enhanced resolution imaging of non- +fluorescent, coherently scattering samples,” Biomedical optics ex- +press 3, 1841–1854 (2012). +20W. Lambert, L. A. Cobus, M. Couade, M. Fink, and A. Aubry, +“Reflection matrix approach for quantitative imaging of scatter- +ing media,” Physical Review X 10, 021048 (2020). +21R. M. Mersereau and A. V. Oppenheim, “Digital reconstruction +of multidimensional signals from their projections,” Proceedings +of the IEEE 62, 1319–1338 (1974). +22S.-R. Zhao and H. Halling, “A new fourier method for fan beam +reconstruction,” in 1995 IEEE Nuclear Science Symposium and +Medical Imaging Conference Record, Vol. 2 (IEEE, 1995) pp. +1287–1291. +23W. Lambert, L. A. Cobus, T. Frappart, M. Fink, and A. Aubry, +“Distortion matrix approach for ultrasound imaging of random +scattering media,” Proceedings of the National Academy of Sci- +ences (2020). +24M. Soumekh, Synthetic aperture radar signal processing, Vol. 7 +(New York: Wiley, 1999). +25E. +Abbe, +“Beitr¨age +zur +theorie +des +mikroskops +und +der +mikroskopischen +wahrnehmung,” +Archiv +f¨ur +mikroskopische +Anatomie 9, 413–468 (1873). +26K. Wicker and R. Heintzmann, “Resolving a misconception about +structured illumination,” Nature Photonics 8, 342–344 (2014). +27T. Ilovitsh, A. Ilovitsh, J. Foiret, B. Z. Fite, +and K. W. Fer- +rara, “Acoustical structured illumination for super-resolution ul- +trasound imaging,” Communications biology 1, 1–11 (2018). +28G. Montaldo, M. Tanter, J. Bercoff, N. Benech, +and M. Fink, +“Coherent plane-wave compounding for very high frame rate ul- +trasonography and transient elastography,” IEEE transactions +on ultrasonics, ferroelectrics, and frequency control 56, 489–506 +(2009). +29J. W. Goodman, Introduction to Fourier optics (Roberts and +Company Publishers, 2005). + +6 +SUPPLEMENTARY MATERIAL +A. +Simulation Parameters +This section discusses the simulation parameters used +to simulate the x-scanning and k-scanning systems +(Fig. 2a-b in the main text). +Both systems were simulated using angular spectrum +as a field propagator29, and the lenses were simulated as +a thin parabolic phase mask. The illumination was set to +a wavelength of 450nm, and the object was imaged onto +the detection plane using lenses of 35mm focal length +and 3mm in diameter. The transfer-function extent in +the x-scanning system was controlled via an aperture in +the Fourier-plane of the object. The aperture size is cho- +sen such that the imaging system was of a numerical +aperture (NA) of 0.0045. This extent was preserved in +the k-scanning system using a finite-extent scan of the +Fourier-plane of the object to the same effective NA. +B. +Ultrasound beam-forming +This section discusses the data processing used for +image reconstruction from an experimental ultrasound +echography dataset (Fig. 3 in the main text). +The dataset represents measurements performed on an +acoustic phantom (GAMMEX SONO403) with multi- +plane-wave transmissions. +Data were acquired using a +Verasonics P4–2v probe at a center frequency of f0 = +4.8MHz, having 64 elements with a total aperture size +of D = 19.2mm, connected to a Verasonics Vantage 256 +multi-channel system. The angular range of the steered +plane-waves was determined by the probe’s geometry +via28: +θm = arcsin mλ0 +D +(6) +Where λ0 is the probe’s center wavelength. +These measurements can be represented with a ma- +trix: RF(u, t, θ) where u is the spatial position of the +transducer element on the probe, t is the measurement +time, and θ is the steering angle of the transmitted plane- +wave28. Therefore, RF(u, t, θ) is the field measured at a +transducer in position u at time t when sonicating with a +plane-wave tilted with an angle θ. The temporal Fourier- +transform of this matrix is: RF(u, ω, θ). +Beamforming is done using the time-of-flight (TOF) for +transmission (tx) and detection (rx) for each point in the +imaged medium (x). The transmission time depends on +the steering angle of transmission, and the detection time +depends on the relative position of the transducer from +the point. Assuming a single dimension, for simplicity, +this TOF is: t(x; u, θ) = ttx(x, θ) + trx(x, u). Therefore, +the reconstructed confocal image at position x is a sum- +mation over all measurements at the respective TOF. +Iconf(x) = +�� +RF(u, ttx(x, θ) + trx(x, u), θ)dudθ += +�� +RF(u, t, θ) δ (t − (ttx(x, θ) + trx(x, u))) dudθ +(7) +A reflection matrix (that is used for x-scan ISM image +formation) can be constructed by using a transmission +time for a position that is different from the position for +the detection time20,23: +R(xin, xout) = +�� +RF (u, ttx(xin, θ) + trx(xout, u), θ) dudθ += +�� +RF(u, t, θ) δ (t − (ttx(xin, θ) + trx(xout, u))) dudθ +(8) +This operation is termed Delay-And-Sum (DAS). +In the temporal Fourier-domain, these beamformations +can be performed using phases instead of time-delays: +Iconf(x) = +��� +RF(u, ω, θ)eiω(ttx(x,θ)+trx(x,u))dudθdω +(9a) +R(xout, xin) = +��� +RF(u, ω, θ)eiω(ttx(xin,θ)+trx(xout,u))dudθdω +(9b) +The k-space reflection-matrix, ˜R(kin, kout), is the 2D +spatial Fourier transform of R(xin, xout): +R(kout, kin) = +�� +R(xout, xin)e−ikinxine−ikoutxoutdxindxout +(10) +And, as the RF measurement is not dependant of xin +and xout, Eq. 10 can be written as: +R(kout, kin)= +��� +RF(u, ω, θ) +× +�� +eiωttx(xin,θ)e−ikinxindxin +� +× +�� +eiωtrx(xout,θ)e−ikoutxoutdxout +� +dudθdω +(11) +This means that the k-space reflection-matrix can be +constructed in the temporal Fourier-domain, using the +1D spatial Fourier-transforms of the phases used for the +reflection-matrix construction. +One should determine the field-of-view (FOV) lateral +grid to be the same as the positions of the transduc- +ers (u). This results in the grids of the two 1D spatial + +7 +Fourier-transforms to coincide. +One can leverage this +method to reconstruct a k-scanned ISM image directly +from the echography dataset using only the phases that +correspond to the wave-vectors kin = kout. + diff --git a/mdE3T4oBgHgl3EQfKgkL/content/tmp_files/load_file.txt b/mdE3T4oBgHgl3EQfKgkL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bf54ff8ed212b70da68f1b8301f242cd42162256 --- /dev/null +++ b/mdE3T4oBgHgl3EQfKgkL/content/tmp_files/load_file.txt @@ -0,0 +1,344 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf,len=343 +page_content='K-space interpretation of image-scanning-microscopy Tal I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Sommer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 2 Gil Weinberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content='1 and Ori Katz1 1)Department of Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Hebrew University of Jerusalem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Jerusalem 9190401,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Israel 2)Alexander Grass Center for Bioengineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Hebrew University of Jerusalem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Jerusalem 9190401,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Israel (Dated: 12 January 2023) In recent years,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' image-scanning microscopy (ISM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' also termed pixel-reassignment microscopy) has emerged as a technique that improves the resolution and signal-to-noise compared to confocal and widefield microscopy by employing a detector array at the image plane of a confocal laser scanning microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Here, we present a k-space analysis of coherent ISM, showing that ISM is equivalent to spotlight synthetic-aperture radar (SAR) and analogous to oblique-illumination microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' This insight indicates that ISM can be performed with a single detector placed in the k-space of the sample, which we numerically demonstrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Confocal imaging is a technique that improves axial- sectioning and transverse resolution in optical microscopy compared to conventional ’widefield’ imaging1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' In wide- field imaging, the entire field of view (FOV) is illumi- nated, and all points in the sample are simultaneously imaged to a detector (camera) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Confocal imaging allows an improvement in the transverse and axial reso- lution by illuminating the sample with a scanned focused spot, and detecting the signal emerging from that focal spot with a small ’confocal’ detector conjugated to the illumination spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The effective size of the confocal detec- tor, which determines the spatial filtering and collection efficiency, is set by a small ’confocal pinhole’ placed in front of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' While the resolution improvement in confocal-microscopy increases with decreasing pinhole size3, a smaller pinhole lowers the detection efficiency, resulting in a lower signal-to-noise ratio (SNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' A recently-introduced method termed image-scanning microscopy4–6 (ISM) or pixel-reassignment7 allows full use of all signal photons in a confocal scanning system without sacrificing transverse imaging resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' More- over, depending on the imaging point spread function (PSF), ISM can also provide an improvement in imag- ing resolution8,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' ISM achieves this feat by using an ar- ray of detectors at the image plane instead of the con- ventional single detector of confocal systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The cen- ter detector of the ISM array collects the same infor- mation as would have been collected by a confocal pin- hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' However, the neighboring detectors collect light that is otherwise rejected in a confocal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' To con- struct the ISM image, at each illumination point, rin, the signals from each detector position, rout, are reas- signed to the midpoint between illumination and detec- tion positions2,4: rISM = 1 2(rin + rout).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The reassigned signals are summed over all scan positions, forming an ISM image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' ISM was first implemented in incoherent mi- croscopy modalities4,10,11, and was recently adapted to coherent imaging modalities12–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Here, we analyze coherent-ISM in the spatial Fourier domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Utilizing the projection-slice theorem16 we show that ISM is equivalent to spotlight synthetic- aperture radar (SAR)17,18 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 1), a well-established beam-scanning imaging technique, which utilizes a sin- gle detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' As a direct result, we numerically demon- strate that ISM can be performed with a single detector, and leverage the k-space analysis to highlight the close connection to oblique-illumination microscopy19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' We begin our analysis by representing the signals collected in coherent-ISM using the reflection matrix formalism14,20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' In this formalism, the fields that are col- lected at position rout at the detection plane when the illumination is focused at rin in the object plane are given by the matrix element R(rin, rout) (see coordinate nota- tion in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' To simplify the mathematical deriva- tions and without loss of generality, we consider below one transverse dimension, whose coordinate is given by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Following the pixel reassignment process (xISM = 1 2(xin +xout)), the ISM image formation at a given imag- ing depth, z, is given by: IISM(x) = � xin R(xin, xout = 2x − xin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' z) (1) In this matrical representation, the ISM summation can be interpreted as a sum over the anti-diagonal ele- ments of the reflection matrix (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Such a line summation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 1a, left inset) is analo- gous to a line projection performed in many tomographic imaging techniques such as x-ray CT21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Leveraging the Fourier-slice theorem16,22, the Fourier transform of this line-projection is the main diagonal in the 2D Fourier transform of R(xin, xout;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' z), scaled by a factor of half (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 1a, right inset): F{IISM(x)} = ˜IISM(k) = ˜R(kin = k/2, kout = k/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' z) (2) where F{} is the spatial Fourier-transform, and kin and kout are the k-vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' For monochromatic illumi- nation, this k-space reflection matrix23, ˜R(kin, kout), can be interpreted as the measured reflected plane-wave with a wavevector kout when illuminating the sample with a plane-wave, kin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' This k-space matrix is the 2D Fourier transform of R: arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content='04353v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content='optics] 11 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Coherent Image-Scanning-Microscopy (ISM), obtained by raster-scanning a focused illumination, and its k-space interpretation as spotlight synthetic aperture radar (SAR), obtained by plane wave illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' (a) ISM measurements can be represented by a reflection matrix R(xin, xout) providing the measured signals at coordinates xout for an illumination focused at xin (where the diagonal R(x, x) is the confocal image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' For simplicity, a one-dimensional object and monochromatic illumination are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' An ISM image, IISM(x), is obtained by summing over the anti-diagonal elements of the reflection matrix, IISM(x) = � xin R(xin, xout = 2x − xin), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=', for all points satisfying x = (xout + xin)/2 (white arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' According to the Fourier-slice theorem, this line summation (projection) is the Fourier-pair of the diagonal of the k-space reflection matrix ˜R(kin, kout) = F{R(xin, xout)}, where F{} is a 2D Fourier-transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' These k-space equivalent measurements, ˜R(kin, kout = kin), represent the reflected amplitude of a plane-wave at kout obtained for a plane-wave illumination at kin = kout, for a set of different illumination angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' These are analogous to spotlight SAR measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' (b) Schematic of conventional ISM measurement setup: focused illumination is raster-scanned across the object plane (x-space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' At each illumination position, the reflected fields at all positions around the illumination point are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' (c) Schematic of a setup for k-space ISM (analogous to spotlight SAR measurements): At each measurement position (green and purple beams), the object is illuminated by a tilted plane wave, kin, and the plane-wave that is reflected in the same angle, kout = kin, is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' ˜R(kin, kout;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' z) = �� R(xin, xout;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' z) e−ikinxin e−ikoutxout dxoutdxin (3) The Fourier-slice equivalence of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 2 can be derived by considering the rescaled diagonal of ˜R: ˜R(kin = k/2, kout = k/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' z) = �� R(xin, xout;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' z) e−i 1 2 kxin e−i 1 2 kxout dxoutdxin = �� R(xin, xout;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' z) e−ik xin+xout 2 dxoutdxin (4) Performing inverse Fourier-transform yields: � ˜R(kin = k/2, kout = k/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' z) eikx dk = ��� R(xin, xout;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' z) e−ik xin+xout 2 eikx dkdxoutdxin = �� R(xin, xout;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' z) δ � x − xin + xout 2 � dxoutdxin (5) Which is the same anti-diagonal summation as in ISM image formation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' This mathematical equivalence implies that the same ISM image can be acquired in two different forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' One, conventionally, utilizes a scanning array in the spatial domain, detecting the reflected fields from a spatially- focused scanning illumination beam (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Alter- natively, the sample can be scanned by plane waves at X X X 2222 LASER Kin SCAN3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Numerical comparison of coherent monochromatic imaging via conventional ISM, plane-wave scanning ISM, and confocal imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' (a) In conventional ISM, a focused illumination beam raster scans the target object plane (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' A detector array (CAM) that is conjugated to the object plane detects the reflected fields, and the ISM image is formed by reassignment and summation of the detected signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' A confocal image is formed from the signal emerging only from the illumination point ((xin, yin) = (xout, yout)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' (b) Monochromatic ISM can be performed with a single detector placed at the Fourier plane of the object, by sequentially illuminating the object with tilted plane waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' (c-e) We display a comparison between the detected amplitudes in confocal detection, an x-scanned ISM, and a k-scanned ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' A vertical cross-section (depicted by a white dashed line in the confocal image) is also displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' This comparison shows the equivalence between x-scan ISM and k-scan ISM results, which both result in a super-resolved image compared to the conventional confocal image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' different angles of incidence (given by kin), and a sin- gle detector placed in the far field of the sample that detects the reflected plane wave at the same angle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=', the single Fourier component kout = kin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' This second approach, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 1c, is known as spotlight synthetic aperture radar (SAR) imaging18,24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The scanning plane wave illumination of k-space ISM is closely related to oblique-illumination microscopy19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' In oblique-illumination microscopy (originally proposed by Abbe25) the sample is illuminated by different an- gled plane-waves, and a camera is measuring the result- ing image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' In the k-space, the plane-wave illumination is manifested as a shift of the high frequencies angular- spectrum information of the sample into the passband of the system, allowing its detection26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' In k-space ISM, each illumination is identical to the plane wave illumina- tion of oblique-illumination microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The difference from oblique illumination microscopy is that only a sin- gle Fourier component is detected in each illumination in k-space ISM (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Similar to oblique illumina- tion microscopy, k-space ISM measurements thus yield extended k-space support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The differences between the techniques are the use of a single detector in k-space ISM rather than the detector-array of oblique-illumination mi- croscopy, which requires a larger number of illuminations, and the angular-spectrum reweighting that is required in oblique illumination microscopy19,27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' To support and demonstrate our mathematical deriva- tion, we numerically simulated the coherent imaging of a test sample with conventional ISM (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 2a), and k-space ISM (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' While the two approaches differ in both their illumination scheme (focused vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' plane-wave), and detection scheme (array detection in the object plane vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' a single detector in the k-space), the resulting images are identical and present an improvement over confocal imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The simulated conventional ISM setup (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 2a) con- sists of a 4-f imaging system, where the scanning laser and the detection array (CAM) are both imaged onto the object plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' A focused illumination beam raster scans the target object plane (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' At each illumination posi- tion, (xin, yin), the reflected field across the image plane, (xout, yout), are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The simulated k-space ISM setup (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 2b) is based on Fourier-transforming a point illumination (LASER) and confocal detection, from the detector plane (DET) to the object plane using a lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The point illumination is thus converted to a tilted plane- wave in the object plane, and the detector measures a single Fourier component, kout = kin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Observing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 2b reveals that k-space ISM is, in fact, confocal coherent imaging performed in the k-space domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' See simula- tion parameters in Supplementary Material, Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 2222 G LASER 22 ASER BS Confocal k-scan ISM X-scan ISM 100μm4 In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 3 we compare the two ISM processing ap- proaches and confocal imaging using an experimental ultrasound echography dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The dataset represents measurements performed on an acoustic phantom with multi-plane-wave transmission28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Data were acquired us- ing a Verasonics P4–2v probe at a center frequency of f0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content='8MHz, having 64 elements with a total aperture size of D = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content='2mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The imaged target is composed of five pins (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 3a, red X’s) having a diameter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content='1mm at a depth of 60mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Data were post-processed with the proper phases to perform coherent compounding for ei- ther confocal, x-space ISM14 or k-space ISM (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 3a,b,c, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' These experimental results agree with the analytic and numerical investigations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' See a discussion of the data processing in Supplementary Ma- terial, Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Experimental comparison of confocal ultrasound imaging (a), conventional ISM (b), and k-space ISM (c), of five reflecting pins (marked by red X’s) in an acoustic phan- tom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The same experimental data, acquired by plane wave sonications, were processed offline to generate the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' (a- c) presents the reconstructed intensity (absolute value square of the reconstructed fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The two approaches for ISM re- sults in practically identical images, having the expected no- ticeable resolution improvement over the confocal image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' (d) A cross-section comparison of (a-c) along the dotted line in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' In conclusion, by leveraging the Fourier slice theorem, we show that it is possible to perform coherent ISM with a single detector located at the Fourier plane of the ob- ject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' This allows the same resolution improvement gained in ISM over confocal imaging without requiring a detec- tor array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' This may be interesting for utilizing ISM in new modalities where detector arrays are not accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' In addition, the k-space acquisition and simple (confocal- like) analysis reduce the memory requirements and may be important also for the reduction of computational bur- den.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' We note that a single detector approach will result in a lower SNR than the detector-array approach in most scenarios since only a fraction of the reflected field is cap- tured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' On the other hand, the plane-wave illumination reduces the power density concentrated on the sample, as compared to focused scanned illumination, and thus may allow higher illumination power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work has received funding from The Israel Sci- ence Foundation (Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 1361/18), European Re- search Council (ERC) Horizon 2020 research and innova- tion program (677909), and was supported by the Min- istry of Science and Technology in Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' AUTHOR CONTRIBUTIONS Tal Sommer and Gil Weinberg contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Tal Sommer: Conceptualization (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Data Curation (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Formal Analysis(equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Methodol- ogy (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Writing/Original Draft Preparation (lead);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Writing/Review & Editing (equal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Gil Wein- berg: Conceptualization (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Data Curation (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Formal Analysis(equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Methodology (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Writ- ing/Review & Editing (equal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Ori Katz: Supervision (lead);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Funding Acquisition (lead);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Writing/Review & Editing (equal) DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' SUPPLEMENTARY MATERIAL See supplementary material for the simulation param- eters and a discussion of the data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 1J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Mertz, Introduction to optical microscopy (Cambridge Univer- sity Press, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Sheppard, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Castello, G.' metadata={'source': 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compounding for very high frame rate ul- trasonography and transient elastography,” IEEE transactions on ultrasonics, ferroelectrics, and frequency control 56, 489–506 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 29J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Goodman, Introduction to Fourier optics (Roberts and Company Publishers, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 6 SUPPLEMENTARY MATERIAL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Simulation Parameters This section discusses the simulation parameters used to simulate the x-scanning and k-scanning systems (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 2a-b in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Both systems were simulated using angular spectrum as a field propagator29, and the lenses were simulated as a thin parabolic phase mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The illumination was set to a wavelength of 450nm, and the object was imaged onto the detection plane using lenses of 35mm focal length and 3mm in diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The transfer-function extent in the x-scanning system was controlled via an aperture in the Fourier-plane of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The aperture size is cho- sen such that the imaging system was of a numerical aperture (NA) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content='0045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' This extent was preserved in the k-scanning system using a finite-extent scan of the Fourier-plane of the object to the same effective NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Ultrasound beam-forming This section discusses the data processing used for image reconstruction from an experimental ultrasound echography dataset (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 3 in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The dataset represents measurements performed on an acoustic phantom (GAMMEX SONO403) with multi- plane-wave transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Data were acquired using a Verasonics P4–2v probe at a center frequency of f0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content='8MHz, having 64 elements with a total aperture size of D = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content='2mm, connected to a Verasonics Vantage 256 multi-channel system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The angular range of the steered plane-waves was determined by the probe’s geometry via28: θm = arcsin mλ0 D (6) Where λ0 is the probe’s center wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' These measurements can be represented with a ma- trix: RF(u, t, θ) where u is the spatial position of the transducer element on the probe, t is the measurement time, and θ is the steering angle of the transmitted plane- wave28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Therefore, RF(u, t, θ) is the field measured at a transducer in position u at time t when sonicating with a plane-wave tilted with an angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The temporal Fourier- transform of this matrix is: RF(u, ω, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Beamforming is done using the time-of-flight (TOF) for transmission (tx) and detection (rx) for each point in the imaged medium (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' The transmission time depends on the steering angle of transmission, and the detection time depends on the relative position of the transducer from the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Assuming a single dimension, for simplicity, this TOF is: t(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' u, θ) = ttx(x, θ) + trx(x, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Therefore, the reconstructed confocal image at position x is a sum- mation over all measurements at the respective TOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' Iconf(x) = �� RF(u, ttx(x, θ) + trx(x, u), θ)dudθ = �� RF(u, t, θ) δ (t − (ttx(x, θ) + trx(x, u))) dudθ (7) A reflection matrix (that is used for x-scan ISM image formation) can be constructed by using a transmission time for a position that is different from the position for the detection time20,23: R(xin, xout) = �� RF (u, ttx(xin, θ) + trx(xout, u), θ) dudθ = �� RF(u, t, θ) δ (t − (ttx(xin, θ) + trx(xout, u))) dudθ (8) This operation is termed Delay-And-Sum (DAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' In the temporal Fourier-domain, these beamformations can be performed using phases instead of time-delays: Iconf(x) = ��� RF(u, ω, θ)eiω(ttx(x,θ)+trx(x,u))dudθdω (9a) R(xout, xin) = ��� RF(u, ω, θ)eiω(ttx(xin,θ)+trx(xout,u))dudθdω (9b) The k-space reflection-matrix, ˜R(kin, kout), is the 2D spatial Fourier transform of R(xin, xout): R(kout, kin) = �� R(xout, xin)e−ikinxine−ikoutxoutdxindxout (10) And, as the RF measurement is not dependant of xin and xout, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' 10 can be written as: R(kout, kin)= ��� RF(u, ω, θ) × �� eiωttx(xin,θ)e−ikinxindxin � × �� eiωtrx(xout,θ)e−ikoutxoutdxout � dudθdω (11) This means that the k-space reflection-matrix can be constructed in the temporal Fourier-domain, using the 1D spatial Fourier-transforms of the phases used for the reflection-matrix construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' One should determine the field-of-view (FOV) lateral grid to be the same as the positions of the transduc- ers (u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' This results in the grids of the two 1D spatial 7 Fourier-transforms to coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQfKgkL/content/2301.04353v1.pdf'} +page_content=' One can leverage this method to reconstruct a k-scanned ISM image directly from the echography dataset using only the phases that correspond to the wave-vectors kin = kout.' metadata={'source': 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Lu +ID ,1 M.J. Barlow,2 D. Basler,3 P. Gutfreund,4 O. Holderer,5 A. Ioffe,5 S. Pasini,5 P. +Pistel,6 Z. Salhi,5 K. Zhernenkov,5 B.M. Goodson +ID ,3 W.M. Snow +ID ,1 and E. Babcock +ID 5 +1Indiana University/CEEM, 2401 Milo B. Sampson Lane, Bloomington, IN 47408, USA +2School of Medicine, University of Nottingham, Queens Medical Centre, Nottingham, UK +3School of Chemical and Biomolecular Sciences, Southern Illinois University, Carbondale, IL 62901, USA +4Instut Laue-Langevin, 71 Avenue des Martyrs, CS 20156, 38042 Grenoble Cedex 9, France +5Forschungszentrum J¨ulich GmbH, J¨ulich Centre for Neutron Science +(JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), 85747 Garching, Germany +6Forschungzentrum J¨ulich mbH, ZEA-1, 52425 J¨ulich Germany +(Dated: January 3, 2023) +We present the first measurements of polarized neutron birefringence in transmission through +nuclear-polarized 129Xe and 131Xe gas and determine the neutron incoherent scattering lengths +bi(129Xe) = 0.186 ± (0.021)stat. ± (0.004)syst. fm and bi(131Xe) = 2.09 ± (0.29)stat. ± (0.12)syst. fm +for the first time. These results determine the essential parameter needed for interpretation of spin- +dependent neutron-scattering studies on polarized xenon ensembles, with possible future applications +ranging from tests of time-reversal violation to mode-entangled neutron scattering experiments on +nuclear-polarized systems. +PACS numbers: 11.30.Er, 24.70.+s, 13.75.Cs +This work presents the first measurement of neutron +birefringence in polarized 129Xe and 131Xe nuclei and the +first measurement of the nuclear polarization-dependent +bound scattering length difference ∆b = b+ − b− for +nuclear spin I parallel or antiparallel to the neutron +spin s. +Knowing ∆b one can for the first time now +conduct and interpret spin-dependent neutron scatter- +ing from an ensemble of polarized xenon nuclei using +the well-established theory of Van Hove [1, 2] generalized +for neutron spin-dependent scattering from polarized nu- +clei [3]. Because nuclear-polarized xenon atom ensembles +can be created in conditions where the electron spins do +not dominate the magnetic properties (unlike the great +majority of magnetic systems in condensed matter), our +work enables qualitatively new types of polarized neu- +tron investigations. Highly-polarized ensembles of xenon +gas can be created by spin-exchange optical pumping +(SEOP) [4–12] in volumes high enough to create long- +lived polarized Xe liquids and solids by freezing [13–15] +for exploration of subtle properties of these “pure” spin +systems. The conclusions of this paper describe exam- +ples of possible future polarized neutron investigations +which make essential use of the special properties of po- +larized xenon in quantum entanglement [16, 17] and in +searches for new sources of time reversal violation. These +newly-enabled neutron scientific applications of polarized +129Xe and 131Xe can also complement their many existing +applications in biomedical imaging [4, 10, 12, 18–20] (in- +cluding new MRI/gamma-ray imaging modalities [21]), +NMR spectroscopy [18, 22], fluid dynamics [18, 19, 22], +gas/surface interactions [23–25], studies of Berry geo- +metric phases [26], and searches for CPT/Lorentz viola- +tion [27–32], electric dipole moments [33], and axion-like +particles [34, 35]. +Neutron scattering amplitudes are often expressed in +operator form as b = bc+bi⃗s·⃗I where ⃗s is the neutron spin, +bc = [(I + 1)b+ + Ib−]/(2I + 1) is the spin-independent +coherent scattering length, and the spin-dependent inco- +herent scattering length bi = I +√ +I + 1[b+ − b−]/(2I + 1) +is directly proportional to ∆b. For 129Xe or 3He with nu- +clear spin I = 1/2 the compound neutron-nucleus total +spin J = I ± s can form a triplet, (J = 1) and singlet +(J = 0) total spin state corresponding to the b1 ≡ b+ and +b0 ≡ b− channels, so ∆b = b1 − b0. For I = 3/2 131Xe, +mJ = 2, 1, 0, −1 are possible. ∆b measures the difference +of the bmJ=2 + bmJ=1 and bmJ=0 + bmJ=−1 scattering +amplitudes for spin order characterized by the nuclear +polarization Px = < Iz > /I with no tensor alignment. +A general analysis of neutron spin dynamics in media +with nuclear spin order [36] implies that, for the preci- +sion reached here and for the neutron energies far from +neutron-nucleus resonances used in this work, we can re- +late the spin rotation angle to the scattering length dif- +ference in the usual way. Texts on neutron optics [37] +discuss the statistical weight factors used to derive the +above relations. +We measured ∆b by observing the precession of the +neutron spin as neutrons pass through a polarized nu- +clear target, named “pseudomagnetic precession” [38] in +the literature. Although this phenomenon was initially +described [38, 39] in terms of a fictitious “pseudomagnetic +field” inside the sample, ∆b originates from neutron- +nucleus scattering. The optical theorem [37] relates the +spin dependence of the neutron optical potentials asso- +ciated with the scattering amplitudes b+ and b− to a +two-valued neutron index of refraction (n+,n−) depend- +ing on the relative orientation of the neutron spin and +arXiv:2301.00460v1 [nucl-ex] 1 Jan 2023 + +2 +the nuclear polarization: +n2 +± = 1 − 4π +k2 N(bcoh + b±), +∆n = (n+ − n−) ≈ −2π +k2 N(b+ − b−), +(1) +where N is the number of nuclei per unit volume, k is +the neutron wave number, and the approximation in the +second expression is valid in our case as the neutron in- +dex of refraction is ≃ 1. ∆n makes the medium optically +birefringent for neutrons so that the two helicity com- +ponents of the neutron spin state accumulate different +phases, kn±d, in the forward direction as neutrons prop- +agate a distance d through the target. Therefore neutron +spins orthogonal to the nuclear polarization direction of +the target precess around the nuclear polarization by an +angle φ∗ = k∆nd. +Measurements of neutron birefringence are well-suited +to the Ramsey method of separated oscillatory fields [40, +41]. Previous work [39, 42–45] determined ∆b for several +nuclei dynamically-polarized in the solid state. We used +the neutron spin-echo technique (NSE) [46] to measure +∆b in SEOP cells filled with 3He, 129Xe, or 131Xe (an +earlier measurement in 3He [47] also used this method). +The measurement sequence is similar to spin echo manip- +ulations in nuclear magnetic resonance [48], however the +precession and flipping fields are encountered in space +along the traveling neutron beam, as opposed to time- +dependent fields applied to spins at rest in the lab frame. +In contrast to the Ramsey sequence, NSE uses a π spin +flip at the field symmetry point (Fig. 1) to refocus the +spin precession of neutrons with different velocities so +they are rephased at the polarization analyzer. +Phase +shifts of the interference fringes from the sample are com- +pensated by DC magnetic fields from phase (compensa- +tion) coils which are scanned over several periods about +the compensation point to obtain the NSE signal. The +sensitivity of the measurement is therefore set by the ra- +tio of the field resolution in the compensation coils to the +total field integral of the instrument. Since for the J-NSE +instrument [49] used in this work the phase coil precision +can be nT/m compared to a total field integral on the +instrument of over 1 T/m, very high phase precision is +possible. +The spin-echo condition holds for any group of neu- +tron velocities at B1,echo = L2 +L1 B2 where the number of +forward precessions though field B1 over length L1 in the +first region and back precessions in the second region of +field B2 and length L2 are equal, i.e. φ1(B1,echo) = φ2. +The phase shift accumulated in either region is γmλ +2πℏ B1L1 +where m is the mass of a neutron with de Broglie wave- +length λ and gyromagnetic ratio γ. The additional phase +shift from ∆b modifies the spin echo condition by adding +an extra phase φ∗. The precession caused by the neutron +polarizer +analyzer +2D detector +p/2 +flipper +p/2 +flipper +p +flipper +SEOP +polarizer +B1 +B2 +L1 +L2 +Bphase +Bphase +BSEOP +cell +FIG. 1: A schematic drawing of an NSE instrument +following the description in the text. The first neutron +π/2 flipper sets the neutron spins to precess about the +total field defined by the direction of B1, BSEOP , or B2, +i.e. in the respective regions. The π flipper reverses +neutron precession and the second π/2 flipper and +analyser return the in-phase magnitude of the final +neutron polarization. +birefringence is +φ∗ = − +2I +2I + 1λPxNxdx∆bx += −2 +� +Ix +Ix + 1λPxNxdxbx +i . +(2) +Here I3 = I129 = 1/2 for 3He and 129Xe and I131 = 3/2 +for 131Xe, and Px and Nx are the polarization and num- +ber density of the respective polarized nuclei of atomic +weight x with corresponding scattering length difference +∆bx +i or incoherent scattering length bx +i . +The relevant +product PxNx is determined by NMR calibration mea- +surements using absolute P3N3 of the 3He cell from neu- +tron transmission as a standard. φ∗ is then measured by +the shift of the NSE signal upon reversal of the nuclear +polarization, with all static magnetic fields constant. The +NSE signal i(Bphase) is the transmitted intensity after +the neutron polarization analyzer as a function of the +current in the solenoid phase coil field Bphase. +Fig. 1 shows the self-compensated superconducting +(SC) coil sets for the two precession regions of the J- +NSE [50]. The polarized noble gas samples rest in the +sample region inside a B0 holding field normal to the +neutron beam. Three GE180 SEOP cells [51] produced +in FZ-J¨ulich were used in this experiment: a 5 cm inner +diameter, 4.9 cm long 3He cell, and two Xe cells, both 5 +cm inner diameter and 12.7 cm long. All cells contained +several mg of Rb, the 3He cell had 0.3556 bar of pure 3He +gas with 0.1 bar of N2, the 129Xe cell was filled to 0.3 bar +of 91.18% enrichment 129Xe gas with 0.25 bar of N2, and +the 131Xe cell was filled with 0.20 bar 84.4% enriched +131Xe (Berry and Associates / Icon Isotopes) and 0.2 bar +N2. The 3He and 129Xe cells were prepared in Garch- +ing (FZ-J¨ulich) [52] and the 131Xe cell was prepared and +characterized at Southern Illinois University [53, 54]. +Two frequency-narrowed diode array bars [55] realized + +3 +1500 +1000 +500 +amplitude +25.8 x10 +3 +25.7 +25.6 +25.5 +25.4 +25.3 +25.2 +frequency (Hz) +1600 +1200 +800 +400 +0 +FFT area +15 +10 +5 +0 +FID aquission number +FIG. 2: Example NMR spectra from +Fourier-transformed single-shot FID signals recorded +during the calibration of the 131Xe cell. 131Xe and +129Xe spectra are black and blue, respectively. The +inset shows the NMR FID strengths versus acquisition +number used for the averaging of the 131Xe signal. +in-situ SEOP. A 70 cm diameter Helmholtz coil pair pro- +duced the magnetic field BSEOP normal to the neutron +path. Cell heating and temperature regulation was pro- +vided by AC electric cartridge heaters for the 3He cell +and by flowing air for the two xenon cells. +Nuclear magnetic resonance free-induction decay mea- +surements of the cell magnetizations directly propor- +tional to PxNx used a home-built pulse-receive NMR sys- +tem [55] with a single 2 cm diameter, 300-turn pickup +coil placed on the cell’s surface normal to both the opti- +cal pumping axis and the neutron beam. Since the three +isotopes studied here vary in gyromagnetic ratios by an +order of magnitude (γ3 = −3.243 kHz/g, γ129 = −1.178 +kHz/g, and γ131 = 0.349 kHz/g for 3He, 129Xe, and 131Xe +respectively), the NMR FID calibrations presented an ex- +perimental challenge. +The +129Xe cell magnetization was measured using +NMR which was calibrated to an absolute 3He stan- +dard from neutron transmission during the NSE mea- +surements. The ratio R = Tp/T0 of the neutron trans- +mission through the polarized 3He cell (Tp) to unpolar- +ized (T0) determines cosh−1(R) = +σp +λth λP3N3d3 where +σp = (1−σ1/σun)σun is the polarized 3He spin dependent +neutron absorption cross section, σun the total unpolar- +ized neutron absorption cross section, λth = 1.798 ˚A is +the standard thermal neutron wavelength, λ = 8±0.08 ˚A +was the neutron wavelength of the measurement, P3N3 +is the product of 3He polarization and density, and d3 is +the cell length. We use σp ≃ σun = 5333 ± 7 barn as +done for other 3He neutron spin filter cells [56]. σp is +smaller than σun barn by σ1, which has been estimated +to be 24 barn [57, 58], leading to a small systematic in +P3N3 on the order of -0.45%. Our measurements found +R = 1.2506 ± 0.0030 giving cosh−1R = 0.6939 ± 0.0040, +which gives P3N3 = (0.609 ± 0.145) × 1024 m−3 or +P3N3 = 0.251 ± 0.006 bar. +Separate neutron trans- +mission measurements of this cell using neutron time- +of-flight [59, 60] determined N3 = 0.3556 ± 0.0011 bar +at 298.5 K in the cell center where d3 = 4.8 ± 0.1 cm +and characterized the shape of its rounded ends. d3 is +inferred from measurements of the cell’s external length +and assumptions of the glass thickness resulting in the +given error. This density implies P3 = 70.6±1.6%. Only +the product P3N3 is needed for absolute calibration of +our NMR system so the error in d3 is not propagated +to the bi results, but the Xe cell lengths are needed to +solve for ∆b of 129Xe and 131Xe. The observed 3He NMR +signal was stable to better than 0.3%. +The neutron wavelength distribution transmitted by +the velocity selector is fit by a unity-normalized triangu- +lar function convoluted with a cosine function. The re- +sulting form is i = i0−A0·tri(Iphase)·cos (ω(Iphase − I0)), +where the triangular function tri has a width fixed by the +velocity selector, i is the neutron signal intensity, i0 is a +constant intensity offset, A0 is the maximum amplitude +of the spin echo oscillation (which occurs at the echo con- +dition Iphase = I0), and ω is the angular frequency of the +oscillation. All pixels of the 2D position sensitive neu- +tron detector are analyzed individually and the results +averaged. +The data were taken in defined time-ordered sequences +of alternating up and down target polarization for all +three nuclei. Since the 3He pseudomagnetic precession +angle has been measured previously [47] and was known +to be large compared to our expected effects, we used +this data to check the experimental procedure and appa- +ratus. The up/down polarization states for the polarized +xenon targets were switched by reversing the pump-laser +polarization by turning the quarter-wave plates without +any other changes. The nuclear polarization is reversed +by SEOP on a timescale near the T1 relaxation time of +129Xe, about 5 min. for our cell, so one 20 min NSE scan +was skipped after each wave plate change. +For 131Xe, +T1 ≃ 30 s [53, 61] is much shorter than the scan time and +the polarization buildup time is negligible. +We analyzed the individual NSE scans in single detec- +tor pixels for each run. The spin echo stationary phase +point varies slightly across the neutron beam due to the +small spatial variations in the field integral, and the spin +echo phase drifts slowly over timescales long compared to +the target spin flip due to very small changes in the total +field over time. To improve the signal/noise ratio for fit- +ting the pixel spin echo scans, we applied a Fast Fourier +Transform (FFT) frequency filter to the spin echo data. +A frequency bandpass around the main frequency peak +having a wide frequency range compared to the distribu- +tion of frequencies associated with the fractional neutron +wavelength distribution of 10% was used. To fit the FFT- +filtered NSE signals, we first fix the wavelength distribu- + +4 +tion width and compute the average spin echo frequency +ω, which is related to the offset in Iphase for each mea- +surement, and was ω = 35.92, 35.28, and 35.42 Hz for +129Xe, 131Xe, and 3He respectively. Then we allow the +remaining three parameters, N0, A0, and φ0 to vary. +The precession angle is extracted from the relative +phase shift between oppositely-polarized nuclear target +states with the NSE scans performed using alternating +groups of polarizations. The absolute phase φ was well +represented by a square wave on top of a slowly varying +linear instrumental phase drift (Fig. 3). This resulted in +precession angles averaged over the active detector pixels +of 4.05◦ ±0.43◦ for the 129Xe target and 3.05◦ ±0.36◦ for +the 131Xe target. +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +Time (hours) +4 +2 +0 +2 + (Degrees) +: 4.05±0.43 +(a) 129Xe +Model +Data +3 +2 +1 +0 +1 +2 +3 += +fit (Degrees) +0 +1 +2 +3 +4 +5 +6 +Count +(c) 129Xe += 1.09 + +0 = -0.000336 +0 +10 +20 +30 +40 +Time (hours) +4 +3 +2 +1 +0 +1 +2 +3 +4 + (Degrees) +: 3.05±0.36 +(b) 131Xe +Model +Data +3 +2 +1 +0 +1 +2 +3 += +fit (Degrees) +0 +2 +4 +6 +8 +10 +Count +(d) 131Xe += 1.27 + +0 = 0.00232 +FIG. 3: (a,b) Plots of relative NSE phases of 129Xe and +131Xe versus experiment time. The data were evenly +distributed into groups of alternating Px. The fist point +after switching of P129 is excluded from analysis due to +the finite polarization build up time constant, whereas +all scans for 131Xe can be included due to its fast +polarization buildup. (c,d) Bar graphs show the +distribution of phases about the mean value from the fit. +We then use Eq. 2 to compute the incoherent scat- +tering lengths bi. Since the NMR calibration of the Xe +measures magnetization proportional to PxNx, any error +in Nx for Xe drops out for the determination of bi. Using +the NMR calibrations, we determine PxNx for the two +xenon isotopes as follows. For I = 1/2 129Xe and 3He +and for NMR FID performed in the very low tip angle +limit (i.e. << 90◦), +P129N129 = P3N3 +� γ3 +γ129 +�2 S129V3 +S3V129 +, +(3) +where the ratio of gyromagnetic ratios is squared to scale +for both the coil pickup and tipping pulse at fixed tip +parameters, Sx is the NMR strength of the respective +noble gas isotope and V129/V3 = 4.085 was the increase +in tip amplitude for 129Xe to obtain a good signal/noise +ratio (S/N). Using this relation P129N129 = 1.62±0.04× +isotope +129Xe +131Xe +δbi +0.112 +0.150 +δbi stat. +0.110 +0.139 +stat. error source +δφ +0.106 +0.118 +δ/δR(cosh−1(R)) +0.0057 +0.0057 +δ(S3/S129) +0.028 +0.028 +δ(S′ +129/S′ +131) +0.067 +δbi syst. +0.023 +0.055 +syst. error source +σ1 +-0.0045 -0.0045 +δd3 +0.021 +0.021 +δd129 or δd131 +0.0079 +0.0079 +repeatability Xe pol. +0.05 +TABLE I: The relative error contributions divided +between systematic and statistical sources. The +statistical error of the φ measurement dominates. The +estimate of σ1 = 24 barn from [57, 58] was used; the - +denotes a one-sided systematic that would lower the +reported bi values. +1024 m−3. The 0.3 bar total Xe pressure measured during +cell filling implies P129 = 17.6%. +The 131Xe NMR calibration could not be performed +during the neutron experiment with our standard pickup +coil as the lower 131Xe polarization and the small 131Xe +gyromagnetic ratio lead to very weak signals; also NSE +signals could not be obtained at the high B0 field re- +quired to obtain the cross-calibration NMR frequency of +25.6 kHz chosen to reach high enough signal to noise +ratio. Therefore 131Xe polarization was calibrated in a +separate measurement after the NSE experiment, leaving +the SEOP apparatus and conditions unchanged. Using +an NMR coil with a 6-fold higher quality factor, the NMR +calibration was performed with a maximum π/2 tip angle +for both Xe isotopes, so one factor of the ratio of gyro- +magnetic ratios drops out of the calibration calculation. +Additionally one needs to account for the ratio of the +different nuclear spins. The relation for the calibration +131Xe to 129Xe becomes: +P131N131 = P129N129 +�γ129 +γ131 +� �I129 +I131 +� S′ +131 +S′ +129 +, +(4) +where S′ +x denote the signals obtained for the π/2 tip +angles used for this step. +The result is P131N131 = +0.0498P129N129 = 8.1 ± 0.2 × 1022 m−3. Given the 0.20 +bar total Xe pressure, P131 = 1.96%. Neither the spe- +cific number densities nor the isotopic concentrations of +the xenon isotopes are needed for the neutron scattering +length determination with our method using calibrated +NMR. +We also briefly measured b3 +i to compare with previ- +ous results. This measurement is calibrated absolutely +from the polarization dependent 3He neutron absorption +cross section [60]. +Our value of 2.280 ± 0.020(stat.) + + +5 +0.015(syst.) fm for b3 +i agrees with previous work [47, 57] +and is determined with much higher precision than our +bi values for 131Xe and 129Xe. Since P3N3 is determined +by direct measurement of the neutron transmission ratio +through a polarized/unpolarized cell for this particular +beam, all of the bi values are independent of the detailed +shape or mean value of the neutron wavelength distribu- +tion. +Combining Eqs. (2), (3) and (4), we can write the mag- +nitudes of incoherent scattering lengths for the xenon iso- +topes in terms of directly-measured experimental quan- +tities as: +b129 +i += +√ +3 +2 +�γ129 +γ3 +�2 +φ129 +cosh−1(R) +d3 +d129 +S3V129 +S129V3 +σp +λth +(5) +and +b131 +i += 3 +2 +� +5 +3 +|γ129γ131| +γ2 +3 +φ131 +cosh−1 (R) +d3 +d131 +S3V129 +S129V3 +S′ +129 +S′ +131 +σp +λth +(6) +where cosh−1 (R) = 0.6939 ± 0.0040 at the center of the +3He cell. The last term in the products account for the +3He triplet absorption cross section σ1 (i.e. total spin +of the neutron+3He of 1) compared to the total neutron +wavelength dependent unpolarized absorption cross sec- +tion (this factor is equivalent to Eq. 25 in [47]). We used +a previous estimate for the 3He triplet cross section of +σ1 = 24 barn [57, 58]. Other possible corrections due to +cell geometry or neutron wavelength distribution [47] are +negligible for this work. A soon-to-be-submitted work on +bi for 3He [62] wll discuss them. +The values of the incoherent scattering lengths are thus +b129 +i += −0.186 ± (0.021)stat. ± (0.004)syst.fm +and +b131 +i += 2.09 ± (0.29)stat. ± (0.12)syst.fm. +Signs of the scattering lengths are determined from the +spin directions in the SEOP setup. The statistical errors +10% and 12% for 129Xe and 131Xe, respectively, come +from the scatter of the phase shift fits shown in Fig. 3. +These values for bi are in line with those of other nu- +clei. We are not aware of any simple argument that can +explain why |b131 +i +is a factor of 11 larger than |b129 +i +. +With bi(129Xe) measured in this work, we could probe +the degree of entanglement of polarized 129Xe spins gen- +erated in atomic collisions in SEOP systems using po- +larized, mode-entangled neutron beams to measure spin- +spin correlation functions as entanglement witnesses for +the xenon spin states. +A recently-developed quantita- +tive theory for the scattering of mode-entangled neutron +beams from spin-correlated dimers [63] can be extended +to polarized xenon gas, which can be accurately mod- +eled as an ideal gas with an analytical expression for the +neutron dynamic structure factor. +SEOP collisions of +I = 1/2 3He and 129Xe atoms with properly-prepared +polarized alkali atoms can generate a calculable degree +of entanglement in the nuclear spins under certain con- +ditions according to recent work [16, 17]. +The result- +ing long-lived entanglement in the nuclear spin system is +of interest for optical quantum memories [64–66]. The +quantum decoherence of mode-entangled neutron beams +passing through dense matter is so small that the mea- +surement of neutron entanglement witnesses for Bell and +GHZ inequalities are unaffected [67–69]. The transverse +spatial separation between the two opposite-spin sub- +beams created in devices like neutron Wollaston prisms +coincides with the range of mean free paths of the polar- +ized 129Xe gas atoms accessible in SEOP cells. +Polarized 131Xe nuclei could be used in a search for +new sources of time reversal (T) violation in neutron- +nucleus interactions. T violation from some new interac- +tion beyond the Standard Model of particles and inter- +actions is one of the highest intellectual priorities in nu- +clear/particle/astrophysics, and could shed light on the +matter-antimatter asymmetry in the universe according +to the Sakharov argument [70]. The forward scattering +amplitude of polarized neutrons in a polarized nuclear +target can possess a parity (P)-odd and T-odd term of +the form ⃗sn · ( ⃗kn × ⃗I) where ⃗sn is the neutron spin, ⃗kn +is the neutron momentum, and ⃗I is the nuclear polar- +ization. Compound neutron-nucleus resonance reactions +are known to greatly amplify parity violation in neutron- +nucleus interactions [71, 72]. +A 4% P-odd asymmetry +was measured in the 3.2 eV p-wave resonance in 131Xe, +an amplification compared to nucleon-nucleon P-odd am- +plitudes of almost 106. +The theory which successfully +predicted this phenomenon long ago [73, 74] implies that +P-odd and T-odd interactions between nucleons beyond +the Standard Model should also be amplified by a similar +factor [75–77]. Neutron transmission measurements in- +volving such coherent neutron-nucleus interactions could +provide null tests for time-reversal invariance that are +free from contamination by final state interactions [78]. +Advances in neutron polarization technology and source +brightness added to progress in SEOP polarization of +131Xe suffice to conduct a sensitive search for axion-like +particle (ALP) exchange [77], which is poorly constrained +by EDM searches for ALP masses above 10 meV [79], +because here the Standard Model axion relation between +axion mass and coupling constant does not apply [80]. +H. Lu and W. M. Snow acknowledge support from US +National Science Foundation (NSF) grants PHY-1913789 +and PHY-2209481 and the Indiana University Center for + +6 +Spacetime Symmetries. +H. Lu received a Short-Term +Grant, 2019 no. 57442045 from DAAD the German Aca- +demic Exchange Service. D. Basler and B.M. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Germany 6Forschungzentrum J¨ulich mbH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' ZEA-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 52425 J¨ulich Germany (Dated: January 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 2023) We present the first measurements of polarized neutron birefringence in transmission through nuclear-polarized 129Xe and 131Xe gas and determine the neutron incoherent scattering lengths bi(129Xe) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='186 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='021)stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='004)syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' fm and bi(131Xe) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='09 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='29)stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='12)syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' fm for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' These results determine the essential parameter needed for interpretation of spin- dependent neutron-scattering studies on polarized xenon ensembles, with possible future applications ranging from tests of time-reversal violation to mode-entangled neutron scattering experiments on nuclear-polarized systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' PACS numbers: 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='Er, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='+s, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='Cs This work presents the first measurement of neutron birefringence in polarized 129Xe and 131Xe nuclei and the first measurement of the nuclear polarization-dependent bound scattering length difference ∆b = b+ − b− for nuclear spin I parallel or antiparallel to the neutron spin s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Knowing ∆b one can for the first time now conduct and interpret spin-dependent neutron scatter- ing from an ensemble of polarized xenon nuclei using the well-established theory of Van Hove [1, 2] generalized for neutron spin-dependent scattering from polarized nu- clei [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Because nuclear-polarized xenon atom ensembles can be created in conditions where the electron spins do not dominate the magnetic properties (unlike the great majority of magnetic systems in condensed matter), our work enables qualitatively new types of polarized neu- tron investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Highly-polarized ensembles of xenon gas can be created by spin-exchange optical pumping (SEOP) [4–12] in volumes high enough to create long- lived polarized Xe liquids and solids by freezing [13–15] for exploration of subtle properties of these “pure” spin systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The conclusions of this paper describe exam- ples of possible future polarized neutron investigations which make essential use of the special properties of po- larized xenon in quantum entanglement [16, 17] and in searches for new sources of time reversal violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' These newly-enabled neutron scientific applications of polarized 129Xe and 131Xe can also complement their many existing applications in biomedical imaging [4, 10, 12, 18–20] (in- cluding new MRI/gamma-ray imaging modalities [21]), NMR spectroscopy [18, 22], fluid dynamics [18, 19, 22], gas/surface interactions [23–25], studies of Berry geo- metric phases [26], and searches for CPT/Lorentz viola- tion [27–32], electric dipole moments [33], and axion-like particles [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Neutron scattering amplitudes are often expressed in operator form as b = bc+bi⃗s·⃗I where ⃗s is the neutron spin, bc = [(I + 1)b+ + Ib−]/(2I + 1) is the spin-independent coherent scattering length, and the spin-dependent inco- herent scattering length bi = I √ I + 1[b+ − b−]/(2I + 1) is directly proportional to ∆b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' For 129Xe or 3He with nu- clear spin I = 1/2 the compound neutron-nucleus total spin J = I ± s can form a triplet, (J = 1) and singlet (J = 0) total spin state corresponding to the b1 ≡ b+ and b0 ≡ b− channels, so ∆b = b1 − b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' For I = 3/2 131Xe, mJ = 2, 1, 0, −1 are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' ∆b measures the difference of the bmJ=2 + bmJ=1 and bmJ=0 + bmJ=−1 scattering amplitudes for spin order characterized by the nuclear polarization Px = < Iz > /I with no tensor alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' A general analysis of neutron spin dynamics in media with nuclear spin order [36] implies that, for the preci- sion reached here and for the neutron energies far from neutron-nucleus resonances used in this work, we can re- late the spin rotation angle to the scattering length dif- ference in the usual way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Texts on neutron optics [37] discuss the statistical weight factors used to derive the above relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' We measured ∆b by observing the precession of the neutron spin as neutrons pass through a polarized nu- clear target, named “pseudomagnetic precession” [38] in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Although this phenomenon was initially described [38, 39] in terms of a fictitious “pseudomagnetic field” inside the sample, ∆b originates from neutron- nucleus scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The optical theorem [37] relates the spin dependence of the neutron optical potentials asso- ciated with the scattering amplitudes b+ and b− to a two-valued neutron index of refraction (n+,n−) depend- ing on the relative orientation of the neutron spin and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='00460v1 [nucl-ex] 1 Jan 2023 2 the nuclear polarization: n2 ± = 1 − 4π k2 N(bcoh + b±), ∆n = (n+ − n−) ≈ −2π k2 N(b+ − b−), (1) where N is the number of nuclei per unit volume, k is the neutron wave number, and the approximation in the second expression is valid in our case as the neutron in- dex of refraction is ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' ∆n makes the medium optically birefringent for neutrons so that the two helicity com- ponents of the neutron spin state accumulate different phases, kn±d, in the forward direction as neutrons prop- agate a distance d through the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Therefore neutron spins orthogonal to the nuclear polarization direction of the target precess around the nuclear polarization by an angle φ∗ = k∆nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Measurements of neutron birefringence are well-suited to the Ramsey method of separated oscillatory fields [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Previous work [39, 42–45] determined ∆b for several nuclei dynamically-polarized in the solid state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' We used the neutron spin-echo technique (NSE) [46] to measure ∆b in SEOP cells filled with 3He, 129Xe, or 131Xe (an earlier measurement in 3He [47] also used this method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The measurement sequence is similar to spin echo manip- ulations in nuclear magnetic resonance [48], however the precession and flipping fields are encountered in space along the traveling neutron beam, as opposed to time- dependent fields applied to spins at rest in the lab frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' In contrast to the Ramsey sequence, NSE uses a π spin flip at the field symmetry point (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 1) to refocus the spin precession of neutrons with different velocities so they are rephased at the polarization analyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Phase shifts of the interference fringes from the sample are com- pensated by DC magnetic fields from phase (compensa- tion) coils which are scanned over several periods about the compensation point to obtain the NSE signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The sensitivity of the measurement is therefore set by the ra- tio of the field resolution in the compensation coils to the total field integral of the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Since for the J-NSE instrument [49] used in this work the phase coil precision can be nT/m compared to a total field integral on the instrument of over 1 T/m, very high phase precision is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The spin-echo condition holds for any group of neu- tron velocities at B1,echo = L2 L1 B2 where the number of forward precessions though field B1 over length L1 in the first region and back precessions in the second region of field B2 and length L2 are equal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' φ1(B1,echo) = φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The phase shift accumulated in either region is γmλ 2πℏ B1L1 where m is the mass of a neutron with de Broglie wave- length λ and gyromagnetic ratio γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The additional phase shift from ∆b modifies the spin echo condition by adding an extra phase φ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The precession caused by the neutron polarizer analyzer 2D detector p/2 flipper p/2 flipper p flipper SEOP polarizer B1 B2 L1 L2 Bphase Bphase BSEOP cell FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 1: A schematic drawing of an NSE instrument following the description in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The first neutron π/2 flipper sets the neutron spins to precess about the total field defined by the direction of B1, BSEOP , or B2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' in the respective regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The π flipper reverses neutron precession and the second π/2 flipper and analyser return the in-phase magnitude of the final neutron polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' birefringence is φ∗ = − 2I 2I + 1λPxNxdx∆bx = −2 � Ix Ix + 1λPxNxdxbx i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' (2) Here I3 = I129 = 1/2 for 3He and 129Xe and I131 = 3/2 for 131Xe, and Px and Nx are the polarization and num- ber density of the respective polarized nuclei of atomic weight x with corresponding scattering length difference ∆bx i or incoherent scattering length bx i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The relevant product PxNx is determined by NMR calibration mea- surements using absolute P3N3 of the 3He cell from neu- tron transmission as a standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' φ∗ is then measured by the shift of the NSE signal upon reversal of the nuclear polarization, with all static magnetic fields constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The NSE signal i(Bphase) is the transmitted intensity after the neutron polarization analyzer as a function of the current in the solenoid phase coil field Bphase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 1 shows the self-compensated superconducting (SC) coil sets for the two precession regions of the J- NSE [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The polarized noble gas samples rest in the sample region inside a B0 holding field normal to the neutron beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Three GE180 SEOP cells [51] produced in FZ-J¨ulich were used in this experiment: a 5 cm inner diameter, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='9 cm long 3He cell, and two Xe cells, both 5 cm inner diameter and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='7 cm long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' All cells contained several mg of Rb, the 3He cell had 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='3556 bar of pure 3He gas with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='1 bar of N2, the 129Xe cell was filled to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='3 bar of 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='18% enrichment 129Xe gas with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='25 bar of N2, and the 131Xe cell was filled with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='20 bar 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='4% enriched 131Xe (Berry and Associates / Icon Isotopes) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='2 bar N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The 3He and 129Xe cells were prepared in Garch- ing (FZ-J¨ulich) [52] and the 131Xe cell was prepared and characterized at Southern Illinois University [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Two frequency-narrowed diode array bars [55] realized 3 1500 1000 500 amplitude 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='8 x10 3 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='7 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='6 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='4 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='3 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='2 frequency (Hz) 1600 1200 800 400 0 FFT area 15 10 5 0 FID aquission number FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 2: Example NMR spectra from Fourier-transformed single-shot FID signals recorded during the calibration of the 131Xe cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 131Xe and 129Xe spectra are black and blue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The inset shows the NMR FID strengths versus acquisition number used for the averaging of the 131Xe signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' in-situ SEOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' A 70 cm diameter Helmholtz coil pair pro- duced the magnetic field BSEOP normal to the neutron path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Cell heating and temperature regulation was pro- vided by AC electric cartridge heaters for the 3He cell and by flowing air for the two xenon cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Nuclear magnetic resonance free-induction decay mea- surements of the cell magnetizations directly propor- tional to PxNx used a home-built pulse-receive NMR sys- tem [55] with a single 2 cm diameter, 300-turn pickup coil placed on the cell’s surface normal to both the opti- cal pumping axis and the neutron beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Since the three isotopes studied here vary in gyromagnetic ratios by an order of magnitude (γ3 = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='243 kHz/g, γ129 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='178 kHz/g, and γ131 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='349 kHz/g for 3He, 129Xe, and 131Xe respectively), the NMR FID calibrations presented an ex- perimental challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The 129Xe cell magnetization was measured using NMR which was calibrated to an absolute 3He stan- dard from neutron transmission during the NSE mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The ratio R = Tp/T0 of the neutron trans- mission through the polarized 3He cell (Tp) to unpolar- ized (T0) determines cosh−1(R) = σp λth λP3N3d3 where σp = (1−σ1/σun)σun is the polarized 3He spin dependent neutron absorption cross section, σun the total unpolar- ized neutron absorption cross section, λth = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='798 ˚A is the standard thermal neutron wavelength, λ = 8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='08 ˚A was the neutron wavelength of the measurement, P3N3 is the product of 3He polarization and density, and d3 is the cell length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' We use σp ≃ σun = 5333 ± 7 barn as done for other 3He neutron spin filter cells [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' σp is smaller than σun barn by σ1, which has been estimated to be 24 barn [57, 58], leading to a small systematic in P3N3 on the order of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='45%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Our measurements found R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='2506 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='0030 giving cosh−1R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='6939 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='0040, which gives P3N3 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='609 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='145) × 1024 m−3 or P3N3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='251 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='006 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Separate neutron trans- mission measurements of this cell using neutron time- of-flight [59, 60] determined N3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='3556 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='0011 bar at 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='5 K in the cell center where d3 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='1 cm and characterized the shape of its rounded ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' d3 is inferred from measurements of the cell’s external length and assumptions of the glass thickness resulting in the given error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' This density implies P3 = 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Only the product P3N3 is needed for absolute calibration of our NMR system so the error in d3 is not propagated to the bi results, but the Xe cell lengths are needed to solve for ∆b of 129Xe and 131Xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The observed 3He NMR signal was stable to better than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The neutron wavelength distribution transmitted by the velocity selector is fit by a unity-normalized triangu- lar function convoluted with a cosine function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The re- sulting form is i = i0−A0·tri(Iphase)·cos (ω(Iphase − I0)), where the triangular function tri has a width fixed by the velocity selector, i is the neutron signal intensity, i0 is a constant intensity offset, A0 is the maximum amplitude of the spin echo oscillation (which occurs at the echo con- dition Iphase = I0), and ω is the angular frequency of the oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' All pixels of the 2D position sensitive neu- tron detector are analyzed individually and the results averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The data were taken in defined time-ordered sequences of alternating up and down target polarization for all three nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Since the 3He pseudomagnetic precession angle has been measured previously [47] and was known to be large compared to our expected effects, we used this data to check the experimental procedure and appa- ratus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The up/down polarization states for the polarized xenon targets were switched by reversing the pump-laser polarization by turning the quarter-wave plates without any other changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The nuclear polarization is reversed by SEOP on a timescale near the T1 relaxation time of 129Xe, about 5 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' for our cell, so one 20 min NSE scan was skipped after each wave plate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' For 131Xe, T1 ≃ 30 s [53, 61] is much shorter than the scan time and the polarization buildup time is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' We analyzed the individual NSE scans in single detec- tor pixels for each run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The spin echo stationary phase point varies slightly across the neutron beam due to the small spatial variations in the field integral, and the spin echo phase drifts slowly over timescales long compared to the target spin flip due to very small changes in the total field over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' To improve the signal/noise ratio for fit- ting the pixel spin echo scans, we applied a Fast Fourier Transform (FFT) frequency filter to the spin echo data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' A frequency bandpass around the main frequency peak having a wide frequency range compared to the distribu- tion of frequencies associated with the fractional neutron wavelength distribution of 10% was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' To fit the FFT- filtered NSE signals, we first fix the wavelength distribu- 4 tion width and compute the average spin echo frequency ω, which is related to the offset in Iphase for each mea- surement, and was ω = 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='92, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='28, and 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='42 Hz for 129Xe, 131Xe, and 3He respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Then we allow the remaining three parameters, N0, A0, and φ0 to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The precession angle is extracted from the relative phase shift between oppositely-polarized nuclear target states with the NSE scans performed using alternating groups of polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The absolute phase φ was well represented by a square wave on top of a slowly varying linear instrumental phase drift (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' This resulted in precession angles averaged over the active detector pixels of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='05◦ ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='43◦ for the 129Xe target and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='05◦ ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='36◦ for the 131Xe target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='5 Time (hours) 4 2 0 2 (Degrees) : 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='43 (a) 129Xe Model Data 3 2 1 0 1 2 3 = fit (Degrees) 0 1 2 3 4 5 6 Count (c) 129Xe = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='09 0 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='000336 0 10 20 30 40 Time (hours) 4 3 2 1 0 1 2 3 4 (Degrees) : 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='36 (b) 131Xe Model Data 3 2 1 0 1 2 3 = fit (Degrees) 0 2 4 6 8 10 Count (d) 131Xe = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='27 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='00232 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 3: (a,b) Plots of relative NSE phases of 129Xe and 131Xe versus experiment time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The data were evenly distributed into groups of alternating Px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The fist point after switching of P129 is excluded from analysis due to the finite polarization build up time constant, whereas all scans for 131Xe can be included due to its fast polarization buildup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' (c,d) Bar graphs show the distribution of phases about the mean value from the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' We then use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 2 to compute the incoherent scat- tering lengths bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Since the NMR calibration of the Xe measures magnetization proportional to PxNx, any error in Nx for Xe drops out for the determination of bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Using the NMR calibrations, we determine PxNx for the two xenon isotopes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' For I = 1/2 129Xe and 3He and for NMR FID performed in the very low tip angle limit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' << 90◦), P129N129 = P3N3 � γ3 γ129 �2 S129V3 S3V129 , (3) where the ratio of gyromagnetic ratios is squared to scale for both the coil pickup and tipping pulse at fixed tip parameters, Sx is the NMR strength of the respective noble gas isotope and V129/V3 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='085 was the increase in tip amplitude for 129Xe to obtain a good signal/noise ratio (S/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Using this relation P129N129 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='62±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='04× isotope 129Xe 131Xe δbi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='150 δbi stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='139 stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' error source δφ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='118 δ/δR(cosh−1(R)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='0057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='0057 δ(S3/S129) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='028 δ(S′ 129/S′ 131) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='067 δbi syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='055 syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' error source σ1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='0045 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='0045 δd3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='021 δd129 or δd131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='0079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='0079 repeatability Xe pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='05 TABLE I: The relative error contributions divided between systematic and statistical sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The statistical error of the φ measurement dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The estimate of σ1 = 24 barn from [57, 58] was used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' the - denotes a one-sided systematic that would lower the reported bi values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 1024 m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='3 bar total Xe pressure measured during cell filling implies P129 = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The 131Xe NMR calibration could not be performed during the neutron experiment with our standard pickup coil as the lower 131Xe polarization and the small 131Xe gyromagnetic ratio lead to very weak signals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' also NSE signals could not be obtained at the high B0 field re- quired to obtain the cross-calibration NMR frequency of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='6 kHz chosen to reach high enough signal to noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Therefore 131Xe polarization was calibrated in a separate measurement after the NSE experiment, leaving the SEOP apparatus and conditions unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Using an NMR coil with a 6-fold higher quality factor, the NMR calibration was performed with a maximum π/2 tip angle for both Xe isotopes, so one factor of the ratio of gyro- magnetic ratios drops out of the calibration calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Additionally one needs to account for the ratio of the different nuclear spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The relation for the calibration 131Xe to 129Xe becomes: P131N131 = P129N129 �γ129 γ131 � �I129 I131 � S′ 131 S′ 129 , (4) where S′ x denote the signals obtained for the π/2 tip angles used for this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The result is P131N131 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='0498P129N129 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='2 × 1022 m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Given the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='20 bar total Xe pressure, P131 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='96%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Neither the spe- cific number densities nor the isotopic concentrations of the xenon isotopes are needed for the neutron scattering length determination with our method using calibrated NMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' We also briefly measured b3 i to compare with previ- ous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' This measurement is calibrated absolutely from the polarization dependent 3He neutron absorption cross section [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Our value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='280 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='020(stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=') + 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='015(syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=') fm for b3 i agrees with previous work [47, 57] and is determined with much higher precision than our bi values for 131Xe and 129Xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Since P3N3 is determined by direct measurement of the neutron transmission ratio through a polarized/unpolarized cell for this particular beam, all of the bi values are independent of the detailed shape or mean value of the neutron wavelength distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' (2), (3) and (4), we can write the mag- nitudes of incoherent scattering lengths for the xenon iso- topes in terms of directly-measured experimental quan- tities as: b129 i = √ 3 2 �γ129 γ3 �2 φ129 cosh−1(R) d3 d129 S3V129 S129V3 σp λth (5) and b131 i = 3 2 � 5 3 |γ129γ131| γ2 3 φ131 cosh−1 (R) d3 d131 S3V129 S129V3 S′ 129 S′ 131 σp λth (6) where cosh−1 (R) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='6939 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='0040 at the center of the 3He cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The last term in the products account for the 3He triplet absorption cross section σ1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' total spin of the neutron+3He of 1) compared to the total neutron wavelength dependent unpolarized absorption cross sec- tion (this factor is equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 25 in [47]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' We used a previous estimate for the 3He triplet cross section of σ1 = 24 barn [57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Other possible corrections due to cell geometry or neutron wavelength distribution [47] are negligible for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' A soon-to-be-submitted work on bi for 3He [62] wll discuss them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The values of the incoherent scattering lengths are thus b129 i = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='186 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='021)stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='004)syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='fm and b131 i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='09 ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='29)stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' ± (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='12)syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Signs of the scattering lengths are determined from the spin directions in the SEOP setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The statistical errors 10% and 12% for 129Xe and 131Xe, respectively, come from the scatter of the phase shift fits shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' These values for bi are in line with those of other nu- clei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' We are not aware of any simple argument that can explain why |b131 i is a factor of 11 larger than |b129 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' With bi(129Xe) measured in this work, we could probe the degree of entanglement of polarized 129Xe spins gen- erated in atomic collisions in SEOP systems using po- larized, mode-entangled neutron beams to measure spin- spin correlation functions as entanglement witnesses for the xenon spin states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' A recently-developed quantita- tive theory for the scattering of mode-entangled neutron beams from spin-correlated dimers [63] can be extended to polarized xenon gas, which can be accurately mod- eled as an ideal gas with an analytical expression for the neutron dynamic structure factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' SEOP collisions of I = 1/2 3He and 129Xe atoms with properly-prepared polarized alkali atoms can generate a calculable degree of entanglement in the nuclear spins under certain con- ditions according to recent work [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The result- ing long-lived entanglement in the nuclear spin system is of interest for optical quantum memories [64–66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The quantum decoherence of mode-entangled neutron beams passing through dense matter is so small that the mea- surement of neutron entanglement witnesses for Bell and GHZ inequalities are unaffected [67–69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The transverse spatial separation between the two opposite-spin sub- beams created in devices like neutron Wollaston prisms coincides with the range of mean free paths of the polar- ized 129Xe gas atoms accessible in SEOP cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Polarized 131Xe nuclei could be used in a search for new sources of time reversal (T) violation in neutron- nucleus interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' T violation from some new interac- tion beyond the Standard Model of particles and inter- actions is one of the highest intellectual priorities in nu- clear/particle/astrophysics, and could shed light on the matter-antimatter asymmetry in the universe according to the Sakharov argument [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The forward scattering amplitude of polarized neutrons in a polarized nuclear target can possess a parity (P)-odd and T-odd term of the form ⃗sn · ( ⃗kn × ⃗I) where ⃗sn is the neutron spin, ⃗kn is the neutron momentum, and ⃗I is the nuclear polar- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Compound neutron-nucleus resonance reactions are known to greatly amplify parity violation in neutron- nucleus interactions [71, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' A 4% P-odd asymmetry was measured in the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='2 eV p-wave resonance in 131Xe, an amplification compared to nucleon-nucleon P-odd am- plitudes of almost 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' The theory which successfully predicted this phenomenon long ago [73, 74] implies that P-odd and T-odd interactions between nucleons beyond the Standard Model should also be amplified by a similar factor [75–77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Neutron transmission measurements in- volving such coherent neutron-nucleus interactions could provide null tests for time-reversal invariance that are free from contamination by final state interactions [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Advances in neutron polarization technology and source brightness added to progress in SEOP polarization of 131Xe suffice to conduct a sensitive search for axion-like particle (ALP) exchange [77], which is poorly constrained by EDM searches for ALP masses above 10 meV [79], because here the Standard Model axion relation between axion mass and coupling constant does not apply [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Lu and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Snow acknowledge support from US National Science Foundation (NSF) grants PHY-1913789 and PHY-2209481 and the Indiana University Center for 6 Spacetime Symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Lu received a Short-Term Grant, 2019 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 57442045 from DAAD the German Aca- demic Exchange Service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Basler and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Goodson acknowledge support from the NSF (CHE-1905341), DoD (W81XWH-15-1-0272, W81XWH2010578), and a Cot- trell Scholar SEED Award from Research Corporation for Science Advancement, and thank T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Gafar, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Kraft, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Korando, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Ruffing for shipment of the SEOP cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' We acknowledge G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Schrank for discussions and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Huber for detailed discussions of NIST work on b3 i and estimates of σ1 for 3He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Van Hove, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 95, 249 (1954).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Lovesey, Theory of Neutron Scattering from Con- densed Matter (Oxford, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Schermer and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Blume, Phys.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Mehring, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 72, 3921 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' [27] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Bear, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' E.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 85, 5038 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' [28] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Cane, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Bear, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Phillips, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Rosen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Smallwood, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Stoner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Walsworth, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Kostelecky, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 93, 230801 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Scharth, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Schmidt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Schnabel, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Sobolev, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Tullney, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' D 98, 036003 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' [33] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Sachdeva, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Fan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Babcock, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Griffith, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Larsen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Mirijanian, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Fu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Smith, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Snow, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Yan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Walker, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAyT4oBgHgl3EQflvja/content/2301.00460v1.pdf'} +page_content=' 111, 102001 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manuscript no. silicates_milkyway +©ESO 2023 +January 5, 2023 +The intense production of silicates during the final AGB phases of +intermediate mass stars +E. Marini1, F. Dell’Agli1, D. Kamath2, 3, 1, P. Ventura1, L. Mattsson, T. Marchetti4, D. A. García-Hernández5, 6, +R. Carini1, M. Fabrizio1, 7 and S. Tosi1, 8 +1 INAF, Observatory of Rome, Via Frascati 33, 00077 Monte Porzio Catone (RM), Italy +2 School of Mathematical and Physical Sciences, Macquarie University, Sydney, NSW, Australia +3 Astronomy, Astrophysics and Astrophotonics Research Centre, Macquarie University, Sydney, NSW, Australia +4 European Southern Observatory, Karl-Schwarzschild-Strasse 2, D-85748 Garching bei München, Germany +5 Instituto de Astrofísica de Canarias (IAC), E-38200 La Laguna, Tenerife, Spain +6 Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain +7 Space Science Data Center, via del Politecnico snc, 00133 Roma, Italy +8 Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Via della Vasca Navale 84, 00100, Roma, Italy +ABSTRACT +Context. The formation of silicates in circumstellar envelopes of stars evolving through the asymptotic giant branch (AGB) is still +highly debated given the uncertainties affecting stellar evolution modelling, the description of the dust formation process, and the +capability of silicate grains to accelerate stellar outflows via radiation pressure. +Aims. We study the formation of dust in the winds of intermediate mass (M ≥ 4 M⊙) stars of solar metallicity while evolving through +the AGB phase. We tested the different treatments of the mass-loss mechanism by this class of stars, with the aim of assessing their +contribution to the general enrichment of silicates of the interstellar medium of galaxies and, on more general grounds, to the silicates +budget of the Universe. +Methods. We consider a sub-sample of AGB stars, whose spectral energy distribution (SED) is characterised by deep absorption +features at 10 µm and 18 µm, which can be regarded as the class of stars providing the most relevant contribution to the silicates’ +production across the Universe. Results from stellar evolution and dust formation modelling were used to fit the observed SED and to +reproduce, at the same time, the detected pulsation periods and the derived surface chemical composition. This analysis leads to the +derivation of tight constraints on the silicates’ production rates experienced by these sources during the final AGB stages. +Results. Two out of the four sources investigated are interpreted as stars currently undergoing hot bottom burning (HBB), evolving +through phases close to the stage when the mass-loss rate is largest. The remaining two stars are likely evolving through the very +final AGB phases, after HBB was turned off by the gradual consumption of the convective mantle. Mass-loss rates of the order of +1 × 10−4 M⊙/yr to 2 × 10−4 M⊙/yr are required when looking for consistency with the observational evidence. These results indicate +the need for a revision of the silicate yields by intermediate mass stars, which are found to be ∼ 3 times higher than previously +determined. +Key words. stars: AGB and post-AGB – stars: abundances – stars: evolution – stars: winds and outflows +1. Introduction +Silicate dust grains have been detected in a wide variety of envi- +ronments, ranging from nearby protoplanetary disks (Maaskant +et al. 2015) to active galactic nuclei (AGNs, Xie et al. 2017) +and distant quasars (Pennock et al. 2022). These particles play +an important role in the cosmic life cycle of matter (Henning +2010) since they regulate the thermal structure of the dense and +cold phases of the interstellar and circumstellar dust popula- +tions. Furthermore, silicate grains contribute to the interstellar +extinction and emit thermal radiation at IR and millimetre wave- +lengths. Their mid-IR spectral features have an important diag- +nostic value for constraining both the chemical composition of +dust and the grain size distribution. The analysis of these fea- +tures provides information about the thermal and density struc- +ture of circumstellar disks and envelopes and the toroidal struc- +tures around AGNs (Granato & Danese 1994; Shi et al. 2006; +Xie et al. 2017). +Asymptotic giant branch (AGB) stars are probably the most +efficient manufacturers of silicates in the Universe (Ferrarotti & +Gail 2006). Those providing the most relevant contribution to the +overall silicate budget of the interstellar medium are those with +a mass in the 4 M⊙ ≤ M ≤ 8 M⊙ range, known as intermediate +mass stars (Ferrarotti & Gail 2006; Ventura et al. 2014). This +class of objects has attracted a great deal of interest from the +scientific community since it was shown that their contribution +to dust production in the Universe cannot be neglected, even in +early epochs (Valiante et al. 2009, 2017). +To be able to study dust production by AGB stars, some re- +search groups presented updated models of the AGB phase in +which the evolution of the central star is coupled to the descrip- +tion of the dust formation process and the relative impact on +the wind (Ventura et al. 2012, 2014; Nanni et al. 2013, 2014), +following the schematisation proposed by the Heidelberg group +(Ferrarotti & Gail 2002, 2006). These models have been suc- +cessfully applied to study the evolved stellar populations of the +Magellanic Clouds (Dell’Agli et al. 2014b, 2015a,b; Nanni et +al. 2016, 2018, 2019b) and other galaxies in the Local Group +(Dell’Agli et al. 2016, 2018a, 2019). +Article number, page 1 of 12 +arXiv:2301.01647v1 [astro-ph.SR] 4 Jan 2023 + +A&A proofs: manuscript no. silicates_milkyway +Despite these steps forward, the contribution from intermedi- +ate mass stars to the overall silicate budget of individual galaxies, +and more generally of the Universe, is still highly debated. While +we expect negligible production in metal-poor environments, the +amount of silicates produced by stellar populations of sub-solar, +solar, and super-solar chemical composition is still extremely un- +certain, and the results found in the literature differ considerably +(Schneider et al. 2014; Ventura et al. 2014, 2020). This is partly +due to the uncertainties in AGB modelling, which concern the lu- +minosities and the mass-loss rates reached by intermediate mass +AGBs (Karakas & Lattanzio 2014), and the possibility that these +stars reach the C-star stage towards the end of the AGB lifetime +(Ventura et al. 2018). A further source of uncertainty, owing to +the formation process of dust particles, is whether chemisputter- +ing or vaporisation is the mechanism responsible for the destruc- +tion of silicate grains (Nanni et al. 2013). Finally, while the for- +mation of carbon dust was shown to lead to efficient acceleration +of the outflow via radiation pressure on solid grains, the forma- +tion of silicate particles results in insufficient radiative pressure +to drive a wind (Höfner 2008); this introduces further uncertain- +ties on the dynamical description of the outflow of oxygen-rich +AGBs, which has a bearing on the silicate formation process. We +note that very recent results for M-type AGB stars (Sandin et al. +2023) obtained with the new T-800 code (see Sandin & Mattsson +2020, for more details) suggest that even accounting for gas-dust +drift to the picture would not lead to higher mass-loss rates for +oxygen-rich stars. +In this work we focus on a sample of Galactic AGB stars, +whose spectral energy distributions (SEDs) exhibit deep absorp- +tion features at 10 µm and 18 µm, which witness the presence of +large amounts of silicates in the circumstellar envelope. These +stars descend from intermediate mass progenitors, and can there- +fore be considered as representative of the stars providing the +largest contribution to the overall release of silicates in the in- +terstellar medium. A thorough characterisation of these objects +proves crucial to deduce the rates at which mass is lost from +intermediate mass AGBs, the efficiency of the dust formation, +and – on more general grounds – to assess the silicate budget +expected in galaxies and in the Universe. +To this aim, we followed the approach by Marini et al. (2020, +2021) to study dusty, evolved stars in the Large Magellanic +Cloud (LMC), where results from stellar evolution and dust for- +mation modelling were used to build a sequence of synthetic +spectra, which allow us to describe how the SED of stars with +different mass evolves during the AGB phase. Comparison with +the observations was used in an attempt to identify the evolution- +ary phase that is currently experienced by the individual sources, +to characterise the progenitors, and to validate the model adopted +to describe the dust formation process in the winds of intermedi- +ate mass AGB stars. +The present investigation, based on the hypothesis that the +four objects examined here are single stars, offers a complemen- +tary characterisation to the one suggested by Decin et al. (2019), +who propose that two out of the four stars investigated belong +to binary systems, on the basis of Atacama Large Millime- +ter/submillimeter Array (ALMA) observations of low-excitation +rotational lines of 12CO. This issue is connected to the more gen- +eral argument of understanding the origin and the current evolu- +tionary status of extreme OH/IR stars in the Galaxy. +The paper is structured as follows. The sample of stars dis- +cussed in the present work is described in section 2. In section +3 the numerical and physical ingredients used to model the evo- +lution of the star and to produce the synthetic SED are given. +The conclusions drawn from the analysis of the SED of the sub- +sample examined are given in section 4. The main aspects of the +AGB evolution of intermediate mass stars is given in section 5, +while the role of the description of mass loss is discussed in sec- +tion 6. Section 7 is devoted to the characterisation of the sources +investigated here. The implications of the present study for the +dust yields of intermediate mass stars is commented on in sec- +tion 8. Finally, the conclusions are given in section 9. +2. The selected sample +The present work is based on the sample of spectra reprocessed +and re-normalised by Sloan et al. (2003), which are included +in the archive for the Infrared Space Observatory (ISO). This +database contains the observations of a wide variety of sources +taken with the Short Wavelength Spectrometer (SWS) in full- +scan mode, covering the 2.4-45.4 µm wavelength range. The +stars observed have been classified in different classes accord- +ing to the morphology of their spectra by Kraemer et al. (2002). +For the present study, we focus on those classified as AGB stars +and exhibiting the deepest silicate absorption features at 10 µm +and 18 µm and we consider four of these objects for which the +pulsation periods and the surface chemical composition, in par- +ticular the 12C/13C ratio (Justtanont et al. 2013), are available. +This information is crucial to characterise these sources in terms +of the initial mass and formation epoch of the progenitors, as +well as the mineralogy and radial distribution of the dust in their +circumstellar envelope. The abundances’ and pulsation periods’ +measurements are even more essential in this context since these +stars are almost invisible in the optical; this strongly affects the +possibility of an accurate measurement of their parallaxes by +Gaia. +3. Numerical and physical inputs +The study presented here is based on results from stellar evolu- +tion modelling (see section 3.1), which was required to calculate +the time variation of the main physical and chemical quantities +of intermediate mass stars during the AGB lifetime. Dust forma- +tion modelling (see section 3.2) was used to calculate the dust +production rate of the various dust species and the optical depth +of the star during the AGB evolution. Finally, results from radia- +tion transfer were used to build the synthetic SED, which is to be +compared with the observations. In the following sub-sections, +we give a brief description of the numerical and physical ingre- +dients adopted for each of the three tasks above. +3.1. Stellar evolution modelling +We calculated evolutionary sequences of M ≥ 4 M⊙ stars of solar +metallicity, evolved from the pre-main sequence until the almost +complete ejection of the external mantle. To this aim we used the +ATON code for stellar evolution, described in detail in Ventura +et al. (1998). We briefly discuss the physical ingredients most +relevant to this work, namely the description of convection and +of mass loss. +The temperature gradient within regions unstable to convec- +tion was calculated via the Full Spectrum of Turbulence (FST) +model (Canuto & Mazzitelli 1991). Nuclear burning and mixing +of chemicals are self-consistently coupled by means of a diffu- +sive approach, according to the schematisation by Cloutman & +Eoll (1976). The overshoot of convective eddies within radia- +tively stable regions was modelled by assuming that the velocity +of convective elements decays exponentially beyond the neutral- +ity point, which is fixed via the Schwartzschild criterion. The +Article number, page 2 of 12 + +E. Marini et al.: The intense production of silicates from AGBs +e-folding distance of the velocity decays during the hydrogen +and helium core burning phases and during the AGB phase was +taken as 0.02HP and 0.002HP, respectively (HP is the pressure +scale height at the formal convective border). The latter values +reflect the calibrations based on the observed width of the main +sequence of open clusters and on the luminosity function of the +LMC carbon stars, discussed in Ventura et al. (1998) and Ventura +et al. (2014), respectively. +Mass loss during the evolutionary stages preceding the AGB +phase was modelled by the classic Reimers’ formula. The treat- +ment of pre-AGB mass loss is of minor importance here as little +mass is expected to be lost during the red giant branch phase. Re- +garding the AGB evolution, to model mass loss, we considered +the treatments by Blöcker (1995) (hereafter Blo95) and the clas- +sic description by Vassiliadis & Wood (1993, hereafter VW93). +In the first case, the mass-loss rate was modelled with the for- +mula +˙M = 4.83 × 10−22ηRL3.7RM−3.1 +.(1) +The free parameter entering equation (1) was set to ηR = +0.02, following the calibration based on the luminosity function +of lithium-rich stars in the LMC, given in Ventura et al. (2000). +The VW93 mass-loss prescription is simply eq. 5 in VW93. The +period of the star, entering the VW93 recipe, was calculated by +means of equation 4 in VW93. During the super-wind phase, we +assumed ˙M = β(L/cvexp), following Vassiliadis & Wood (1993). +We note that β represents the average number of scattering of a +single photon released by the stellar photosphere by dust parti- +cles, under the assumption that the wind is driven by the radia- +tion incident on dust. A natural first assumption is β = 1, which +corresponds to a single scattering. On the other hand, β was +shown to increase with opacity (Lefevre 1989), which is con- +sistent with the possibility that photons are exposed to multiple +scattering when travelling through a high dust density medium. +Detailed radiative transfer computations by Lefevre (1989) show +that β ∼ 2 when ˙M ∼ 10−4 ˙M⊙/yr. Observational evidences +that the largest mass-loss rates experienced by intermediate mass +AGBs exceed the single scattering radiation limit are presented +in van Loon et al. (1999). We explore the 1 ≤ β ≤ 3 range con- +sistent with the study by Knapp (1986). +3.2. Dust production +The analysis of the IR spectra of the extremely obscured oxygen- +rich stars presented in this work requires knowledge of the dust +that formed in the circumstellar envelope. To this aim, we mod- +elled the dust formation and the growth of the dust particles fol- +lowing Ferrarotti & Gail (2006), according to which the dust +forms in a stationary wind that is expanding radially from the +photosphere of the star. We refer readers to Ventura et al. (2012) +for all the relevant equations. The input parameters required to +describe dust formation at a given evolutionary phase of the star +are mass, luminosity, mass-loss rate, effective temperature, and +surface chemical composition. All of these quantities were ob- +tained by the modelling of the central star, as described earlier in +this section. +The key factor affecting the mineralogy of the dust that +formed is the C/O ratio, thanks to the high stability of the CO +molecule (Sharp & Huebner 1990). For the winds of M-type +AGB stars, we assume that the dust species that formed are sili- +cates, alumina dust (Al2O3), and solid iron. While Al2O3 is the +most stable compound, forming closer (∼ 2 stellar radii) to the +surface of the star (Dell’Agli 2012), the species that formed in +the largest quantities, thus providing the dominant contribution +to the acceleration of the outflow via radiation pressure, are sili- +cates. +The dynamics of the wind is described by the momentum +equation, where the acceleration is determined by the competi- +tion between gravity and radiation pressure on the newly formed +dust grains. To calculate the extinction coefficients entering the +momentum equation and the synthetic spectra, as described in +next section, we used the following optical constants: for Al2O3 +and solid iron, we used the extinction coefficients by Koike et +al. (1995) and Ordal et al. (1988), respectively; for amorphous +silicates, we considered Draine & Lee (1984), Suh (1999), Os- +senkopf et al. (1992), and Dorschner et al. (1995); whereas, for +crystalline silicates, we adopted both the coefficients by Jaeger +et al. (1994) and the crystalline olivine provided by the DUSTY +library, which in turn was taken from the Jena-St. Petersburg +database of optical constants1. +The amount of dust that formed is tightly connected with the +mass-loss rate (Ferrarotti & Gail 2006). This is because accord- +ing to the mass conservation law, the mass-loss rate is propor- +tional to the density of the wind (see equation 4 in Ventura et al. +2012): the larger the ˙M, the larger the gas density of the outflow +will be and also the higher the number of molecules available to +condense into dust will be. Therefore, for a given surface chem- +ical composition, the evolutionary phases during which the stars +experience the highest mass-loss rates are when they produce +the largest quantities of dust. The result of the modelling of dust +formation in the outflow is the thermodynamical and chemical +stratification of the wind, the dust composition, the sizes of the +different dust species, the asymptotic velocity, and the optical +depth, which we calculated at the wavelength λ = 10 µm (τ10). +3.3. Spectral energy distribution +The characterisation of the stars presented in this work requires +the interpretation of their observed SED, considering that the +depth and the shape of the different spectral features is extremely +sensitive to the mineralogy and to the amount of dust present in +the circumstellar envelope. This task is based on the comparison +of the observed spectrum with the synthetic SED, calculated by +means of the code DUSTY (Nenkova et al. 1999). +DUSTY calculates the SED of the radiation released from +a stellar source, after being scattered, absorbed, and re-emitted +by a dusty region, spherically distributed around the central star. +The input necessary for the calculation of the synthetic SED are +the mineralogy and grain size distribution, the optical depth τ10, +the dust temperature at the inner boundary of the dusty region, +and the radial distribution of the gas density. All of these in- +puts are provided on the basis of the results obtained by dust +formation modelling as described in the previous sub-section. +The spectrum emerging from the photosphere of the star2 must +also be indicated; however, in the present case, the results are +substantially unaffected by this latter input. This is because the +dusty region is optically thick, thus the reprocessing of the ra- +diation coming from the central star keeps no memory of the +incoming radiation from the stellar photosphere. This allows to +use DUSTY in the modality in which the mass loss is taken from +the results of stellar evolution modelling and so the density dis- +tribution of the outflow. From this we obtained insights about the +dust mineralogy and the optical depth of the individuals sources, +1 https://www.astro.uni-jena.de/Laboratory/OCDB/ +2 i.e. the SED found by interpolation in surface gravity, effective tem- +perature, and C/O ratios among NextGen atmospheres (Hauschildt et al. +1999) of solar metallicity. +Article number, page 3 of 12 + +A&A proofs: manuscript no. silicates_milkyway +by looking for consistency between the observed and synthetic +SED. +To put further constrains on the physical properties of the +central stars, for each source we also ran DUSTY in the modality +where the density distribution is not provided a priori, rather it is +derived by the code by means of the hydrodynamic calculations +applied to the wind. This allows for a self-consistent determina- +tion of the mass-loss rate and of the terminal outflow velocities. +A more exhaustive description is found in Nenkova et al. (1999). +4. The analysis of the ISO spectra +Fig. 1 shows the interpretation of the ISO spectra of the four stars +analysed in this work. As described in section 3.3, the identifica- +tion of the synthetic SED that best reproduces the ISO spectrum +leads to a robust derivation of the wind properties (e.g. optical +depth and mass-loss rates), which we used to characterise the +sources. For the stars considered in this work, we find values +of optical depth 8<τ10<16, which is required to reproduce the +depth of both the silicate features at 10 and 18 µm, the slope +of the continuum in the spectral region λ<8 µm, and the large +IR emission at λ>12 µm. This part of the spectrum, as well as +the depth of the 18 µm feature also allow for the determination +of the dust temperature Tdust, which is found to be in the range +800-1100 K, as reported in Fig. 1. +Regarding the dust mineralogy, the best agreement with the +observations is found assuming the following mineralogy: a +dominant contribution by silicates (∼80%, of which 5-10% are +under the form of crystalline), completed by smaller fractions of +solid iron (10-20%) and alumina dust (∼5%). For a given dust +species, the choice of the optical constants on which the cal- +culation of the extinction coefficients is based strongly affects +the morphology of the synthetic SED. Silicates are generally the +main dust components in the winds of M-type stars; therefore, +they make a major contribution in this regard. We therefore ex- +plored different possibilities for this species, namely the optical +constants by Draine & Lee (1984, D-L), Ossenkopf et al. (1992, +Oss), and Dorschner et al. (1995, Dor), in order to identify which +optical constants allow for the best fit of the observations. We +further considered the optical constants presented by Suh (1999) +with the specific aim of fitting the observations of AGB stars. +The results of this analysis are shown in Fig. 2, which reports +a comparison of four of our predicted SEDs with the ISO spec- +trum of OH 30.1-0.7, taken as an example. For each of the afore- +mentioned optical constants, we show our best-fit model, charac- +terised by τ10=11 for Suh and Dor and τ10=14 for DL and Oss; +and Tdust=1050K and a mineralogy which is dominated by sili- +cates in all cases (75-90%), with smaller percentages of alumina +dust and solid iron. +The poor agreement between the model spectra calculated +with D-L, Oss, and Dor and the observations makes evident the +limitations of these optical constants when we tried to fit such +obscured sources. The main discrepancies are as follows: i) the +continuum at λ<8 µm is not well reproduced and the flux in the +spectral region at 20 µm<λ<30 µm is overestimated, deviating +from the observations of a factor ∼2 in the former case and be- +tween ∼5-15% in the latter, depending on the wavelength; ii) +D-L and Oss SEDs do not reproduce the depth or the shape of +the λ=18 µm absorption feature at all; and iii) the flux in the far- +IR (λ>30 µm) is underestimated in all three cases by ∼15% and +the synthetic SEDs show a steeper spectral slope than observed. +Overall we may conclude that the Suh optical constants are +the only ones that allow for the global shape of the observed +spectrum to be reproduced, leading us to adopt these coefficients +Table 1. Four sources analysed in this work with the isotopic carbon ra- +tios derived by Justtanont et al. (2013) and the observed periods with the +following references: 1-Olivier et al. (2001); 2-Engels & Bunzel (2015); +3-Suh & Kim (2002); 4-van Langevelde et al. (1990); 5-Groenewegen +(2022); and 6-Wolak et al. (2013). +Source Name +12C/13C +P[days] +Ref. +RAFGL 5379 +27±11 +1440 +1 +OH 26.5+0.6 +30±16 +1591 +2 +1559±7 +3 +1589±42 +4 +OH 30.1-0.7 +4±1 +2171 +1 +1952±46 +5 +2013±243 +4 +OH 127.8-0.0 +2±1 +1590 +2 +1600 +6 +in the analysis of the stars presented this work. However, the Suh +(1999) data are empirical and in some way designed to reproduce +observed spectra. Thus, there is a need to systematically consider +a wide range of combinations of laboratory measurements and +theoretical calculations in the future. +Once the best input parameters for DUSTY were identified, +we were able to determine the mass-loss rates of these stars by +running DUSTY in the alternative modality described in the pre- +vious section, thus finding values of ˙M in the 1-2×10−4 M⊙/yr +range. An additional outcome of the SED analysis would be the +determination of the luminosity, as long as the distance of the +star is known. Unfortunately, this step was not possible in the +case of the present work since we cross-matched our candidates +with the latest data release of the satellite Gaia (Gaia DR3, Gaia +Collaboration et al. 2016, 2022). However, given the faintness of +such obscured sources in the optical regime, we found that only +one of the stars, RAFGL 5379, is included in the Gaia DR3 cat- +alogue. The quoted parallax for RAFGL 5379 is negative with +large uncertainties, (-0.122±0.553) mas, preventing us from es- +timating its distance3 and consequently the luminosity. +5. The AGB evolution and dust formation of +intermediate mass stars +Results from stellar evolution and dust formation modelling +(Ventura et al. 2014, 2018) lead us to consider the stars of in- +termediate mass (M ≥ 4 M⊙) as the best candidates in the inter- +pretation of the sources analysed here, based on the fact that they +are expected to produce the highest amounts of silicates in their +winds. The evolution of intermediate mass AGB stars of solar +metallicity was studied by Ventura et al. (2018), who discuss the +main evolutionary properties of these stars, the efficiency of the +dust production mechanism in their circumstellar envelope, and +the uncertainties associated with the description of their AGB +evolution, primarily connected to the still limited knowledge on +convective instability and mass-loss mechanisms. +The evolution of M ≥ 4 M⊙ stars is characterised by the +ignition of hot bottom burning (HBB) at the base of the convec- +tive envelope (Sackmann & Boothroyd 1991). This mechanism +3 Positive parallaxes with relative errors below ∼20% can be inverted +to derive a distance (e.g. Bailer-Jones 2015). In all the other cases, a +Bayesian approach can be used to infer distances (e.g. Bailer-Jones et +al. 2021; Andriantsaralaza et al. 2022, for an application to AGB stars). +In our case, the signal-to-noise of the only Gaia DR3 parallax available +is so low that we decided not to implement this, so as to not bias our +results as to the choice of the adopted prior. +Article number, page 4 of 12 + +E. Marini et al.: The intense production of silicates from AGBs +Fig. 1. ISO spectra (black lines) of the four stars considered in this work. The best fit, indicated with the red line, was obtained by assuming τ10, +Tdust, and dust composition indicated in the panels. +is due to the partial overlapping of the convective mantle with +the H-burning shell, which triggers the activation of advanced +proton-capture nucleosynthesis in the inner regions of the sur- +face convective zone. The activation of HBB drives the evolution +of the surface chemical composition of the star, which reflects +the equilibria of the CNO nucleosyntyhesis. HBB also affects the +luminosity of the star, which after the beginning of HBB grows +faster than in the earlier AGB phases (Blöcker & Schoenberner +1991; Ventura & D’Antona 2005). The ignition of HBB requires +core masses above ∼ 0.8 M⊙ (Ventura et al. 2013), which is the +reason why it is experienced only by stars of intermediate mass. +Fig. 3 shows the time variation of the main physical quan- +tities of M ≥ 4 M⊙ model stars of solar metallicity presented +in Ventura et al. (2018) during the AGB phase. For clarity con- +cerns, we show only the results of stars with an initial mass of +4, 5, and 6 M⊙. In looking at the top left panel of Fig. 3, one can +recognise the typical behaviour of the luminosity of these stars +(Ventura et al. 2022): the initial phase, during which the luminos- +ity increases owing to the growth of the core mass, is followed +by a second phase, during which the luminosity decreases, as +HBB is gradually extinguished. The peak luminosity of the stars +increases with the initial mass as the core mass of the stars dur- +ing the AGB phase is higher, the larger the initial mass (Ventura +Article number, page 5 of 12 + +A&A proofs: manuscript no. silicates_milkyway +Fig. 2. ISO spectrum of OH 30.1-0.7 (solid black line), together with +four best-fit model spectra calculated adopting the following optical +constants for silicates: Suh (1999) (solid red line), Ossenkopf et al. +(1992) (dashed blue line), Dorschner et al. (1995) (dash-dotted green +line), and Draine & Lee (1984) (dotted yellow line). The DUSTY in- +puts used for each model spectrum are reported in the text. +et al. 2013; Karakas & Lattanzio 2014). This affects the duration +of the AGB phase, which is anti-correlated with the initial mass: +the range of the timescales of the AGB evolution increases from +∼ 2×105 yr, for 8 M⊙ model stars, to ∼ 3×106 yr, for M = 4 M⊙ +(Ventura et al. 2018). +The mass-loss rate experienced by the stars, shown in the +top right panel of Fig. 3, scales approximately with the lumi- +nosity. This is due to the tight relationship between the luminos- +ity and the mass-loss rate in the Blöcker (1995) treatment (see +equation 1). We note, in particular, the decrease in the mass- +loss rate characterising the very final AGB phases, during which +˙M ∼ 10−5 M⊙/yr. The maximum mass-loss rate also changes +with the initial mass: specifically for the model stars shown in +Fig. 3, the peak values of ˙M are 3×10−5 M⊙/yr, 7×10−5 M⊙/yr, +and 10−4 M⊙/yr for the 4, 5, and 6 M⊙ model stars, respectively. +The variation of the pulsation period of the star, shown in the +bottom left panel of Fig. 3, is not correlated with luminosity as +much as the mass-loss rate is. This is because the star continues +to expand even after the luminosity peak is reached, a behaviour +typical of stars surrounded by a convective mantle that is pro- +gressively lost via stellar wind. The contraction phase starts only +at the very end of the AGB evolution, when the residual mass of +the envelope drops below ∼ 0.2 M⊙, and the CNO cycle is no +longer sufficiently efficient to support the star on the energetic +side. As shown in Fig. 3, the period of these stars grows during +the first part of the AGB evolution, from a few hundred days (d) +until exceeding 2000 d, then it decreases to ∼ 1500 d during the +very final AGB evolutionary stages. +To describe the variation of the surface chemical composi- +tion, we show the variation of the 12C/13C carbon ratio in the bot- +tom right panel of the same figure. The onset of HBB is clearly +visible in the drop of the carbon ratio taking place during the +first AGB phases, which continues until the equilibrium value +12C/13C∼ 4 is reached: This demonstrates the full effect of HBB +in modifying the surface chemistry. In Fig. 3 one can recognise +the effects of a third dredge-up (TDU) in the fast increase in +12C/13C that follows each thermal pulse, and in the increase in +the carbon ratio that characterises the final AGB phases, after +HBB was turned off. Further effects of the ignition of HBB are +the synthesis of nitrogen and sodium, which during the AGB +lifetime increase by a factor of ∼ 5 and ∼ 3, respectively, and +the destruction of the surface 18O, the most fragile among the +oxygen isotopes. Intermediate mass stars evolve through the +so-called lithium-rich phase, during which large quantities of +lithium are synthesized by the Cameron & Fowler (1971) mech- +anism. The duration of the lithium-rich phase depends on the rate +at which the surface 3He is consumed, and it accounts for ∼ 50%, +∼ 40%, and ∼ 30% of the AGB phase, for the model stars of +initial mass 4 M⊙, 5 M⊙, and 6 M⊙, respectively. The HBB ex- +perienced by solar-metallicity stars is soft, thus the temperatures +reached by the innermost regions of the convective envelope are +not sufficiently hot to activate more advanced nucleosynthesis, +typical of lower metallicity stars (Dell’Agli et al. 2018b): Neither +the depletion of 16O and 24Mg, nor the increase in the surface +abundances of aluminium and silicon takes place in the model +stars examined here. +Regarding dust, the study by Ventura et al. (2018) suggests +that most of the dust that formed in the circumstellar envelope +of massive AGBs is composed of silicates (70% − 80%), with +alumina dust and solid iron making up the remaining 20%−30%. +The fraction of gaseous silicon that condensed into dust is in +the 20% − 30% range, whereas the fraction of aluminium that +condensed into Al2O3 ranges from ∼ 30% (initial and final AGB +phases) to ∼ 80% (in correspondence of the luminosity peak). +The typical gas-to-dust ratio found in Ventura et al. (2018) is in +the 500 − 1000 range. +6. The role of mass loss +The empirical mass-loss treatment proposed by VW93 is based +on an empirical formula relating the mass-loss rate to the pul- +sation period. According to VW93 the mass-loss rate increases +with the pulsation period, until reaching the super-wind phase, +when the radiation-driven mass-loss rate is adopted, according +to the expression ˙M = βL/(vexpc), discussed in section 3.1. +The differences between the results obtained with the Blo95 +and VW93 treatments applied to the modelling of a 5 M⊙ star are +shown in Fig. 4. For the VW93 case, we considered β = 1 and a +further case in which we set β = 3, starting from the maximum +luminosity phase. +As shown in the middle panel of Fig. 4, the model star cal- +culated with the Blo95 treatment experiences higher mass-loss +rates during the first part of the AGB evolution, thus the en- +velope is expelled faster. This makes the duration of the AGB +phase of VW93 model longer, which allows for higher growth +of the core mass, and thus higher peak luminosities. The more +rapid consumption of the envelope is also the reason why, in the +β = 1 case, the luminosity reached is higher than for β = 3. +The Blo95 model star evolves at larger periods compared to the +VW93 ones: This is once more due to the higher mass loss ex- +perienced during the initial part of the AGB phase, which makes +the star reach a more expanded configuration, and hence experi- +ence longer pulsation periods. The Blo95 and the VW93 β = 1 +model stars experience similar peak mass-loss rates, of the order +Article number, page 6 of 12 + +E. Marini et al.: The intense production of silicates from AGBs +& +Fig. 3. AGB evolution of the main physical and chemical properties of the model stars with a solar metallicity with an initial mass of 4 M⊙ (blue +lines), 5 M⊙ (red), and 6 M⊙ (black), as a function of time, counted since the beginning of the AGB phase. The behaviour of luminosity (top left +panel), mass-loss rate (top right), pulsation period (bottom left), surface 12C/13C (bottom right) are shown. The shadings in the bottom left panel +indicate the range of values of the pulsation periods taken from the literature of: OH 30.1-0.7 (grey), 127.8-0.0 and OH 26.5+0.6 (yellow), and +RAFGL 5379 (magenta). +of 10−4 M⊙/yr; on the other hand, in the VW93 β = 3 case, the +largest mass-loss rate experienced is 2 × 10−4 M⊙/yr. +These rates of mass loss are at odds with the results from +dust formation in the winds of M-type stars in which stellar pul- +sation and associated shocks are properly considered (Bladh et +al. 2013, 2019) and which rarely form outflows of more than +˙M ∼ 10−5 M⊙ yr−1. However, these works are mostly based on +smaller luminosities than those invoked here (L ≥ 40000 L⊙). +More detailed investigations of dust production in the present +luminosity domain is required before solid conclusions can be +drawn in this regard. +Significant differences among the mass-loss rates of the var- +ious model stars are found during the very final phases, when +the Blo95 model star experiences ˙M values being significantly +smaller than the W93 counterparts, owing to the previously dis- +cussed sensitivity of the Blo95 treatment on the luminosity of +the star. The full comparison among the results obtained by dif- +ferent descriptions of mass loss is shown in Fig. 5, where the +time variation of the mass-loss rate, pulsation period, dust pro- +Article number, page 7 of 12 + +A&A proofs: manuscript no. silicates_milkyway +Fig. 4. Time variation during the AGB phase of the luminosity (left panel), mass-loss rate (middle), and pulsation period (right) of solar metallicity, +model stars of initial mass 5 M⊙, in which mass loss was modelled according to Blöcker (1995) (black lines) and to Vassiliadis & Wood (1993), +where the scattering parameter β was set to 1 (green lines) and 3 (red). +duction rate, and optical depth at 10 µm of intermediate mass +AGBs are shown. We restrict the comparison to the Blo95 and +VW93 model stars, calculated with β = 3. +The evolutionary timescales of the 4 M⊙ and 5 M⊙ model +stars obtained by modelling mass loss with Blo95 and VW93 +are similar: The depletion of the envelope during the first part of +the AGB phase is faster in the Blo95 case, but this is counterbal- +anced by the higher luminosity attained by the VW93 models, +which renders the timescales of the final phases shorter. In the +6 M⊙ case,the differences between the luminosity of the Blo95 +and VW93 models arise from the initial AGB phases, making +the duration of the AGB phase of the VW93 model shorter. +Significant differences are found in the mass-loss rates expe- +rienced. The VW93 rates are orders of magnitude smaller than +Blo95 during the initial AGB phase, until the peak luminosity +was reached. When the super-wind takes over the peak ˙M values +of the VW93 model, stars are ∼ 3 times higher than Blo95. +These differences in the mass-loss rate are the result of dif- +ferences in the dust production rate of the stars, for what con- +cerns both the general behaviour across the AGB lifetime and +the largest value reached. As shown in the bottom left panel of +Fig. 5, in the VW93 case, dust production is negligible during +the first part of the AGB phase, then dust is produced at rates of +the order of 5 × 10−7 − 10−6 M⊙/yr after the super-wind takes +over. Almost all the dust released by the stars is produced during +the last four to five inter-pulse phases. Conversely, the behaviour +of ˙Mdust with the AGB time is smoother in the Blo95 case, and +the largest rates reached are ∼ 10−7 M⊙/yr. The gas-to-dust ra- +tio during the phases when the dust production rate is largest is +∼ 300 in the VW93 case, which has yet to be compared to the +minimum value of ∼ 500 found by Ventura et al. (2018). +The expected IR excess of the stars, here represented by τ10, +is tightly connected to the dust production rate. This can be seen +by comparing the behaviour of ˙Mdust with the time variation of +τ10, shown in the bottom right panel of Fig. 5. We note, in partic- +ular, the significant difference in the largest τ10 values, attained +during the final AGB phases: In the VW93 model, star τ10 is in +the 7 − 15 range; whereas, in the Blo95 case, we find τ10 < 3. +7. The characterisation of the sample sources +Table 1 lists the pulsation periods and the isotopic carbon ra- +tios of the four stars in the present study. The sources are char- +acterised by periods in the 1440 d < P < 2000 d range, as +well as optical depths derived from SED fitting (see section 4), +8 < τ10 < 16. These results are not consistent with the outcome +of modelling based on Blöcker (1995), thus we subsequently fo- +cus on the results based on the VW93 treatment of mass loss. +7.1. Stars experiencing hot bottom burning +OH 127.8-0.0 and OH 30.1-0.7 exhibit isotopic carbon ratios +12C/13C= 2 ± 1 and 12C/13C= 4 ± 1, respectively, in agreement +with the values expected when CNO nucleosynthesis reaches +equilibrium; the periods from the literature are ∼ 1600 d for OH +127.8-0.0 and ∼ 2000 d for OH 30.1-0.7, whereas the optical +depths derived from SED fitting are τ10 = 11 (OH 30.1-0.7) and +τ10 = 13 (OH 127.8-0.0). Based on the results shown in Fig. 5, +we conclude that these sources are currently evolving through +the AGB phases during which the dust production (hence mass- +loss) rate are close to the maximum values, and they are currently +experiencing HBB. Unfortunately, as can be seen in Fig. 5, this +information proves insufficient to assess the mass of the pro- +genitor (hence the formation epoch) for OH 30.1-0.7 since all +the model stars of initial mass in the 4 − 6 M⊙ range evolve +through phases characterised by the quantities given above. The +accurate determination of the distance would be crucial to this +scope, as the luminosity at which the star is expected to evolve +at the periods and optical depths given above is ∼ 3.5 × 104 L⊙, +∼ 5 × 104 L⊙, or ∼ 7 × 104 L⊙, according to whether the initial +mass of the star is 4 M⊙, 5 M⊙, or 6 M⊙, respectively. Inversely, +if the luminosity is known, it is possible to give an estimation +of the distance of the star4. If our understanding is correct, we +would expect the following distances depending on luminosity: +∼3.5kpc (∼ 3.5 × 104 L⊙), ∼4.2kpc (∼ 5 × 104 L⊙), and ∼5kpc +(∼ 7 × 104 L⊙). +On the other hand, the lower pulsation period of 127.8-0.0 (∼ +1600d) indicates that this star cannot descend from a progenitor +with an initial mass ∼4 M⊙ because such periods correspond, in +this case, to very low values of mass-loss rate (<∼ 10−6 M⊙/yr) +and the theoretical optical depths (see the bottom right panel of +Fig. 5) would be too low in comparison to those found by SED +fitting. Therefore, we can conclude that 127.8-0.0 descends from +4 This can be done by fixing the luminosity and shifting the synthetic +SED until it matches the observed spectrum. +Article number, page 8 of 12 + +E. Marini et al.: The intense production of silicates from AGBs +& +Fig. 5. Time variation of the mass-loss rate (top left panel), pulsation period (top right), dust production rate (bottom left), and optical depth +τ10 (bottom right) of model stars with an initial mass of 4 M⊙ (black lines and circles), 5 M⊙ (red lines and squares), and 6 M⊙ (blue lines and +triangles), calculated with the mass-loss prescription by Blöcker (1995). The points in the bottom panels refer to the values in the middle of each +inter-pulse phase. Green points refer to the inter-pulse phases, and indicate results from modelling based on the VW93 description of mass loss, +where the value β = 3 was considered for the scattering parameter. The shadings in the top right panel are the same as in Fig. 3. +stars of initial mass M>4 M⊙, and we expect luminosities de- +pending on the mass of the progenitor, which are ∼ 5 × 104 L⊙ +(M=4 M⊙) and ∼ 7 × 104 L⊙ (M=6 M⊙). These would corre- +spond to distances of ∼5kpc and ∼6kpc, respectively. The above +discussed interpretation given for OH 127.8-0.0 and OH 30.1-0.7 +is consistent with the mass-loss rates derived from SED fitting, +which are in the 1.5 − 2.5 × 10−4 M⊙/yr range5, also consistent +with the results shown in the top left panel of Fig. 5. +5 The mass-loss rate derived from SED fitting via the DUSTY code +depends on the luminosity of the star (Nenkova et al. 1999). When con- +sidering the luminosities ∼ 3.5×104 L⊙, ∼ 5×104 L⊙, and ∼ 7×104 L⊙ +given in the text, we find that ˙M = 1.5, 2.0, 2.5 × 10−4 M⊙/yr. +Article number, page 9 of 12 + +A&A proofs: manuscript no. silicates_milkyway +7.2. Stars evolving through the very late AGB phases +The interpretation of RAFGL 5379 and OH 26.5+0.6 is more +tricky because the given 12C/13C (27±11 and 30±16, respec- +tively) are significantly higher than the equilibrium values. The +possibility that they are evolving through the initial AGB phase, +when HBB has not yet started, can be ruled out since the periods +would be much shorter than those measured (see Fig. 5). A fur- +ther possibility is that these stars have just experienced a thermal +pulse (TP), after which the action of TDU favours the increase +in the surface 12C, and thus an off-equilibrium 12C/13C; while +this possibility cannot be ruled out, we consider it extremely +unlikely since re-ignition of HBB favours a fast destruction of +12C in favour of 13C, thus the phase during which 12C/13C is +off-equilibrium lasts only for a very small fraction of the whole +inter-pulse. +We believe it is more plausible that these two stars are ex- +periencing late AGB phases, during which HBB is turned off: +as shown in the bottom right panel of Fig. 3, the occurrence of +a couple of TDU events is sufficient to raise the carbon isotopic +ratio above the equilibrium values. This is in agreement with one +of the possibilities invoked for RAFGL 5379 by de Vries et al. +(2014). The measured period of the two stars, 1440 d for RAFGL +5379 and around 1600 d for OH 26.5+0.6, further support this +hypothesis (see Fig. 3). The optical depth and the mass-loss rates +derived from SED fitting also support this picture. +A further confirmation that RAFGL 5379 and OH 26.5+0.6 +experienced HBB is given by the studies based on Herschel ob- +servations by Justtanont et al. (2013, 2015), who could not detect +any H2O18 line, despite the strong water emission lines in H2O16 +and H2O17. As discussed in section 5, intermediate mass stars of +solar metallicity experience strong depletion of the surface 18O +and negligible reduction of the surface 16O and 17O. +We note that unlike the two sources discussed above for +which no or poor predictions regarding the luminosity was pos- +sible, for RAFGL 5379 and OH 26.5+0.6 we can claim values +of the order of 2 × 104 L⊙, independently of the initial mass. +Indeed, as can be seen in Fig. 5, the luminosities obtained by +the stars during the very final AGB phase, after HBB was turned +off, tend to converge towards the previously given value, sub- +stantially independently of the initial mass of the stars. Consid- +ering this luminosity, the estimated distance in this case would +be ∼1.2kpc for RAFGL 5379 and ∼1.8kpc for OH 26.5+0.6. The +mass-loss rate derived from SED fitting, when this luminosity is +used, is 10−4 M⊙/yr, consistent with the results shown in the top +left panel of Fig. 5. +7.3. The binary possibility +In a study based on results from ALMA observations of low- +excitation rotational lines of 12CO, Decin et al. (2019) proposed +that OH 26.5+0.6 and OH 30.1-07 are part of binary systems. +This conclusion is motivated by the detection of an incomplete +ring-like pattern, which is characteristic of a shell-like spiral, in +turn connected to the orbital motion of the AGB star around the +centre of mass of a binary system. On general grounds, bina- +rity offers an interesting explanation to interpret the morphology +of the gaseous emission of the circumstellar envelope extreme +OH/IR stars. +If the stars belong to binary systems, the large dust produc- +tion rates would be related to significant equatorial density en- +hancements, triggered by the gravitational attraction of the com- +panion star. In this case the largest single scattering mass-loss +rates required would be of the order of a few 10−5 M⊙/yr, a few +times smaller than those needed in the case of single stars. +On the statistical side, we do not see any significant short- +comings from either possibilities. Binarity is found to be a rather +common feature of intermediate mass stars, with percentages es- +timated to be 30 − 40%. On the other hand, in the single star +hypothesis, we note from the results shown in Fig. 5 that the +time interval during which the stars experience a super-wind-like +mass loss, with ˙M > 10−4 M⊙/yr, is 20-40 kyr, which represents +10 − 50% of the entire AGB lifetime, with the percentage be- +ing higher the larger the initial mass of the star. We therefore +expect that the fraction of extreme oxygen-rich stars are a non- +negligible percentage of the whole sample of O-rich stars of in- +termediate mass. +We see in Fig. 5 that a few thermal pulses take place each +2-4 kyr during the phase when the stars experience intense mass +loss. Dust formation is temporarily halted during thermal pulses +as the star readjusts on a more compact configuration, so as to +counterbalance the effects of the CNO activity turning off, and +the larger photospheric temperatures inhibit the formation of sili- +cates in the wind. The stars interpreted as currently experiencing +strong mass loss can be evolving through any of the inter-pulse +phases of the super-wind evolution. Therefore, while the phase +during which the stars are expected to produce dust with large +rates lasts a few thousand years, we expect that the dust responsi- +ble for the IR excess observed nowadays was not released earlier +than 2-3 kyr ago, that is when the last TP took place. This under- +standing is consistent with the results by de Vries et al. (2014), +who found that the phase of intense dust production in these stars +did not start earlier than 200-1000 yr ago, and was preceded by +a phase during which dust production was negligible. +Presently, we may consider the possibility that part of the +most extreme OH/IR stars in the Milky Way are single objects +currently evolving through the final AGB stages, and the remain- +ing fraction belong to binary systems. Further observations of a +wider sample of this class of objects are required before it can +be assessed whether binarity is in fact a common rule for these +objects and, more generally, to establish the fraction of OH/IR +stars that are part of binary systems. +8. A revision of the dust yields from intermediate +mass AGBs? +From the results shown in the bottom left panel of Fig. 5, it +is clear that the predictions regarding dust production between +the VW93 and the Blo95 models are significantly different. The +Blo95 dust production rates are higher than VW93 for most of +the AGB lifetime. However, this has very little effect on the over- +all dust yield of intermediate mass stars because most of the dust +is released during the phases when the mass-loss rate is largest: +during these phases, the VW93 ˙Mdust values are a factor ∼ 3 +higher than the Blo95 ones. +For the three models discussed in the previous section, we +find that the dust yields are 0.012 M⊙ for M = 4 M⊙ and +0.021 M⊙ for M = 5, 6 M⊙. The dust yield for the 7 M⊙ model +star (not shown for clarity concerns in Fig. 5) is 0.025 M⊙. The +dust released by the stars is mostly composed of silicates, which +account for 85%, with smaller contributions from alumina dust +(10%) and solid iron (5%). These percentages are similar to +those found by Ventura et al. (2018); on the other hand, the over- +all dust budget expected using VW95 mass loss is ∼ 3 times +larger than that of Ventura et al. (2018) in which the Blo95 pre- +scription was adopted. +Article number, page 10 of 12 + +E. Marini et al.: The intense production of silicates from AGBs +Table 2. Dust yields (solar masses) from stars of a different mass by various research groups. +M/M⊙ +This work +Ventura et al. (2018) +Nanni et al. (2013) +Ferrarotti & Gail (2006) +4 +1.2 × 10−2 +2.9 × 10−3 +4 × 10−4 +1.2 × 10−2 +5 +2.1 × 10−2 +4.5 × 10−3 +9.3 × 10−3 +1.5 × 10−2 +6 +2.1 × 10−2 +6.2 × 10−3 +1.4 × 10−2 +- +7 +2.5 × 10−2 +8.4 × 10−3 +- +1.8 × 10−2 +In Table 2 the dust yields from intermediate mass stars of +solar metallicity found in the present work are compared with +those published in Ventura et al. (2018), and with results from +Nanni et al. (2013) and Ferrarotti & Gail (2006)6. Our yields +are the largest among the solar metallicity dust yields published +so far, indicating that the contribution to the overall silicate dust +production by intermediate mass stars has been underestimated +so far. We note that the largest amount of carbon dust in Ventura +et al. (2018), released by the 3 M⊙ model star, is ∼ 0.015 M⊙; +therefore, we find that at solar metallicities the quantity of sili- +cates produced by intermediate mass AGB stars is slightly higher +than the amount of carbon that formed in the wind of low-mass +stars. +The main results of the present work concern the mass-loss +rates suffered by intermediate mass stars during the final AGB +evolutionary stages, during and after the phases close to the lu- +minosity peak experienced by these stars. With the VW93 mass- +loss prescription, this pertains to the AGB evolution from the +point when the superwind sets in. +Regardless of the prescription of mass-loss adopted, the ef- +fects on the gas yields are negligible. The time between the acti- +vation of HBB and the achievement of the luminosity peak is suf- +ficiently long, such that the surface chemistry of the stars reached +the equilibrium distribution of the proton-capture nucleosynthe- +sis corresponding to the temperature attained at the bottom of +the convective mantle. In this context, the chemical composition +of the gas ejected into the interstellar medium is independent on +the timescale with which the gas itself is released, which is de- +termined by the mass-loss rate. We may safely conclude that the +present investigation has led us to pose some questions regard- +ing the reliability of the predicted silicate yields from intermedi- +ate mass stars, but no conclusions regarding the gas yields from +these objects can be drawn based on the present analysis. +9. Conclusions +We have studied a sample of Galactic AGB stars, whose SEDs +exhibit deep absorption features centred at 10 µm and 18 µm, as- +sociated with silicate dust. The study of these stars, for which the +pulsation periods and some information on the surface chemical +composition are available, is based on stellar evolution and dust +formation modelling, and supported by results from radiative +transfer modelling. These ingredients have been used to charac- +terise the individual objects in an attempt to determine the mass +and formation epoch of the progenitors, and, on more general +grounds, to understand the silicate budget expected from AGB +stars. +The sources investigated are interpreted as the progeny of +intermediate mass stars, currently evolving through advanced +AGB phases, either currently experiencing HBB or after HBB +6 In the comparison with results from different studies, it must be con- +sidered that the metallicity adopted here (Z=0.014) is smaller than the +value used in Ventura et al. (2018) (Z=0.018) and that used by Nanni et +al. (2013) and Ferrarotti & Gail (2006) (Z = 0.02) +turned off by the gradual loss of the external envelope. The study +of their SED indicates mass-loss rates in the 1 − 2 × 10−4 M⊙/yr +range, consistent with the radiation pressure wind description in +which photons experience multiple scattering processes. These +rates of mass loss of intermediate mass AGBs of solar metallic- +ity are significantly higher (a factor ∼ 3) than those published. +Provided that this picture is correct, we deduce that mass +loss suffered by intermediate mass stars and the dust produc- +tion mechanism remain highly efficient until the very final AGB +phases, preceding the contraction to the post-AGB phase. These +results indicate the need of an upwards revision of the dust +yields by intermediate mass stars, which are now found in the +0.012−0.025 M⊙ range, mostly under the form of silicates, with +a smaller contribution from alumina dust (∼ 10%) and solid iron +(∼ 5%). +Acknowledgements. EM acknowledges support from the INAF research project +’LBT - Supporto Arizona Italia’. D.K. acknowledges the support of the Aus- +tralian Research Council (ARC) Discovery Early Career Research Award (DE- +CRA) grant (DE190100813) and the Australian Research Council Centre of +Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through +project number CE170100013. MF acknowledges financial support from the +ASI-INAF agreement n. 2022-14-HH.0. This research has made use of the Ga- +iaPortal catalogues access tool, ASI?Space Science Data Center, Rome, Italy +(http://gaiaportal.ssdc.asi.it). +References +Andriantsaralaza M., Ramstedt S., Vlemmings W. H. T., et al., 2022, arXiv, +arXiv:2209.03906 +Bailer-Jones C. A. L., 2015, PASP, 127, 994 +Bailer-Jones C. A. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Karl-Schwarzschild-Strasse 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' D-85748 Garching bei München,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Germany 5 Instituto de Astrofísica de Canarias (IAC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' E-38200 La Laguna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Tenerife,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Spain 6 Departamento de Astrofísica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Universidad de La Laguna (ULL),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' E-38206 La Laguna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Tenerife,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Spain 7 Space Science Data Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' via del Politecnico snc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 00133 Roma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Italy 8 Dipartimento di Matematica e Fisica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Università degli Studi Roma Tre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Via della Vasca Navale 84,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 00100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Roma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Italy ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The formation of silicates in circumstellar envelopes of stars evolving through the asymptotic giant branch (AGB) is still highly debated given the uncertainties affecting stellar evolution modelling, the description of the dust formation process, and the capability of silicate grains to accelerate stellar outflows via radiation pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' We study the formation of dust in the winds of intermediate mass (M ≥ 4 M⊙) stars of solar metallicity while evolving through the AGB phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' We tested the different treatments of the mass-loss mechanism by this class of stars, with the aim of assessing their contribution to the general enrichment of silicates of the interstellar medium of galaxies and, on more general grounds, to the silicates budget of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' We consider a sub-sample of AGB stars, whose spectral energy distribution (SED) is characterised by deep absorption features at 10 µm and 18 µm, which can be regarded as the class of stars providing the most relevant contribution to the silicates’ production across the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Results from stellar evolution and dust formation modelling were used to fit the observed SED and to reproduce, at the same time, the detected pulsation periods and the derived surface chemical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This analysis leads to the derivation of tight constraints on the silicates’ production rates experienced by these sources during the final AGB stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Two out of the four sources investigated are interpreted as stars currently undergoing hot bottom burning (HBB), evolving through phases close to the stage when the mass-loss rate is largest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The remaining two stars are likely evolving through the very final AGB phases, after HBB was turned off by the gradual consumption of the convective mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Mass-loss rates of the order of 1 × 10−4 M⊙/yr to 2 × 10−4 M⊙/yr are required when looking for consistency with the observational evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' These results indicate the need for a revision of the silicate yields by intermediate mass stars, which are found to be ∼ 3 times higher than previously determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' stars: AGB and post-AGB – stars: abundances – stars: evolution – stars: winds and outflows 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Introduction Silicate dust grains have been detected in a wide variety of envi- ronments, ranging from nearby protoplanetary disks (Maaskant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2015) to active galactic nuclei (AGNs, Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2017) and distant quasars (Pennock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' These particles play an important role in the cosmic life cycle of matter (Henning 2010) since they regulate the thermal structure of the dense and cold phases of the interstellar and circumstellar dust popula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Furthermore, silicate grains contribute to the interstellar extinction and emit thermal radiation at IR and millimetre wave- lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Their mid-IR spectral features have an important diag- nostic value for constraining both the chemical composition of dust and the grain size distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The analysis of these fea- tures provides information about the thermal and density struc- ture of circumstellar disks and envelopes and the toroidal struc- tures around AGNs (Granato & Danese 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Asymptotic giant branch (AGB) stars are probably the most efficient manufacturers of silicates in the Universe (Ferrarotti & Gail 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Those providing the most relevant contribution to the overall silicate budget of the interstellar medium are those with a mass in the 4 M⊙ ≤ M ≤ 8 M⊙ range, known as intermediate mass stars (Ferrarotti & Gail 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This class of objects has attracted a great deal of interest from the scientific community since it was shown that their contribution to dust production in the Universe cannot be neglected, even in early epochs (Valiante et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2009, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' To be able to study dust production by AGB stars, some re- search groups presented updated models of the AGB phase in which the evolution of the central star is coupled to the descrip- tion of the dust formation process and the relative impact on the wind (Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2012, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Nanni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2013, 2014), following the schematisation proposed by the Heidelberg group (Ferrarotti & Gail 2002, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' These models have been suc- cessfully applied to study the evolved stellar populations of the Magellanic Clouds (Dell’Agli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2014b, 2015a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Nanni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2016, 2018, 2019b) and other galaxies in the Local Group (Dell’Agli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2016, 2018a, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Article number, page 1 of 12 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='01647v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='SR] 4 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' silicates_milkyway Despite these steps forward, the contribution from intermedi- ate mass stars to the overall silicate budget of individual galaxies, and more generally of the Universe, is still highly debated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' While we expect negligible production in metal-poor environments, the amount of silicates produced by stellar populations of sub-solar, solar, and super-solar chemical composition is still extremely un- certain, and the results found in the literature differ considerably (Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2014, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This is partly due to the uncertainties in AGB modelling, which concern the lu- minosities and the mass-loss rates reached by intermediate mass AGBs (Karakas & Lattanzio 2014), and the possibility that these stars reach the C-star stage towards the end of the AGB lifetime (Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' A further source of uncertainty, owing to the formation process of dust particles, is whether chemisputter- ing or vaporisation is the mechanism responsible for the destruc- tion of silicate grains (Nanni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Finally, while the for- mation of carbon dust was shown to lead to efficient acceleration of the outflow via radiation pressure on solid grains, the forma- tion of silicate particles results in insufficient radiative pressure to drive a wind (Höfner 2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' this introduces further uncertain- ties on the dynamical description of the outflow of oxygen-rich AGBs, which has a bearing on the silicate formation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' We note that very recent results for M-type AGB stars (Sandin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2023) obtained with the new T-800 code (see Sandin & Mattsson 2020, for more details) suggest that even accounting for gas-dust drift to the picture would not lead to higher mass-loss rates for oxygen-rich stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' In this work we focus on a sample of Galactic AGB stars, whose spectral energy distributions (SEDs) exhibit deep absorp- tion features at 10 µm and 18 µm, which witness the presence of large amounts of silicates in the circumstellar envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' These stars descend from intermediate mass progenitors, and can there- fore be considered as representative of the stars providing the largest contribution to the overall release of silicates in the in- terstellar medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' A thorough characterisation of these objects proves crucial to deduce the rates at which mass is lost from intermediate mass AGBs, the efficiency of the dust formation, and – on more general grounds – to assess the silicate budget expected in galaxies and in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' To this aim, we followed the approach by Marini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2020, 2021) to study dusty, evolved stars in the Large Magellanic Cloud (LMC), where results from stellar evolution and dust for- mation modelling were used to build a sequence of synthetic spectra, which allow us to describe how the SED of stars with different mass evolves during the AGB phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Comparison with the observations was used in an attempt to identify the evolution- ary phase that is currently experienced by the individual sources, to characterise the progenitors, and to validate the model adopted to describe the dust formation process in the winds of intermedi- ate mass AGB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The present investigation, based on the hypothesis that the four objects examined here are single stars, offers a complemen- tary characterisation to the one suggested by Decin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2019), who propose that two out of the four stars investigated belong to binary systems, on the basis of Atacama Large Millime- ter/submillimeter Array (ALMA) observations of low-excitation rotational lines of 12CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This issue is connected to the more gen- eral argument of understanding the origin and the current evolu- tionary status of extreme OH/IR stars in the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The sample of stars dis- cussed in the present work is described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' In section 3 the numerical and physical ingredients used to model the evo- lution of the star and to produce the synthetic SED are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The conclusions drawn from the analysis of the SED of the sub- sample examined are given in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The main aspects of the AGB evolution of intermediate mass stars is given in section 5, while the role of the description of mass loss is discussed in sec- tion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Section 7 is devoted to the characterisation of the sources investigated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The implications of the present study for the dust yields of intermediate mass stars is commented on in sec- tion 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Finally, the conclusions are given in section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The selected sample The present work is based on the sample of spectra reprocessed and re-normalised by Sloan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2003), which are included in the archive for the Infrared Space Observatory (ISO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This database contains the observations of a wide variety of sources taken with the Short Wavelength Spectrometer (SWS) in full- scan mode, covering the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='4-45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='4 µm wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The stars observed have been classified in different classes accord- ing to the morphology of their spectra by Kraemer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' For the present study, we focus on those classified as AGB stars and exhibiting the deepest silicate absorption features at 10 µm and 18 µm and we consider four of these objects for which the pulsation periods and the surface chemical composition, in par- ticular the 12C/13C ratio (Justtanont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2013), are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This information is crucial to characterise these sources in terms of the initial mass and formation epoch of the progenitors, as well as the mineralogy and radial distribution of the dust in their circumstellar envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The abundances’ and pulsation periods’ measurements are even more essential in this context since these stars are almost invisible in the optical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' this strongly affects the possibility of an accurate measurement of their parallaxes by Gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Numerical and physical inputs The study presented here is based on results from stellar evolu- tion modelling (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='1), which was required to calculate the time variation of the main physical and chemical quantities of intermediate mass stars during the AGB lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Dust forma- tion modelling (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='2) was used to calculate the dust production rate of the various dust species and the optical depth of the star during the AGB evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Finally, results from radia- tion transfer were used to build the synthetic SED, which is to be compared with the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' In the following sub-sections, we give a brief description of the numerical and physical ingre- dients adopted for each of the three tasks above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Stellar evolution modelling We calculated evolutionary sequences of M ≥ 4 M⊙ stars of solar metallicity, evolved from the pre-main sequence until the almost complete ejection of the external mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' To this aim we used the ATON code for stellar evolution, described in detail in Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' We briefly discuss the physical ingredients most relevant to this work, namely the description of convection and of mass loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The temperature gradient within regions unstable to convec- tion was calculated via the Full Spectrum of Turbulence (FST) model (Canuto & Mazzitelli 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Nuclear burning and mixing of chemicals are self-consistently coupled by means of a diffu- sive approach, according to the schematisation by Cloutman & Eoll (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The overshoot of convective eddies within radia- tively stable regions was modelled by assuming that the velocity of convective elements decays exponentially beyond the neutral- ity point, which is fixed via the Schwartzschild criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The Article number, page 2 of 12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Marini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' : The intense production of silicates from AGBs e-folding distance of the velocity decays during the hydrogen and helium core burning phases and during the AGB phase was taken as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='02HP and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='002HP, respectively (HP is the pressure scale height at the formal convective border).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The latter values reflect the calibrations based on the observed width of the main sequence of open clusters and on the luminosity function of the LMC carbon stars, discussed in Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (1998) and Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2014), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Mass loss during the evolutionary stages preceding the AGB phase was modelled by the classic Reimers’ formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The treat- ment of pre-AGB mass loss is of minor importance here as little mass is expected to be lost during the red giant branch phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Re- garding the AGB evolution, to model mass loss, we considered the treatments by Blöcker (1995) (hereafter Blo95) and the clas- sic description by Vassiliadis & Wood (1993, hereafter VW93).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' In the first case, the mass-loss rate was modelled with the for- mula ˙M = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='83 × 10−22ηRL3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='7RM−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (1) The free parameter entering equation (1) was set to ηR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='02, following the calibration based on the luminosity function of lithium-rich stars in the LMC, given in Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The VW93 mass-loss prescription is simply eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 5 in VW93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The period of the star, entering the VW93 recipe, was calculated by means of equation 4 in VW93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' During the super-wind phase, we assumed ˙M = β(L/cvexp), following Vassiliadis & Wood (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' We note that β represents the average number of scattering of a single photon released by the stellar photosphere by dust parti- cles, under the assumption that the wind is driven by the radia- tion incident on dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' A natural first assumption is β = 1, which corresponds to a single scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' On the other hand, β was shown to increase with opacity (Lefevre 1989), which is con- sistent with the possibility that photons are exposed to multiple scattering when travelling through a high dust density medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Detailed radiative transfer computations by Lefevre (1989) show that β ∼ 2 when ˙M ∼ 10−4 ˙M⊙/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Observational evidences that the largest mass-loss rates experienced by intermediate mass AGBs exceed the single scattering radiation limit are presented in van Loon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' We explore the 1 ≤ β ≤ 3 range con- sistent with the study by Knapp (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Dust production The analysis of the IR spectra of the extremely obscured oxygen- rich stars presented in this work requires knowledge of the dust that formed in the circumstellar envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' To this aim, we mod- elled the dust formation and the growth of the dust particles fol- lowing Ferrarotti & Gail (2006), according to which the dust forms in a stationary wind that is expanding radially from the photosphere of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' We refer readers to Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2012) for all the relevant equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The input parameters required to describe dust formation at a given evolutionary phase of the star are mass, luminosity, mass-loss rate, effective temperature, and surface chemical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' All of these quantities were ob- tained by the modelling of the central star, as described earlier in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The key factor affecting the mineralogy of the dust that formed is the C/O ratio, thanks to the high stability of the CO molecule (Sharp & Huebner 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' For the winds of M-type AGB stars, we assume that the dust species that formed are sili- cates, alumina dust (Al2O3), and solid iron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' While Al2O3 is the most stable compound, forming closer (∼ 2 stellar radii) to the surface of the star (Dell’Agli 2012), the species that formed in the largest quantities, thus providing the dominant contribution to the acceleration of the outflow via radiation pressure, are sili- cates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The dynamics of the wind is described by the momentum equation, where the acceleration is determined by the competi- tion between gravity and radiation pressure on the newly formed dust grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' To calculate the extinction coefficients entering the momentum equation and the synthetic spectra, as described in next section, we used the following optical constants: for Al2O3 and solid iron, we used the extinction coefficients by Koike et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (1995) and Ordal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (1988), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' for amorphous silicates, we considered Draine & Lee (1984), Suh (1999), Os- senkopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (1992), and Dorschner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' whereas, for crystalline silicates, we adopted both the coefficients by Jaeger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (1994) and the crystalline olivine provided by the DUSTY library, which in turn was taken from the Jena-St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Petersburg database of optical constants1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The amount of dust that formed is tightly connected with the mass-loss rate (Ferrarotti & Gail 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This is because accord- ing to the mass conservation law, the mass-loss rate is propor- tional to the density of the wind (see equation 4 in Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2012): the larger the ˙M, the larger the gas density of the outflow will be and also the higher the number of molecules available to condense into dust will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Therefore, for a given surface chem- ical composition, the evolutionary phases during which the stars experience the highest mass-loss rates are when they produce the largest quantities of dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The result of the modelling of dust formation in the outflow is the thermodynamical and chemical stratification of the wind, the dust composition, the sizes of the different dust species, the asymptotic velocity, and the optical depth, which we calculated at the wavelength λ = 10 µm (τ10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Spectral energy distribution The characterisation of the stars presented in this work requires the interpretation of their observed SED, considering that the depth and the shape of the different spectral features is extremely sensitive to the mineralogy and to the amount of dust present in the circumstellar envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This task is based on the comparison of the observed spectrum with the synthetic SED, calculated by means of the code DUSTY (Nenkova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' DUSTY calculates the SED of the radiation released from a stellar source, after being scattered, absorbed, and re-emitted by a dusty region, spherically distributed around the central star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The input necessary for the calculation of the synthetic SED are the mineralogy and grain size distribution, the optical depth τ10, the dust temperature at the inner boundary of the dusty region, and the radial distribution of the gas density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' All of these in- puts are provided on the basis of the results obtained by dust formation modelling as described in the previous sub-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The spectrum emerging from the photosphere of the star2 must also be indicated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' however, in the present case, the results are substantially unaffected by this latter input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This is because the dusty region is optically thick, thus the reprocessing of the ra- diation coming from the central star keeps no memory of the incoming radiation from the stellar photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This allows to use DUSTY in the modality in which the mass loss is taken from the results of stellar evolution modelling and so the density dis- tribution of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' From this we obtained insights about the dust mineralogy and the optical depth of the individuals sources, 1 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='uni-jena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='de/Laboratory/OCDB/ 2 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' the SED found by interpolation in surface gravity, effective tem- perature, and C/O ratios among NextGen atmospheres (Hauschildt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 1999) of solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Article number, page 3 of 12 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' silicates_milkyway by looking for consistency between the observed and synthetic SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' To put further constrains on the physical properties of the central stars, for each source we also ran DUSTY in the modality where the density distribution is not provided a priori, rather it is derived by the code by means of the hydrodynamic calculations applied to the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This allows for a self-consistent determina- tion of the mass-loss rate and of the terminal outflow velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' A more exhaustive description is found in Nenkova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The analysis of the ISO spectra Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 1 shows the interpretation of the ISO spectra of the four stars analysed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' As described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='3, the identifica- tion of the synthetic SED that best reproduces the ISO spectrum leads to a robust derivation of the wind properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' optical depth and mass-loss rates), which we used to characterise the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' For the stars considered in this work, we find values of optical depth 8<τ10<16, which is required to reproduce the depth of both the silicate features at 10 and 18 µm, the slope of the continuum in the spectral region λ<8 µm, and the large IR emission at λ>12 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This part of the spectrum, as well as the depth of the 18 µm feature also allow for the determination of the dust temperature Tdust, which is found to be in the range 800-1100 K, as reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Regarding the dust mineralogy, the best agreement with the observations is found assuming the following mineralogy: a dominant contribution by silicates (∼80%, of which 5-10% are under the form of crystalline), completed by smaller fractions of solid iron (10-20%) and alumina dust (∼5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' For a given dust species, the choice of the optical constants on which the cal- culation of the extinction coefficients is based strongly affects the morphology of the synthetic SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Silicates are generally the main dust components in the winds of M-type stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' therefore, they make a major contribution in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' We therefore ex- plored different possibilities for this species, namely the optical constants by Draine & Lee (1984, D-L), Ossenkopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (1992, Oss), and Dorschner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (1995, Dor), in order to identify which optical constants allow for the best fit of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' We further considered the optical constants presented by Suh (1999) with the specific aim of fitting the observations of AGB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The results of this analysis are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2, which reports a comparison of four of our predicted SEDs with the ISO spec- trum of OH 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='7, taken as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' For each of the afore- mentioned optical constants, we show our best-fit model, charac- terised by τ10=11 for Suh and Dor and τ10=14 for DL and Oss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' and Tdust=1050K and a mineralogy which is dominated by sili- cates in all cases (75-90%), with smaller percentages of alumina dust and solid iron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The poor agreement between the model spectra calculated with D-L, Oss, and Dor and the observations makes evident the limitations of these optical constants when we tried to fit such obscured sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The main discrepancies are as follows: i) the continuum at λ<8 µm is not well reproduced and the flux in the spectral region at 20 µm<λ<30 µm is overestimated, deviating from the observations of a factor ∼2 in the former case and be- tween ∼5-15% in the latter, depending on the wavelength;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' ii) D-L and Oss SEDs do not reproduce the depth or the shape of the λ=18 µm absorption feature at all;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' and iii) the flux in the far- IR (λ>30 µm) is underestimated in all three cases by ∼15% and the synthetic SEDs show a steeper spectral slope than observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Overall we may conclude that the Suh optical constants are the only ones that allow for the global shape of the observed spectrum to be reproduced, leading us to adopt these coefficients Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Four sources analysed in this work with the isotopic carbon ra- tios derived by Justtanont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2013) and the observed periods with the following references: 1-Olivier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2-Engels & Bunzel (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 3-Suh & Kim (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 4-van Langevelde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (1990);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 5-Groenewegen (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' and 6-Wolak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Source Name 12C/13C P[days] Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' RAFGL 5379 27±11 1440 1 OH 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='6 30±16 1591 2 1559±7 3 1589±42 4 OH 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='7 4±1 2171 1 1952±46 5 2013±243 4 OH 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='0 2±1 1590 2 1600 6 in the analysis of the stars presented this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' However, the Suh (1999) data are empirical and in some way designed to reproduce observed spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Thus, there is a need to systematically consider a wide range of combinations of laboratory measurements and theoretical calculations in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Once the best input parameters for DUSTY were identified, we were able to determine the mass-loss rates of these stars by running DUSTY in the alternative modality described in the pre- vious section, thus finding values of ˙M in the 1-2×10−4 M⊙/yr range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' An additional outcome of the SED analysis would be the determination of the luminosity, as long as the distance of the star is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Unfortunately, this step was not possible in the case of the present work since we cross-matched our candidates with the latest data release of the satellite Gaia (Gaia DR3, Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2016, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' However, given the faintness of such obscured sources in the optical regime, we found that only one of the stars, RAFGL 5379, is included in the Gaia DR3 cat- alogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The quoted parallax for RAFGL 5379 is negative with large uncertainties, (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='122±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='553) mas, preventing us from es- timating its distance3 and consequently the luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The AGB evolution and dust formation of intermediate mass stars Results from stellar evolution and dust formation modelling (Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2014, 2018) lead us to consider the stars of in- termediate mass (M ≥ 4 M⊙) as the best candidates in the inter- pretation of the sources analysed here, based on the fact that they are expected to produce the highest amounts of silicates in their winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The evolution of intermediate mass AGB stars of solar metallicity was studied by Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2018), who discuss the main evolutionary properties of these stars, the efficiency of the dust production mechanism in their circumstellar envelope, and the uncertainties associated with the description of their AGB evolution, primarily connected to the still limited knowledge on convective instability and mass-loss mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The evolution of M ≥ 4 M⊙ stars is characterised by the ignition of hot bottom burning (HBB) at the base of the convec- tive envelope (Sackmann & Boothroyd 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This mechanism 3 Positive parallaxes with relative errors below ∼20% can be inverted to derive a distance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Bailer-Jones 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' In all the other cases, a Bayesian approach can be used to infer distances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Andriantsaralaza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2022, for an application to AGB stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' In our case, the signal-to-noise of the only Gaia DR3 parallax available is so low that we decided not to implement this, so as to not bias our results as to the choice of the adopted prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Article number, page 4 of 12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Marini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' : The intense production of silicates from AGBs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' ISO spectra (black lines) of the four stars considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The best fit, indicated with the red line, was obtained by assuming τ10, Tdust, and dust composition indicated in the panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' is due to the partial overlapping of the convective mantle with the H-burning shell, which triggers the activation of advanced proton-capture nucleosynthesis in the inner regions of the sur- face convective zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The activation of HBB drives the evolution of the surface chemical composition of the star, which reflects the equilibria of the CNO nucleosyntyhesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' HBB also affects the luminosity of the star, which after the beginning of HBB grows faster than in the earlier AGB phases (Blöcker & Schoenberner 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Ventura & D’Antona 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The ignition of HBB requires core masses above ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='8 M⊙ (Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2013), which is the reason why it is experienced only by stars of intermediate mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 3 shows the time variation of the main physical quan- tities of M ≥ 4 M⊙ model stars of solar metallicity presented in Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2018) during the AGB phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' For clarity con- cerns, we show only the results of stars with an initial mass of 4, 5, and 6 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' In looking at the top left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 3, one can recognise the typical behaviour of the luminosity of these stars (Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2022): the initial phase, during which the luminos- ity increases owing to the growth of the core mass, is followed by a second phase, during which the luminosity decreases, as HBB is gradually extinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The peak luminosity of the stars increases with the initial mass as the core mass of the stars dur- ing the AGB phase is higher, the larger the initial mass (Ventura Article number, page 5 of 12 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' silicates_milkyway Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' ISO spectrum of OH 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='7 (solid black line), together with four best-fit model spectra calculated adopting the following optical constants for silicates: Suh (1999) (solid red line), Ossenkopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (1992) (dashed blue line), Dorschner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (1995) (dash-dotted green line), and Draine & Lee (1984) (dotted yellow line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The DUSTY in- puts used for each model spectrum are reported in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Karakas & Lattanzio 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This affects the duration of the AGB phase, which is anti-correlated with the initial mass: the range of the timescales of the AGB evolution increases from ∼ 2×105 yr, for 8 M⊙ model stars, to ∼ 3×106 yr, for M = 4 M⊙ (Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The mass-loss rate experienced by the stars, shown in the top right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 3, scales approximately with the lumi- nosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This is due to the tight relationship between the luminos- ity and the mass-loss rate in the Blöcker (1995) treatment (see equation 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' We note, in particular, the decrease in the mass- loss rate characterising the very final AGB phases, during which ˙M ∼ 10−5 M⊙/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The maximum mass-loss rate also changes with the initial mass: specifically for the model stars shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 3, the peak values of ˙M are 3×10−5 M⊙/yr, 7×10−5 M⊙/yr, and 10−4 M⊙/yr for the 4, 5, and 6 M⊙ model stars, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The variation of the pulsation period of the star, shown in the bottom left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 3, is not correlated with luminosity as much as the mass-loss rate is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This is because the star continues to expand even after the luminosity peak is reached, a behaviour typical of stars surrounded by a convective mantle that is pro- gressively lost via stellar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The contraction phase starts only at the very end of the AGB evolution, when the residual mass of the envelope drops below ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='2 M⊙, and the CNO cycle is no longer sufficiently efficient to support the star on the energetic side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 3, the period of these stars grows during the first part of the AGB evolution, from a few hundred days (d) until exceeding 2000 d, then it decreases to ∼ 1500 d during the very final AGB evolutionary stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' To describe the variation of the surface chemical composi- tion, we show the variation of the 12C/13C carbon ratio in the bot- tom right panel of the same figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The onset of HBB is clearly visible in the drop of the carbon ratio taking place during the first AGB phases, which continues until the equilibrium value 12C/13C∼ 4 is reached: This demonstrates the full effect of HBB in modifying the surface chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 3 one can recognise the effects of a third dredge-up (TDU) in the fast increase in 12C/13C that follows each thermal pulse, and in the increase in the carbon ratio that characterises the final AGB phases, after HBB was turned off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Further effects of the ignition of HBB are the synthesis of nitrogen and sodium, which during the AGB lifetime increase by a factor of ∼ 5 and ∼ 3, respectively, and the destruction of the surface 18O, the most fragile among the oxygen isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Intermediate mass stars evolve through the so-called lithium-rich phase, during which large quantities of lithium are synthesized by the Cameron & Fowler (1971) mech- anism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The duration of the lithium-rich phase depends on the rate at which the surface 3He is consumed, and it accounts for ∼ 50%, ∼ 40%, and ∼ 30% of the AGB phase, for the model stars of initial mass 4 M⊙, 5 M⊙, and 6 M⊙, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The HBB ex- perienced by solar-metallicity stars is soft, thus the temperatures reached by the innermost regions of the convective envelope are not sufficiently hot to activate more advanced nucleosynthesis, typical of lower metallicity stars (Dell’Agli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2018b): Neither the depletion of 16O and 24Mg, nor the increase in the surface abundances of aluminium and silicon takes place in the model stars examined here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Regarding dust, the study by Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2018) suggests that most of the dust that formed in the circumstellar envelope of massive AGBs is composed of silicates (70% − 80%), with alumina dust and solid iron making up the remaining 20%−30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The fraction of gaseous silicon that condensed into dust is in the 20% − 30% range, whereas the fraction of aluminium that condensed into Al2O3 ranges from ∼ 30% (initial and final AGB phases) to ∼ 80% (in correspondence of the luminosity peak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The typical gas-to-dust ratio found in Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2018) is in the 500 − 1000 range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The role of mass loss The empirical mass-loss treatment proposed by VW93 is based on an empirical formula relating the mass-loss rate to the pul- sation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' According to VW93 the mass-loss rate increases with the pulsation period, until reaching the super-wind phase, when the radiation-driven mass-loss rate is adopted, according to the expression ˙M = βL/(vexpc), discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The differences between the results obtained with the Blo95 and VW93 treatments applied to the modelling of a 5 M⊙ star are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' For the VW93 case, we considered β = 1 and a further case in which we set β = 3, starting from the maximum luminosity phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' As shown in the middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 4, the model star cal- culated with the Blo95 treatment experiences higher mass-loss rates during the first part of the AGB evolution, thus the en- velope is expelled faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This makes the duration of the AGB phase of VW93 model longer, which allows for higher growth of the core mass, and thus higher peak luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The more rapid consumption of the envelope is also the reason why, in the β = 1 case, the luminosity reached is higher than for β = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The Blo95 model star evolves at larger periods compared to the VW93 ones: This is once more due to the higher mass loss ex- perienced during the initial part of the AGB phase, which makes the star reach a more expanded configuration, and hence experi- ence longer pulsation periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The Blo95 and the VW93 β = 1 model stars experience similar peak mass-loss rates, of the order Article number, page 6 of 12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Marini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' : The intense production of silicates from AGBs & Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' AGB evolution of the main physical and chemical properties of the model stars with a solar metallicity with an initial mass of 4 M⊙ (blue lines), 5 M⊙ (red), and 6 M⊙ (black), as a function of time, counted since the beginning of the AGB phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The behaviour of luminosity (top left panel), mass-loss rate (top right), pulsation period (bottom left), surface 12C/13C (bottom right) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The shadings in the bottom left panel indicate the range of values of the pulsation periods taken from the literature of: OH 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='7 (grey), 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='0 and OH 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='6 (yellow), and RAFGL 5379 (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' of 10−4 M⊙/yr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' on the other hand, in the VW93 β = 3 case, the largest mass-loss rate experienced is 2 × 10−4 M⊙/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' These rates of mass loss are at odds with the results from dust formation in the winds of M-type stars in which stellar pul- sation and associated shocks are properly considered (Bladh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2013, 2019) and which rarely form outflows of more than ˙M ∼ 10−5 M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' However, these works are mostly based on smaller luminosities than those invoked here (L ≥ 40000 L⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' More detailed investigations of dust production in the present luminosity domain is required before solid conclusions can be drawn in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Significant differences among the mass-loss rates of the var- ious model stars are found during the very final phases, when the Blo95 model star experiences ˙M values being significantly smaller than the W93 counterparts, owing to the previously dis- cussed sensitivity of the Blo95 treatment on the luminosity of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The full comparison among the results obtained by dif- ferent descriptions of mass loss is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 5, where the time variation of the mass-loss rate, pulsation period, dust pro- Article number, page 7 of 12 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' silicates_milkyway Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Time variation during the AGB phase of the luminosity (left panel), mass-loss rate (middle), and pulsation period (right) of solar metallicity, model stars of initial mass 5 M⊙, in which mass loss was modelled according to Blöcker (1995) (black lines) and to Vassiliadis & Wood (1993), where the scattering parameter β was set to 1 (green lines) and 3 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' duction rate, and optical depth at 10 µm of intermediate mass AGBs are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' We restrict the comparison to the Blo95 and VW93 model stars, calculated with β = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The evolutionary timescales of the 4 M⊙ and 5 M⊙ model stars obtained by modelling mass loss with Blo95 and VW93 are similar: The depletion of the envelope during the first part of the AGB phase is faster in the Blo95 case, but this is counterbal- anced by the higher luminosity attained by the VW93 models, which renders the timescales of the final phases shorter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' In the 6 M⊙ case,the differences between the luminosity of the Blo95 and VW93 models arise from the initial AGB phases, making the duration of the AGB phase of the VW93 model shorter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Significant differences are found in the mass-loss rates expe- rienced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The VW93 rates are orders of magnitude smaller than Blo95 during the initial AGB phase, until the peak luminosity was reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' When the super-wind takes over the peak ˙M values of the VW93 model, stars are ∼ 3 times higher than Blo95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' These differences in the mass-loss rate are the result of dif- ferences in the dust production rate of the stars, for what con- cerns both the general behaviour across the AGB lifetime and the largest value reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' As shown in the bottom left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 5, in the VW93 case, dust production is negligible during the first part of the AGB phase, then dust is produced at rates of the order of 5 × 10−7 − 10−6 M⊙/yr after the super-wind takes over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Almost all the dust released by the stars is produced during the last four to five inter-pulse phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Conversely, the behaviour of ˙Mdust with the AGB time is smoother in the Blo95 case, and the largest rates reached are ∼ 10−7 M⊙/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The gas-to-dust ra- tio during the phases when the dust production rate is largest is ∼ 300 in the VW93 case, which has yet to be compared to the minimum value of ∼ 500 found by Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The expected IR excess of the stars, here represented by τ10, is tightly connected to the dust production rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This can be seen by comparing the behaviour of ˙Mdust with the time variation of τ10, shown in the bottom right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' We note, in partic- ular, the significant difference in the largest τ10 values, attained during the final AGB phases: In the VW93 model, star τ10 is in the 7 − 15 range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' whereas, in the Blo95 case, we find τ10 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The characterisation of the sample sources Table 1 lists the pulsation periods and the isotopic carbon ra- tios of the four stars in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The sources are char- acterised by periods in the 1440 d < P < 2000 d range, as well as optical depths derived from SED fitting (see section 4), 8 < τ10 < 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' These results are not consistent with the outcome of modelling based on Blöcker (1995), thus we subsequently fo- cus on the results based on the VW93 treatment of mass loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Stars experiencing hot bottom burning OH 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='0 and OH 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='7 exhibit isotopic carbon ratios 12C/13C= 2 ± 1 and 12C/13C= 4 ± 1, respectively, in agreement with the values expected when CNO nucleosynthesis reaches equilibrium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' the periods from the literature are ∼ 1600 d for OH 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='0 and ∼ 2000 d for OH 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='7, whereas the optical depths derived from SED fitting are τ10 = 11 (OH 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='7) and τ10 = 13 (OH 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Based on the results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 5, we conclude that these sources are currently evolving through the AGB phases during which the dust production (hence mass- loss) rate are close to the maximum values, and they are currently experiencing HBB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Unfortunately, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 5, this information proves insufficient to assess the mass of the pro- genitor (hence the formation epoch) for OH 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='7 since all the model stars of initial mass in the 4 − 6 M⊙ range evolve through phases characterised by the quantities given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The accurate determination of the distance would be crucial to this scope, as the luminosity at which the star is expected to evolve at the periods and optical depths given above is ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='5 × 104 L⊙, ∼ 5 × 104 L⊙, or ∼ 7 × 104 L⊙, according to whether the initial mass of the star is 4 M⊙, 5 M⊙, or 6 M⊙, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Inversely, if the luminosity is known, it is possible to give an estimation of the distance of the star4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' If our understanding is correct, we would expect the following distances depending on luminosity: ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='5kpc (∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='5 × 104 L⊙), ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='2kpc (∼ 5 × 104 L⊙), and ∼5kpc (∼ 7 × 104 L⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' On the other hand, the lower pulsation period of 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='0 (∼ 1600d) indicates that this star cannot descend from a progenitor with an initial mass ∼4 M⊙ because such periods correspond, in this case, to very low values of mass-loss rate (<∼ 10−6 M⊙/yr) and the theoretical optical depths (see the bottom right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 5) would be too low in comparison to those found by SED fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Therefore, we can conclude that 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='0 descends from 4 This can be done by fixing the luminosity and shifting the synthetic SED until it matches the observed spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Article number, page 8 of 12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Marini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' : The intense production of silicates from AGBs & Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Time variation of the mass-loss rate (top left panel), pulsation period (top right), dust production rate (bottom left), and optical depth τ10 (bottom right) of model stars with an initial mass of 4 M⊙ (black lines and circles), 5 M⊙ (red lines and squares), and 6 M⊙ (blue lines and triangles), calculated with the mass-loss prescription by Blöcker (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The points in the bottom panels refer to the values in the middle of each inter-pulse phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Green points refer to the inter-pulse phases, and indicate results from modelling based on the VW93 description of mass loss, where the value β = 3 was considered for the scattering parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The shadings in the top right panel are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' stars of initial mass M>4 M⊙, and we expect luminosities de- pending on the mass of the progenitor, which are ∼ 5 × 104 L⊙ (M=4 M⊙) and ∼ 7 × 104 L⊙ (M=6 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' These would corre- spond to distances of ∼5kpc and ∼6kpc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The above discussed interpretation given for OH 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='0 and OH 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='7 is consistent with the mass-loss rates derived from SED fitting, which are in the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='5 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='5 × 10−4 M⊙/yr range5, also consistent with the results shown in the top left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 5 The mass-loss rate derived from SED fitting via the DUSTY code depends on the luminosity of the star (Nenkova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' When con- sidering the luminosities ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='5×104 L⊙, ∼ 5×104 L⊙, and ∼ 7×104 L⊙ given in the text, we find that ˙M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='5 × 10−4 M⊙/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Article number, page 9 of 12 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' silicates_milkyway 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Stars evolving through the very late AGB phases The interpretation of RAFGL 5379 and OH 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='6 is more tricky because the given 12C/13C (27±11 and 30±16, respec- tively) are significantly higher than the equilibrium values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The possibility that they are evolving through the initial AGB phase, when HBB has not yet started, can be ruled out since the periods would be much shorter than those measured (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' A fur- ther possibility is that these stars have just experienced a thermal pulse (TP), after which the action of TDU favours the increase in the surface 12C, and thus an off-equilibrium 12C/13C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' while this possibility cannot be ruled out, we consider it extremely unlikely since re-ignition of HBB favours a fast destruction of 12C in favour of 13C, thus the phase during which 12C/13C is off-equilibrium lasts only for a very small fraction of the whole inter-pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' We believe it is more plausible that these two stars are ex- periencing late AGB phases, during which HBB is turned off: as shown in the bottom right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 3, the occurrence of a couple of TDU events is sufficient to raise the carbon isotopic ratio above the equilibrium values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This is in agreement with one of the possibilities invoked for RAFGL 5379 by de Vries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The measured period of the two stars, 1440 d for RAFGL 5379 and around 1600 d for OH 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='6, further support this hypothesis (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The optical depth and the mass-loss rates derived from SED fitting also support this picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' A further confirmation that RAFGL 5379 and OH 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='6 experienced HBB is given by the studies based on Herschel ob- servations by Justtanont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2013, 2015), who could not detect any H2O18 line, despite the strong water emission lines in H2O16 and H2O17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' As discussed in section 5, intermediate mass stars of solar metallicity experience strong depletion of the surface 18O and negligible reduction of the surface 16O and 17O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' We note that unlike the two sources discussed above for which no or poor predictions regarding the luminosity was pos- sible, for RAFGL 5379 and OH 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='6 we can claim values of the order of 2 × 104 L⊙, independently of the initial mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Indeed, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 5, the luminosities obtained by the stars during the very final AGB phase, after HBB was turned off, tend to converge towards the previously given value, sub- stantially independently of the initial mass of the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Consid- ering this luminosity, the estimated distance in this case would be ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='2kpc for RAFGL 5379 and ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='8kpc for OH 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The mass-loss rate derived from SED fitting, when this luminosity is used, is 10−4 M⊙/yr, consistent with the results shown in the top left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The binary possibility In a study based on results from ALMA observations of low- excitation rotational lines of 12CO, Decin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2019) proposed that OH 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='6 and OH 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='1-07 are part of binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This conclusion is motivated by the detection of an incomplete ring-like pattern, which is characteristic of a shell-like spiral, in turn connected to the orbital motion of the AGB star around the centre of mass of a binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' On general grounds, bina- rity offers an interesting explanation to interpret the morphology of the gaseous emission of the circumstellar envelope extreme OH/IR stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' If the stars belong to binary systems, the large dust produc- tion rates would be related to significant equatorial density en- hancements, triggered by the gravitational attraction of the com- panion star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' In this case the largest single scattering mass-loss rates required would be of the order of a few 10−5 M⊙/yr, a few times smaller than those needed in the case of single stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' On the statistical side, we do not see any significant short- comings from either possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Binarity is found to be a rather common feature of intermediate mass stars, with percentages es- timated to be 30 − 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' On the other hand, in the single star hypothesis, we note from the results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 5 that the time interval during which the stars experience a super-wind-like mass loss, with ˙M > 10−4 M⊙/yr, is 20-40 kyr, which represents 10 − 50% of the entire AGB lifetime, with the percentage be- ing higher the larger the initial mass of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' We therefore expect that the fraction of extreme oxygen-rich stars are a non- negligible percentage of the whole sample of O-rich stars of in- termediate mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' We see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 5 that a few thermal pulses take place each 2-4 kyr during the phase when the stars experience intense mass loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Dust formation is temporarily halted during thermal pulses as the star readjusts on a more compact configuration, so as to counterbalance the effects of the CNO activity turning off, and the larger photospheric temperatures inhibit the formation of sili- cates in the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The stars interpreted as currently experiencing strong mass loss can be evolving through any of the inter-pulse phases of the super-wind evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Therefore, while the phase during which the stars are expected to produce dust with large rates lasts a few thousand years, we expect that the dust responsi- ble for the IR excess observed nowadays was not released earlier than 2-3 kyr ago, that is when the last TP took place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This under- standing is consistent with the results by de Vries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2014), who found that the phase of intense dust production in these stars did not start earlier than 200-1000 yr ago, and was preceded by a phase during which dust production was negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Presently, we may consider the possibility that part of the most extreme OH/IR stars in the Milky Way are single objects currently evolving through the final AGB stages, and the remain- ing fraction belong to binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Further observations of a wider sample of this class of objects are required before it can be assessed whether binarity is in fact a common rule for these objects and, more generally, to establish the fraction of OH/IR stars that are part of binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' A revision of the dust yields from intermediate mass AGBs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' From the results shown in the bottom left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 5, it is clear that the predictions regarding dust production between the VW93 and the Blo95 models are significantly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The Blo95 dust production rates are higher than VW93 for most of the AGB lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' However, this has very little effect on the over- all dust yield of intermediate mass stars because most of the dust is released during the phases when the mass-loss rate is largest: during these phases, the VW93 ˙Mdust values are a factor ∼ 3 higher than the Blo95 ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' For the three models discussed in the previous section, we find that the dust yields are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='012 M⊙ for M = 4 M⊙ and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='021 M⊙ for M = 5, 6 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The dust yield for the 7 M⊙ model star (not shown for clarity concerns in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 5) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='025 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The dust released by the stars is mostly composed of silicates, which account for 85%, with smaller contributions from alumina dust (10%) and solid iron (5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' These percentages are similar to those found by Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' on the other hand, the over- all dust budget expected using VW95 mass loss is ∼ 3 times larger than that of Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2018) in which the Blo95 pre- scription was adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Article number, page 10 of 12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Marini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' : The intense production of silicates from AGBs Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Dust yields (solar masses) from stars of a different mass by various research groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' M/M⊙ This work Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2018) Nanni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2013) Ferrarotti & Gail (2006) 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='2 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='9 × 10−3 4 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='2 × 10−2 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='1 × 10−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='5 × 10−3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='3 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='5 × 10−2 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='1 × 10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='2 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='4 × 10−2 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='5 × 10−2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='4 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='8 × 10−2 In Table 2 the dust yields from intermediate mass stars of solar metallicity found in the present work are compared with those published in Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2018), and with results from Nanni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2013) and Ferrarotti & Gail (2006)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Our yields are the largest among the solar metallicity dust yields published so far, indicating that the contribution to the overall silicate dust production by intermediate mass stars has been underestimated so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' We note that the largest amount of carbon dust in Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2018), released by the 3 M⊙ model star, is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='015 M⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' therefore, we find that at solar metallicities the quantity of sili- cates produced by intermediate mass AGB stars is slightly higher than the amount of carbon that formed in the wind of low-mass stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The main results of the present work concern the mass-loss rates suffered by intermediate mass stars during the final AGB evolutionary stages, during and after the phases close to the lu- minosity peak experienced by these stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' With the VW93 mass- loss prescription, this pertains to the AGB evolution from the point when the superwind sets in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Regardless of the prescription of mass-loss adopted, the ef- fects on the gas yields are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The time between the acti- vation of HBB and the achievement of the luminosity peak is suf- ficiently long, such that the surface chemistry of the stars reached the equilibrium distribution of the proton-capture nucleosynthe- sis corresponding to the temperature attained at the bottom of the convective mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' In this context, the chemical composition of the gas ejected into the interstellar medium is independent on the timescale with which the gas itself is released, which is de- termined by the mass-loss rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' We may safely conclude that the present investigation has led us to pose some questions regard- ing the reliability of the predicted silicate yields from intermedi- ate mass stars, but no conclusions regarding the gas yields from these objects can be drawn based on the present analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Conclusions We have studied a sample of Galactic AGB stars, whose SEDs exhibit deep absorption features centred at 10 µm and 18 µm, as- sociated with silicate dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The study of these stars, for which the pulsation periods and some information on the surface chemical composition are available, is based on stellar evolution and dust formation modelling, and supported by results from radiative transfer modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' These ingredients have been used to charac- terise the individual objects in an attempt to determine the mass and formation epoch of the progenitors, and, on more general grounds, to understand the silicate budget expected from AGB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The sources investigated are interpreted as the progeny of intermediate mass stars, currently evolving through advanced AGB phases, either currently experiencing HBB or after HBB 6 In the comparison with results from different studies, it must be con- sidered that the metallicity adopted here (Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='014) is smaller than the value used in Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2018) (Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='018) and that used by Nanni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' (2013) and Ferrarotti & Gail (2006) (Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='02) turned off by the gradual loss of the external envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' The study of their SED indicates mass-loss rates in the 1 − 2 × 10−4 M⊙/yr range, consistent with the radiation pressure wind description in which photons experience multiple scattering processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' These rates of mass loss of intermediate mass AGBs of solar metallic- ity are significantly higher (a factor ∼ 3) than those published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Provided that this picture is correct, we deduce that mass loss suffered by intermediate mass stars and the dust produc- tion mechanism remain highly efficient until the very final AGB phases, preceding the contraction to the post-AGB phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' These results indicate the need of an upwards revision of the dust yields by intermediate mass stars, which are now found in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='012−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='025 M⊙ range, mostly under the form of silicates, with a smaller contribution from alumina dust (∼ 10%) and solid iron (∼ 5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' EM acknowledges support from the INAF research project ’LBT - Supporto Arizona Italia’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' acknowledges the support of the Aus- tralian Research Council (ARC) Discovery Early Career Research Award (DE- CRA) grant (DE190100813) and the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' MF acknowledges financial support from the ASI-INAF agreement n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' 2022-14-HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content=' This research has made use of the Ga- iaPortal catalogues access tool, ASI?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='Space Science Data Center, Rome, Italy (http://gaiaportal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='ssdc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAzT4oBgHgl3EQfrv0i/content/2301.01647v1.pdf'} +page_content='asi.' metadata={'source': 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Optimization under Mean-Covariance Ambiguity +Set and Half-Space Support for Bivariate Problems +Jiayi Guoa, Hao Qiua, Zhen Wangb,c, Zizhuo Wangb,∗, Xinxin Zhanga +aSchool of Information Management and Engineering, Shanghai University of Finance and +Economics, Shanghai, 200433, P.R. China +bSchool of Data Science, The Chinese University of Hong Kong, Shen Zhen, Shenzhen, Guangdong, 518172, P.R. +China +cUniversity of Science and Technology of China, Hefei, Anhui, 230026, P.R. China +Abstract +In this paper, we study a bivariate distributionally robust optimization problem with mean- +covariance ambiguity set and half-space support. Under a conventional type of objective function +widely adopted in inventory management, option pricing, and portfolio selection, we obtain closed- +form tight bounds of the inner problem in six different cases. Through a primal-dual approach, +we identify the optimal distributions in each case. As an application in inventory control, we first +derive the optimal order quantity and the corresponding worst-case distribution, extending the +existing results in the literature. Moreover, we show that under the distributionally robust set- +ting, a centralized inventory system does not necessarily reduce the optimal total inventory, which +contradicts conventional wisdom. Furthermore, we identify two effects, a conventional pooling +effect, and a novel shifting effect, the combination of which determines the benefit of incorporat- +ing the covariance information in the ambiguity set. Finally, we demonstrate through numerical +experiments the importance of keeping the covariance information in the ambiguity set instead of +compressing the information into one dimension. +Keywords: +robustness and sensitivity analysis, distributionally robust optimization, bivariate +moment problem, mean-covariance ambiguity set, newsvendor +1. Introduction +In recent years, distributionally robust optimization (DRO) has become a popular approach to +address optimization problems affected by uncertainty. Its applications have received considerable +attention in various fields including economics, management science, and mathematical finance [3]. +∗Corresponding author +Email addresses: guo.jiayi@sufe.edu.cn (Jiayi Guo), qiu.hao@163.sufe.edu.cn (Hao Qiu), +wangzhen@cuhk.edu.cn (Zhen Wang), wangzizhuo@cuhk.edu.cn (Zizhuo Wang ), xinxin_zhang@163.sufe.edu.cn +(Xinxin Zhang) +Preprint submitted to +January 11, 2023 +arXiv:2301.03936v1 [math.OC] 10 Jan 2023 + +The appeal of distributionally robust optimization lies in its flexibility in specifying uncertainty +beyond a fixed probability distribution, as well as in its ability to produce computationally tractable +models [8]. We refer the readers to [34, 12, 17, 5, 35] for some comprehensive reviews for DRO +problems. +The problem studied in distributionally robust optimization is as follows: +inf +y∈Y sup +P∈F +EP [f (y, X)] +(A.1) +where X ∈ Rn is the realization of the uncertainty, y ∈ Y ⊆ Rm is the decision taken before the +realization, and f (y, X) : Rm × Rn → R is the objective function. Given the decision y, the inner +problem of (A.1) evaluates the worse-case objective: +sup +P∈F +EP [f (y, X)] , +(A.2) +where the ambiguity set, denoted by F, characterizes a set of possible distributions X may follow. +In the literature, various characterizations of ambiguity set have been considered, such as support +[14, 28], moments [34, 2], shape [29], and dispersions [20]. Among them, one of the most classical +ambiguity sets is the mean-covariance ambiguity set given as follows: +F = +� +P ∈ M(S) : EP[X] = µ, EP +� +XXT � += Σ +� +, +(A.3) +where µ, Σ are the mean and covariance of the distribution and S is the support. If S = Rn ++ or +S = Rn, then we call the support of F half-space or full-space respectively. The key advantage of +the mean-covariance ambiguity set lies in its simplicity and computational tractability, especially +under the full-space support. Generally speaking, the computational complexity of DRO problem +depends on the complexity of solving the inner problem, which is a semi-infinite linear program +on its probability measure, and NP-hard in general [2]. +In this paper, we consider a class of widely-adopted loss functions for the inner problem as +follows: +ℓ(X) = max{u1wT X + v1, u2wT X + v2}, +(A.4) +with X ∈ Rn, w ∈ Rn ++ and u1, u2, v1, v2 ∈ R. Such a loss function is adopted in a wide range of +applications, including inventory management [34, 11], option pricing [28, 38, 41, 1], and portfolio +selection [10, 27, 26, 40]. Regarding the ambiguity set, we focus on the mean-covariance ambi- +guity set, which is a common choice and has been studied in every application listed above (see, +for instance, [28, 7, 34, 1, 38]). Furthermore, we consider the problem under the half-space sup- +port, which is of relevance under many situations (e.g., nonnegative demand in inventory control +2 + +[18], nonnegative losses in a stop-loss contract [38]). Moreover, by incorporating the half-space +information, the ambiguity sets shrink, resulting in a less conservative optimal solution. +However, for problems with a mean-covariance ambiguity set, the methodology for solving the +DRO problem under half-space support is less well-studied than that under full-space support. +Specifically, when considering the problems with objectives of a linear reward function under the +full-space support, Popescu demonstrates a powerful projection property that can translate an +n−dimensional multivariate inner problem into an equivalent univariate problem [30]. Unfortu- +nately, it is not easy to incorporate half-space support information in such an approach. In fact, +as shown in [1], even determining the feasibility of such a problem under half-space support is +NP-hard. +Therefore, given mean and covariance information, much research on incorporating +half-space support into DRO focus on approximation algorithms. For example, Rujeerapaiboon et +al. [33] study the upper Chebyshev bound of a product of non-negative random variables given +their first two moments. They show that in order to obtain a tractable numerical procedure, the +first two moments should follow a permutation-symmetric structure. Kong et al. [22] formulate a +semidefinite programming relaxation for the appointment scheduling problem given the mean and +covariance information for the nonnegative random service time variables. Natarajan et al. [28] +provide a mathematically tractable lower bound for the expected piecewise linear utility function +by taking a convolution of two problems that are computationally simple through relaxing either +the half-space support to the full-space or the mean-covariance ambiguity set to the one with only +mean information. +In this paper, we further shed light on the DRO problem under mean-covariance ambiguity +set and half-space support. Particularly, we are able to derive an analytical solution to such a +problem for the two-dimensional case. Two-dimensional problem has been extensively studied in +the literature, as it is simple and useful to illustrate the relation between the optimal decisions and +the correlations of two random variables. For example, two-dimensional problems are considered in +the variations of newsvendor model including dual sourcing [39], two markets [23], and two products +[24, 25]. Moreover, the stop-loss problem in option pricing typically involves two types of losses +(i.e., property losses and liability losses) [38]. As far as we know, it is computationally challenging +even to solve two-dimensional DRO problems with mean-covariance ambiguity set under half- +space support. Specifically, a typical approach to solving those problems under half-space support +is to relax the support to the full-space [18, 9, 1]. +Moreover, as the projection theorem [30] +cannot be applied to those problems, both works [9, 1] obtain numerical solutions to the relaxed +problems. Govindarajan et al. achieve analytical solutions to the relaxed problem under strong +3 + +assumptions [18]. Unlike the relaxation of the support, Tian relaxes the two-dimensional problem +into a univariate one and applies the moment approach [38]. To sum up, the analytical result of a +two-dimensional problem with mean-covariance ambiguity set under half-space support is largely +missing in the literature, and our work fills in the gap with a conclusive answer. +We summarize the contributions of this paper as follows: +1. We propose an analytical solution for the two-dimensional DRO problem with mean-covariance +ambiguity set and loss function ℓ under half-space support. Specifically, the optimal value of +this problem can be characterized by six different cases. To obtain the optimal solution in +each case, we extend the primal-dual approach in [19], designed for the univariate moment +problems, to our two-dimensional problems. +2. We extend the loss function ℓ defined in (A.4) to a generalized multi-piece quadratic objective +function and provide a semi-definite programming reformulation for the DRO problem with +mean-covariance ambiguity set and the generalized loss function. +3. We apply our analytical result to inventory control problems, providing the optimal order +quantity and the worst-case distribution, which is an extension of the result in [34]. More- +over, we find that under the DRO setting with mean-covariance ambiguity set, inventory +pooling does not necessarily reduce the optimal total inventory, which contradicts with the +conventional wisdom that pooling can reduce the total inventory. Furthermore, we identify +two effects, a conventional pooling effect and a novel shifting effect, which together deter- +mine the benefit of incorporating the covariance information in the ambiguity set. Finally, +we demonstrate through numerical experiments the importance of keeping the covariance +information in the ambiguity set, instead of compressing the information in one dimension. +The rest of this paper is organized as follows. +In Section 2, we introduce our model. +In +Section 3, we formally present the bivariate moment problem and its closed-form solution. We +also demonstrate some properties of the worst-case distribution and provide a numerical approach +for the generalized objectives. In Section 4, we study an inventory control problem as the outer +problem and provide some managerial insights for adopting the mean-covariance DRO model for +such problems compared to other approaches. We conclude the paper and point out some future +research directions in Section 5. +4 + +2. Model +Consider the following distributionally robust optimization problem +inf +y∈Y sup +P∈F +EP [f (X, y)] +(A.5) +with feasible region Y ⊆ Rm, ambiguity set F, and loss function f : Rn × Rm → R. The loss +function depends both on the decision vector y ∈ Rm and the random vector X ∈ Rn. +We first investigate the inner worst-case expectation problem. For ease of notation, we sup- +press the dependence on the decision variable y. Specifically, we consider the inner worst-case +expectation problem of the form +sup +P∈F +EP +� +ℓ +� +wT X +�� +(A.6) +where the loss function ℓ : R → R is a convex piecewise linear function with the following form +ℓ(x) = max {u1x + v1, u2x + v2} , +and w ∈ Rn ++ is a nonnegative coefficient of the random vector X. The ambiguity set F of the +distribution P consists of nonnegative random vector X with given mean µ and second-order +moment Σ, that is +F = +� +P ∈ M(Rn ++) : EP[X] = µ, EP +� +XXT � += Σ +� +. +(A.7) +The problem of form (A.6) is widely adopted in DRO. We provide the following two applications +as examples. +Example 1. (Multi-dimensional Newsvendor Problem). We consider a multi-dimensional distri- +butionally robust newsvendor problem in a centralized inventory management setting (see [4]) as +follows: +inf +q≥0 +� +sup +P∈F +EP +�� +n +� +i=1 +Xi − q +� ++ +� ++ (1 − η)q +� +, +(A.8) +where we denote (·)+ = max{·, 0} in the rest of this paper. Specifically, q and η are the order +quantity and the critical ratio respectively, and �n +i=1 Xi represents the pooling of uncertain de- +mands. Obviously, the worst-case expectation of the inner problem can be considered as a special +case of problem (A.6) with u1 = 1, v1 = −q, u2 = v2 = 0 and w = 1. +Example 2. (Mean-CVaR Portfolio Selection Problem). We consider a capital market consisting +of n assets whose uncertain returns are captured by an n−dimensional random vector X. +A +5 + +portfolio is encoded by an n−dimensional vector w which has to satisfy some constraints w ∈ W. +The distributionally robust mean-CVaR portfolio selection problem can be formulated as follows: +inf +w∈W +� +sup +P∈F +� +EP +� +−wT X +� ++ ρ P-CVaRα +� +−wT X +� �� +or equivalently, +inf +w∈W +� +− wT µ + ρ inf +τ∈R +� +τ + 1 +α sup +P∈F +EP +�� +−wT X − τ +� ++ +� �� +where the equivalence is formally proved in [32]. Again, the worst-case expectation of the inner +problem can be considered as a special case of problem (A.6) with u1 = −1, v1 = −τ, and +u2 = v2 = 0. +Without loss of generality, we assume u1 < u2 in the rest of this paper. Otherwise, supposing +u1 = u2 = 0, the loss function ℓ(x) can be reduced to a linear function ℓ(x) = ux + max{v1, v2}. +As a result, problem (A.6) has a trivial solution as +EP[ℓ +� +wT X +� +] = u1EP[wT X] + max{v1, v2} = u1wT µ + max{v1, v2}. +In fact, with this assumption, we can simplify the formulation of problem (A.6) by the following +proposition. +Proposition 1. Suppose u1 < u2. We can reformulate the problem supP∈F EP +� +ℓ +� +wT X +�� +in +(A.6) as follows: +(u2 − u1) sup +P∈ ˜ +F +EP +�� +1T ˜X + v2 − v1 +u2 − u1 +� ++ +� ++ u1wT µ + v1 +with the transformed random variable ˜X = w ◦ X and the transformed ambiguity set +˜F = +� +P ∈ M(Rn ++) : EP[ ˜X] = w ◦ µ, EP +� ˜X ˜X +T � += wwT ◦ Σ +� +where the notation ◦ denotes the Hadamard product. +This proposition demonstrates that it is sufficient to solve the simplified problem in the form +of +sup +P∈F +EP +�� +n +� +i=1 +Xi − q +� ++ +� +(A.9) +so as to solve the original problem (A.6) with q = − v2−v1 +u2−u1 . We relegate the proof of Proposition 1 +to Appendix A. +Generally speaking, even determining the feasibility of problem (A.9) is NP-hard as shown in +[1]. For univariate case (n = 1), Scarf derives a closed-form solution in [34], which we state below +for completeness. +6 + +Lemma 1 ([34]). Let the ambiguity set F† = +� +P ∈ M(R+) : EP[X] = µ, EP[X2] = Σ +� +. +1. If 0 ≤ q ≤ Σ +2µ, then the optimal value is +sup +P∈F† EP[(X − q)+] = µ − q · µ2 +Σ +(A.10) +with an optimal distribution X∗ = +� +� +� +� +� +0 +w.p. 1 − µ2 +Σ +Σ +µ +w.p. +µ2 +Σ +. +2. If q > Σ +2µ, then the optimal value is +sup +P∈F† EP[(X − q)+] = 1 +2(Q − q + µ) +(A.11) +with an optimal distribution P∗ = +� +� +� +� +� +q − Q +w.p. +1 +2 + q−µ +2Q +q + Q +w.p. +1 +2 − q−µ +2Q +, where Q = +� +q2 − 2µq + Σ. +In this paper, we study the bivariate case (n = 2) of problem (A.9), which takes the correlation +between two random variables into account. +We provide a closed-form solution for the inner +problem (A.9), based on which an efficient algorithm for the outer DRO problem is also provided. +3. Bivariate Moment Problem +In this section, we study the bivariate case (n = 2) of problem (A.9) stated as follows: +vP (θ; q) = +sup +P∈F(θ) +EP +� +(X1 + X2 − q)+ +� +(A.12) +where θ = (µ1, µ2, Σ11, Σ22, Σ12) represents the moment information of random variables X1 and +X2. For simplicity, we denote (Σ11, Σ22, Σ12) := (aµ2 +1, bµ2 +2, cµ1µ2) for the rest of this paper, and +consider the ambiguity set +F(θ) = +� +� +� P ∈ M(R2 ++) +������ +EP[X1] = µ1, +EP[X2] = µ2 +EP +� +X2 +1 +� += aµ2 +1, EP +� +X2 +2 +� += bµ2 +2, EP [X1X2] = cµ1µ2 +� +� +�. +(A.13) +Thus, we have the correlation ρ = +c−1 +√ +(a−1)(b−1) and the covariance matrix +M = +� +� +(a − 1)µ2 +1 +(c − 1)µ1µ2 +(c − 1)µ1µ2 +(b − 1)µ2 +2 +� +� . +(A.14) +The nonemptyness of ambiguity set F(θ) requires that the covariance matrix M be positive semi- +definite, which means a ≥ 1, b ≥ 1 and (a − 1)(b − 1) ≥ (c − 1)2. Note that X1, X2 ∈ R+ and thus +c ≥ 0. To exclude trivial cases, we focus on a, b > 1. In all, without loss of generality, we assume +that our input parameters satisfy the following assumption in the remaining part of this paper. +7 + +Assumption 1. We assume a > 1, b > 1, c ≥ 0 and (a − 1)(b − 1) ≥ (c − 1)2. +Next, we present the closed-form solution for the bivariate moment problem (A.12). +3.1. Closed-Form Solution +Before establishing our closed-form solution, we first define three terms Qa, Qb, Qc that will be +frequently used in this subsection: +Qa = +� +q2 − 2q a − c +a − 1µ2 + ab − c2 +a − 1 µ2 +2, +Qb = +� +q2 − 2q b − c +b − 1µ1 + ab − c2 +b − 1 µ2 +1, +Qc = +� +q2 − 2q(µ1 + µ2) + aµ2 +1 + bµ2 +2 + 2cµ1µ2. +In the following lemma, we present the conditions that correspond to six different optimal values of +problem (A.12). Specifically, this lemma shows that such six conditions divide the feasible region +into six disjoint sets. +Lemma 2. Every feasible input of (θ, q) satisfies one and only one of the six conditions in the +following table. +Condition 1 +Condition 2 +Condition 3 +Condition 4 +Condition 5 +Condition 6 +Qa ≥ q, +Qb < q, +Qa < q, +Qb < q, +Qa < q, +Qa > |ζa|, +Qb ≥ q +Qb ≤ ζb +Qa ≤ ζa +Qb ≤ −ζb +Qa ≤ −ζa +Qb > |ζb| +where ζa = aµ1 + cµ2 − q and ζb = cµ1 + bµ2 − q. +We relegate the proof of this lemma to Appendix B.1. Below we provide a sample plot as a +graphical illustration. Specifically, we plot Figure A.1 with µ1 = 1, µ2 = 1.5, a+b = 3 and c = 0.7, +when ranging a and q over [1.1, 1.9] and [0, 10] respectively. Figure 1 shows that the feasible region +can be divided into six disjoint subsets, where borderlines are in the form of ¯A, A, ¯B, B, ¯C and C +that are functions of input (θ, q) defined in the proof of this lemma, see equation (B.5) in Appendix +B.1. +For each condition, the following theorem presents a closed-form optimal value of the bivariate +moment problem (A.12). +Theorem 1. Given a non-empty ambiguity set F(θ) and q > 0, the optimal value vP (q; θ) can be +8 + +Figure A.1: The Feasible Region is Divided into Six Conditions. +characterized as +vP (q; θ) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +µ1 + µ2 − q · a + b − 2c +ab − c2 +if Condition 1 holds; +b − 1 +2b +� +(q + Qb) − b − c +b − 1µ1 +� ++ µ1 + µ2 − q +if Condition 2 holds; +a − 1 +2a +� +(q + Qa) − a − c +2a µ2 +� ++ µ1 + µ2 − q +if Condition 3 holds; +b − 1 +2b +� b − c +b − 1µ1 − (q − Qb) +� +if Condition 4 holds; +a − 1 +2a +� a − c +a − 1µ2 − (q − Qa) +� +if Condition 5 holds; +1 +2(Qc − q + µ1 + µ2) +if Condition 6 holds. +We relegate the proof of this theorem to Appendix B. In this theorem, we characterize the +optimal value vP (q; θ) of problem (A.12) by six different cases. To obtain the optimal distributions, +we extend the primal-dual approach in [19], designed for the univariate moment problems, to our +two-dimensional problems, and successfully identify a corresponding optimal distribution for each +case stated in Lemma B.2.1-B.2.6. +There are several interesting observations of the optimal values in Theorem 1. First of all, the +optimal value under condition 6 is the same as that in a univariate pooling problem. Specifically, +let ¯X = X1 +X2 and we consider the univariate problem supP∈ ¯ +F1 EP[( ¯X −q)+] with the ambiguity +set +¯F1 = +� +P ∈ M(R+) : EP[ ¯X] = µ1 + µ2, EP[ ¯X2] = aµ2 +1 + bµ2 +2 + 2cµ1µ2 +� +. +We can verify that the optimal value under condition 6 is the same as that in this univariate +pooling problem via (A.11) in Lemma 1. +9 + +1.9 +3 +1.8 +A +B +B +1.7 +1.6 +6 +81.5 +1.4 +B +1.3 +2 +A +B +5 +1.2 +1.1 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +bSecondly, the optimal value under condition 2 and that under condition 3 are symmetric through +replacing b by a and µ1 by µ2. This symmetric property also applies to the optimal values under +condition 4 and condition 5, while the optimal values under condition 1 and condition 6 are self- +symmetric. +With respect to the optimal distributions, from Lemma B.2.2 and Lemma B.2.4 in Appendix +B, we can see that the optimal distributions under condition 2 and condition 4 share the same +formulation, though their optimal values are different. Specifically, both optimal distributions are +in the form of +� +� +� +� +� +� +� +� +� +� +� +� +� +x(1) = (q − Qb, 0) +w.p. p1 = b−1 +2b + q(b−1)+µ1(c−b) +2bQb +; +x(2) = (q + Qb, 0) +w.p. p2 = b−1 +2b − q(b−1)+µ1(c−b) +2bQb +; +x(3) = (cµ1, bµ2) +w.p. p3 = 1 +b. +To illustrate, we plot the optimal distribution under condition 2 in Figure 2(a). The points on the +0 q-Qb +q +q+Qb +q +cμ1 +bμ2 +Demand In +Item 1 +Demand In +Item 2 +(a) Condition 2 +q-Qb +q+Qb +q +cμ1 +bμ2 +q +0 +Demand In +Item 1 +Demand In +Item 1 +(b) Condition 4 +Figure A.2: The Three Support Points under Conditions 2 and 4 +solid line together with (q − Qb, 0) are potential points in the optimal support, as they satisfy the +complementary slackness conditions. In addition, condition 2 indicates that cµ1+bµ2−q ≥ Qb > 0, +so the objective value at point x(3) is strictly positive with +� +x(3) +1 ++ x(3) +2 +− q +� ++ = cµ1 + bµ2 − q > 0. +We also plot the optimal distribution under condition 4 in Figure 2(b). This condition indicates +that cµ1 + bµ2 − q ≤ −Qb < 0, which implies a zero objective value at x(3). In short, x(3) affects +the objective value under condition 2 but not under condition 4. +In fact, the optimal distributions in other conditions can also be interpreted similarly. Note +that the optimal distributions under conditions 3 and 5 also share the same formulation, as they +10 + +are symmetric to those under condition 2 and condition 4 respectively. Furthermore, notice that +the optimal distribution under condition 2 becomes invalid under condition 1, because x(1) +1 += +q − Qb < 0, which contradicts the non-negativity of random variables. When q shrinks from above +Qb to below Qb, condition 2 is switched to condition 1, which is consistent with the illustration +in Figure A.1 with q crossing A. Consequently, the optimal distribution under condition 1 always +takes (0, 0) in its optimal support from Lemma A.1. Lastly, as the optimal value under condition +6 is the same as that in the univariate pooling problem, it is not surprising to see each support +point x of optimal distributions lay on either x1 + x2 = q − Qc or x1 + x2 = q + Qc from Lemma +B.2.6, which is consistent with the supporting points of the univariate pooling problem via (A.11) +in Lemma 1. +3.2. Extension of the Loss Function +Beyond the two-piecewise linear objectives, recent literature also considers the multi-piece +quadratic loss functions in newsvendor models [15, 16]. +Specifically, quadratic losses are used +to measure the cost severity of critical perishable commodities [15]. Moreover, such multi-piece +quadratic functions are also employed in the robust portfolio selection problem to represent the +lower partial moment as a measure of risk [7]. +In this subsection, we study the bivariate moment problem under half-space support with a +multi-piece quadratic objective function. As it is computationally challenging to obtain such a +solution in closed form, we propose a numerical approach to solve this problem. Specifically, we +provide a semi-definite programming (SDP) formulation by applying the sum-of-squares technique +[31] to our dual problem. We relegate the proof to Appendix C. +Proposition 2. Given a random vector X ∈ M(R2) with its ambiguity set F(θ) defined in (A.13) +and a matrix of coefficients W ∈ R6×K for a K piece-wise quadratic objective, we denote each +piece as follows: +ℓk(X) = w1k + w2kX1 + w3kX2 + w4kX2 +1 + w5kX2 +2 + w6kX1X2 +for k ∈ {1, ..., K}. The following bivariate moment problem +sup +P∈F(θ) +EP +� +max +k=1,...,K {ℓk(X)} +� +can be solved by an SDP: +inf +z,G,H +z1 + µ1z2 + µ2z3 + Σ11z4 + Σ22z5 + Σ12z6 +s.t. +M +� +z − w(k), g(k), h(k)� +⪰ 0, +k = 1, ..., K +11 + +with z ∈ R6, G ∈ R3×K, H ∈ R3×K and the matrix M(˜z, g, h) defined as follows: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +˜z4 +0 +−g1 +0 +−h1 +−g2 +0 +˜z6 + 2g1 +0 +h1 +−h2 +−h3 +−g1 +0 +˜z5 +h2 +0 +−g3 +0 +h1 +h2 +˜z2 + 2g2 +h3 +0 +−h1 +−h2 +0 +h3 +˜z3 + 2g3 +0 +−g2 +−h3 +−g3 +0 +0 +˜z1 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +where w(k), g(k), h(k) are the kth column vectors of matrices W, G, H respectively. +As far as we know, it is mathematically challenging to obtain an exact reformulation for piece- +wise polynomial objectives with either a higher degree or more than two variables. This is because +the sum-of-squares technique may no longer apply in these settings. Specifically, each dual con- +straint of our problem is to ensure that the corresponding polynomial is non-negative. Indeed, our +analysis uses the result by Hilbert in 1888, who showed in [21] that every non-negative polynomial +with n variables and degree 2d could be represented as sums of squares of other polynomials if and +only if either n = 1 or 2d = 2 or n = 2 and 2d = 4. Note that the formulation of our proposition +is a bivariate polynomial (n = 2) with fourth-order (2d = 4). Therefore, the sum of squares tech- +nique can be applied. Similar challenges for a higher degree or more than two variables are also +presented by Bertsimas and Popescu, who briefly review the computational tractability in moment +problems [2]. +4. The Bivariate Newsvendor Problem +In the previous section, we provided a closed-form solution for the inner bivariate moment +problem. In this section, we study the outer DRO problem. Specifically, we consider the DRO +newsvendor problem as follows: +inf +q≥0 +� +sup +P∈F(θ) +EP +� +(X1 + X2 − q)+ +� ++ (1 − η)q +� +, +(A.15) +where F(θ) is the mean-covariance ambiguity set in (A.13), and the critical ratio η is a given +constant in (0, 1). +In the following, we first propose an approach to solve this problem in Section 4.1, which also +reveals the relation between the optimal order q∗ and the moment parameter θ in an analytical +form. In Section 4.2, we analyze the benefit of incorporating the covariance information into the +ambiguity set, as well as the impact on the total inventory. In Section 4.3, we show that it is +12 + +important to keep the covariance information instead of compressing such information into one +dimension. +4.1. Optimal Order Quantity +We introduce the following procedure in the box below to solve problem (A.15). In terms of the +computational techniques to solve q∗ +i in (A.16), we first note that each Ai essentially corresponds +to an interval of q, which can be solved from a quadratic equation. As a result, we obtain each Ai +in a closed form shown in Table B.2 in Appendix B.1. Furthermore, we note that vp(q; θ)+(1−η)q +is the sum of a linear term and a square root of a quadratic term of q, which is strictly convex in q. +Therefore, it is not difficult to obtain the unique stationary points of (A.16) shown in Appendix E. +Since an optimal solution locates at either a stationary point or a boundary point of the interval +Ai, (A.16) can be solved explicitly. To summarize, this framework is capable of solving the optimal +order q∗ systematically. +1. Identify local solutions separately: for each condition i of the six conditions in Theorem 1, solve +q∗ +i = arg min +q∈Ai +{vp(q; θ) + (1 − η)q} +(A.16) +where Ai = {q ∈ R+ : q satisfies condition i}. +2. Obtain global solutions jointly: solve +q∗ = arg min +q∈{q∗ +1,...,q∗ +6} +{vp(q; θ) + (1 − η)q} . +4.2. The Effects of Inventory Pooling +Inventory pooling is frequently employed as an operational strategy to mitigate demand uncer- +tainty: combining inventory allows the company to decrease demand variability, cut operational +costs, and boost profits, especially if the component market demands are negatively correlated [37]. +The pooling strategy often results in a centralized inventory system [4, 13, 36]. We illustrate the +difference between centralized and decentralized structures of inventory systems through Figure +A.3. In Figure 3(a), we show a single supplier serving two demand streams, which we call the +centralized system; in Figure 3(b), we show two suppliers deciding order quantities separately, +which we call the decentralized system. +From the perspective of pooling, we consider our model in (A.15) as a bivariate centralized model +(BCM). In the following, we compare the centralized model with the decentralized model. We first +13 + + +S1 +D1 +D2 +(a) Centralized System + +S1 +S2 +D1 +D2 +(b) Decentralized System +Figure A.3: Two Structures of Inventory Systems +denote individual ambiguity sets of demands X1 and X2 by +F1(θ) = +� +P ∈ M(R+) : EP[X1] = µ1, EP[X2 +1] = Σ11 +� +, +F2(θ) = +� +P ∈ M(R+) : EP[X2] = µ2, EP[X2 +2] = Σ22 +� +, +as the marginal ambiguity sets of F(θ) in (A.15). Note that these marginal ambiguity sets neglect +the covariance information in the original ambiguity set F(θ). Now we are ready to propose the +bivariate decentralized model (BDM) as follows: +inf +q1,q2≥0 {v1(q1; θ) + v2(q2; θ) + (1 − η)(q1 + q2)} , +(A.17) +where v1(q1; θ) and v2(q2; θ) are the optimal values of +sup +P∈F1(θ) +EP[(X1 − q1)+] +and +sup +P∈F2(θ) +EP[(X2 − q2)+] , +respectively. Note that we can decompose the BDM to be the sum of two marginal univariate +DRO problems, i.e., +inf +q1≥0 {v1(q1; θ) + (1 − η)q1} + inf +q2≥0 {v2(q2; θ) + (1 − η)q2} . +Therefore, the optimal solution of the BDM can be obtained through applying Scarf’s result [34] +on each individual univariate problem with +q∗ +i = +� +� +� +� +� +µi + +√ +Σii−µ2 +i +2 +2η−1 +√ +η(1−η), +if Σii−µ2 +i +Σii +< η < 1; +0, +if 0 ≤ η ≤ Σii−µ2 +i +Σii +, +for i = 1, 2. +In the rest of this subsection, we compare the optimal values and optimal solutions between +the BCM and the BDM. By doing so, we demonstrate the role of the correlation coefficient ρ +in the BCM over various critical ratios η. With regard to the optimal values, the BCM always +outperforms the BDM. Intuitively, the BDM is inclined to adopt an overly conservative decision, +14 + +since this model neglects the correlation between demands. As an illustration, we plot the relative +gap of optimal values between the BCM and the BDM over correlation coefficient ρ and critical +ratio η in Figure A.4, when fixing the other moment information with µ1 = 1, µ2 = 1, a = 2, b = 6. +Specifically, this relative gap is defined as the ratio: +κ(ρ, η) = VBDM(θ) − VBCM(θ) +VBCM(θ) +, +where VBCM and VBDM are optimal values of the BCM and the BDM, respectively. In Figure A.4, +Figure A.4: Relative Gap between BCM and BDM +it is not surprising to see κ(ρ, η) is relatively large when ρ is small. This phenomenon is consistent +with the well-known pooling effect, i.e., inventory pooling can significantly reduce operational costs +when demands are negatively correlated. Moreover, when η is small, both models tend to accept +a zero inventory so that κ(ρ, η) is zero as expected. +An unexpected observation is that the BCM also enjoys an advantage when ρ is close to 1 +and η is in the intermediate range. This is due to another effect which we call the shifting effect. +Before introducing the shifting effect, we first illustrate this observation with a concrete example +through further fixing η = 0.5. In such a case, we obtain VBDM = 2. For VBCM(ρ), by the method +introduced in Section 4.1, we have +VBCM(ρ) = 1 +5 +� +5 − 5ρ2 + +√ +5 +10 ρ + 3 +2. +As shown in Figure A.5, VBCM(ρ) is a concave function on [− +√ +5/5, 1]1 with the optimal value +VBCM(ρ∗) = 2 obtained at ρ∗ = +√ +5/5. +The increase of VBCM(ρ) over ρ ∈ [− +√ +5/5, +√ +5/5], as +1If ρ < − +√ +5/5, then Assumption 1 will be violated. +15 + +0.35 +60 +0.36 +0.30 +0.25 +0.20 +0.16 +0.12 +0.08 +0.05 +EO0 +0.25 +0.20 +0.15 +0.12 +0.08 +0.05 +EO0 +0.01 +0.00 +0.30 +10 +0.14 +0.10 +0.07 +0.05 +EO0 +0.01 +0.00 +0.00 +0.01 +0.25 +0.6 +0.09 +0.06 +0.04 +0.02 +0.01 +0.00 +0.00 +0.01 +EO'0 +0.20 +0.07 +0.04 +0.02 +0.01 +0.00 +0.00 +0.00 +0.01 +0.04 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.01 + 0.15 +EO +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.10 +- +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.05 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +-0.26 +-0.12 +0.b2 +a.16 +α.'3 +a.44 +0.58 +a.72 + 0.00 +pFigure A.5: VBDM and VBCM(ρ) when η = 0.5. +expected, is because of the pooling effect. On the other hand, the shifting effect dominates the +pooling effect over ρ ∈ ( +√ +5/5, 1], resulting in a decrease in the objective function of VBCM(ρ). To +explain the shifting effect, we first introduce the following proposition with ρ = 1, whose proof is +relegated to Appendix D. +Proposition 3. Suppose that two nonnegative random variables X1 and X2 satisfy +E[X1] = µ1, E[X2] = µ2, E[X2 +1] = aµ2 +1, E[X2 +2] = bµ2 +2, +and assume 1 ≤ a ≤ b without loss of generality. If X1 and X2 are perfectly correlated with ρ = 1, +then it must be X1 ≥ +� +1 − +� +a−1 +b−1 +� +µ1 ≥ 0. +From this proposition, we can see that a perfect correlation with ρ = 1 still benefits the BCM by +shifting the restriction of uncertain demand X1 from X1 ≥ 0 to X1 ≥ +� +1 − +� +(a − 1)/(b − 1) +� +µ1, +which is considered as the shifting effect. Consequently, this effect tightens the ambiguity set and +thus results in a less conservative decision. +To further illustrate the shifting effect, we note that a large ρ benefits the BCM by reducing +the probability of occurrences of a small demand realization, thus tightening the ambiguity set and +making the decision less conservative. As an illustration, we plot the upper bound of probability +P(X1 ≤ ξ) over correlation ρ in Figure A.6, when fixing the same parameters with µ1 = 1, µ2 = +1, a = 2, b = 6. On the upper left of the red dividing line, the values of maxF∈F(θ) P(X1 ≤ ξ) +remain the same as maxF∈F1(θ) P(X1 ≤ ξ). On the lower right of this red line, we observe that +max +F∈F(θ) P(X1 ≤ ξ) < +max +F∈F1(θ) P(X1 ≤ ξ), +16 + +21 - +VBCM(p) +VBDM +Objective +2D +15 +LB +17 +0.4 +.2 +O.D +0.2 +0.4 +0.6 +0.B +LD +p(a) The Value of maxF ∈F(θ) P(X1 ≤ ξ) +(b) The Value of maxF ∈F1(θ) P(X1 ≤ ξ) +Figure A.6: Shifting Effect +and the shifting effect takes place. Besides, for ξ ≤ 0.5, the shifting effect keeps getting stronger +as the value of ρ increases. In the extreme case when ρ = 1 and ξ ≤ 0.5 < 1 − +√ +0.2, Figure 6(a) +shows that maxF∈F(θ) P(X1 ≤ ξ) = 0, which verifies Proposition 3. +Another counter-intuitive observation is that a centralized system does not necessarily reduce +the optimal total order quantity, i.e., the optimal order quantity q∗ +BCM may be larger than q∗ +BDM +Figure A.7: Gap of Optimal Orders: q∗ +BDM − q∗ +BCM +with q∗ +BDM = q∗ +1 BDM + q∗ +2 BDM. We illustrate this observation in Figure A.7 with the same set of +moment parameters as in Figure A.6. In Figure A.7, a very large η (e.g., η ≥ 0.9) results in the +worst-case demand distribution taking condition 6 in Theorem 1. In such a case, the conventional +17 + +138 +1.20 +1.02 +0.86 +0.70 +0.55 +0.41 +0.27 +0.13 +LD +-1.85 +-1.86 +-1.81 +-1.72 +-1.57 +-1.36 +-1.09 +-0.71 +-0.18 +10 +0.66 +-0.63 +-0.57 +-0.49 +-0.40 +-0.28 +-0.13 +0.05 +0.30 +0.5 +0.6 +-0.43 +-0.38 +-0.32 +-0.25 +-0.17 +-0.08 +EO' +0.15 +0.31 +0.D +-1.33 +-1.28 +-1.21 +-1.15 +-1.08 +-1.00 +-0.92 +-0.83 +-0.73 +-1.07 +-0.00 +-0.00 +-0.00 +-0.00 +-0.00 +-0.00 +-0.00 +-0.59 +0.5 +-0.00 +-0.00 +-0.00 +-0.00 +-0.00 +-0.00 +-0.00 +-0.00 +-0.00 + 1.0 +-0.00 +-0.00 +-0.00 +-0.00 +-0.00 +-0.00 +-0.00 +-0.00 +-0.00 +1.5 +: +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +-0.26 +-0.12 +0.D2 +0.16 +0.3 +0.44 +0.58 +0.72 +0.B6 +pLD +9 +1000 +1.000 +1.000 +1000 +1000 +1.000 +1.000 +1.000 +1000 +0.962 +0.962 +0.962 +0.962 +0.962 +0.962 +0.962 +0.962 +0.962 + 0.B +8 +0.862 +0.862 +0.862 +0.862 +0.862 +0.862 +0.862 +0.862 +0.862 +50 +0.800 +0.800 +0.800 +0.800 +0.800 +0.795 +0.771 +0.733 +0.000 + 0.6 +加s +0.735 +0.735 +0.735 +0.734 +0.712 +0.660 +0.537 +0.409 +0.000 +EO +- 0.671 +0.671 +0.670 +0.656 +0.600 +0.506 +0.341 +0.221 +0.000 + 0.4 +0.610 +0.609 +0.601 +0.574 +0.492 +0.378 +0.222 +0.131 +0.000 +To +0.552 +0.548 +EES'O +0.496 +0.400 +0.284 +0.152 +0.085 +0.000 + 0.2 +00 +0.499 +0.491 +0.471 +0.426 +0.325 +0.218 +0.109 +0.060 +0.000 +0.5 +0.6 +0.7 +0.B +0.9 +0.95 +0.98 +0.99 +1D +p1D +oT +1000 +1.000 +1000 +1000 +1000 +1000 +1000 +1000 +1.000 +0.962 +0.962 +0.962 +0.962 +0.962 +0.962 +0.962 +0.962 +0.962 +- 0.B +0.6 +0.862 +0.862 +0.862 +0.862 +0.862 +0.862 +0.862 +0.862 +0.862 +0.800 +0.800 +0.800 +0.800 +0.800 +0.800 +0.800 +0.800 +0.800 +-0.6 +加sn +0.735 +0.735 +0.735 +0.735 +0.735 +0.735 +0.735 +0.735 +0.735 +E +- 0.671 +0.671 +0.671 +0.671 +0.671 +0.671 +0.671 +0.671 +0.671 +- 0.4 +- 0.610 +0.610 +0.610 +0.610 +0.610 +0.610 +0.610 +0.610 +0.610 +To +0.552 +0.552 +0.552 +0.552 +0.552 +0.552 +0.552 +0.552 +0.552 + 0.2 +00 +0.500 +0.500 +0.500 +0.500 +0.500 +0.500 +0.500 +0.500 +0.500 +0'5 +α'6 +α'7 +0.95 +0.98 +0.99 +LD +0.D +ppooling effects dominate [4, 13]. For an instance with η = 0.9, ρ = 0.3, the centralized optimal order +q∗ +BCM = 5.61, whereas the decentralized orders are q∗ +1 BDM = 2.33 and q∗ +2 BDM = 3.98 respectively +and pooling indeed reduces the optimal total order quantity. +However, for a medium-large η, +the individual optimal order quality could become 0 under sufficiently large demand uncertainty. +For an instance with η = 0.7, ρ = 0.3, the centralized optimal order q∗ +BCM = 1.84, whereas the +decentralized orders are q∗ +1 BDM = 1.44 and q∗ +2 BDM = 0 respectively. In some sense, demand pooling +can reduce demand variability and smooth the change of the optimal orders along with different +critical ratios η, and during this process, the optimal total order quantity under the centralized +system may be larger than that under the decentralized system. +4.3. The Effects of Ambiguity Pooling +For bivariate moment problems, a straightforward technique of relaxation, often considered in +the literature, is to model the problem as a univariate problem [38]. Such relaxation can simplify +the problem definition, reduce the solution complexity, and ease the analysis. More precisely, this +technique of relaxation results in a univariate centralized model (UCM) as follows: +inf +q≥0 +� +sup +P∈ ¯ +F(θ) +EP[( ¯X − q)+] + (1 − η)q +� +(A.18) +where we treat ¯X = X1 + X2 with ¯µ = µ1 + µ2, ¯Σ = Σ11 + 2Σ12 + Σ22, and define +¯F(θ) = {P ∈ M(R+) : EP[ ¯X] = ¯µ, EP +� ¯X2� += ¯Σ}. +Unlike the BDM which neglects the covariance, the UCM compresses the uncertainty of mean +and covariance into information in the one-dimensional space. We call such a maneuver ambiguity +pooling. +In the rest of this subsection, we demonstrate the effects of ambiguity pooling by comparing +the optimal value and ambiguity sets of the BCM and the UCM. In Figure A.8, we plot the +relative gap between the optimal values through fixing the same part of moment parameters with +µ1 = 1, µ2 = 1, a = 2, b = 6. As shown in Figure A.8, the BCM enjoys a 5% − 15% performance +improvement with a medium to large η. It is not surprising to see that the relative gap is zero +when η is small or very large: when η is small, both models will adopt a zero inventory, thus the +gap is zero; when η is very large, condition 6 in Theorem 1 holds, resulting in the same optimal +values for both models. +An essential observation is that the ambiguity pooling process may lead to a loss of information +on the uncertainty under half-space support. To be precise, we consider the following ambiguity +18 + +Figure A.8: Relative Gap between BCM and UCM +set +ˆF(θ) = +� +P ∈ M(R2) : EP[X1 + X2] = ¯µ, EP +� +(X1 + X2)2� += ¯Σ, X1 + X2 ≥ 0 +� +, +and the UCM can be reformulated as follows: +inf +q≥0 +� +sup +P∈ ˆ +F(θ) +EP +� +(X1 + X2 − q)+ +� ++ (1 − η)q +� +. +It is obvious that F(θ) ⊂ ˆF(θ), which means the ambiguity pooling process potentially results in +a more conservative decision. Next, we shall provide an example to demonstrate that the optimal +distribution of problem supP∈ ˆ +F EP[(X1 + X2 − q)+] does not belong to F; in other words, F(θ) is +a proper subset of ˆF(θ). We take µ1 = 1, µ2 = 1, a = 2, b = 6, c = 1 as an example, in which ¯µ = 2 +and ¯Σ = 10. Based on the Scarf bound in Lemma 1, a class of optimal distributions for problem +supP∈ ˆ +F EP[(X1 + X2 − q)+] with total K + 1 mass points can be described as follows: +ˆP∗ = +� +� +� +� +� +(0, 0), +w.p. p1 = 1 − ¯µ2 +¯Σ = 0.6 +(x(k) +1 , x(k) +2 ), +w.p. p(k) +2 , for all k = 1, ..., K +with �K +k=1 p(k) +2 += +¯µ2 +¯Σ = 0.4 and x(k) +1 ++ x(k) +2 += +¯Σ +¯µ = 5 for all k = 1, ..., K. To show ˆP∗ /∈ F by +contradiction, we first assume ˆP∗ ∈ F. Thus, the following constraints must hold for X1: +E[X1] = +K +� +k=1 +x(k) +1 p(k) +2 += µ1 = 1, and E[X2 +1] = 10 − 10 + 2 = 2. +As a result, the second moment of X2 must satisfy +E[X2 +2] = +K +� +k=1 +� ¯Σ +¯µ − x(k) +1 +�2 +p(k) +2 += +¯Σ2 +¯µ2 +K +� +k=1 +p(k) +2 +− 2 +¯Σ +¯µ E[X1] + E[X2 +1] = +� +1 − 2µ1 +¯µ +� +¯Σ + Σ11 = 2. +19 + +60 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.14 +0.00 +0.00 +0.00 +0.00 +0.00 +0.01 +0.01 +0.02 +0.04 + 0.12 +0.04 +0.05 +0.05 +0.06 +0.08 +0.09 +0.11 +0.13 +0.15 +0.10 +0.6 +0.09 +0.10 +0.10 +0.08 +0.07 +0.06 +0.06 +0.07 +0.09 +0.07 +0.04 +0.02 +0.01 +0.00 +0.00 +0.00 +0.01 +0.04 +0.08 +- +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.01 + 0.0-G +EO +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +too- +0.2 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 + 0.02 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +0.00 +-0.26 +-0.12 +0.b2 +a.16 +α'3 +0.44 +0.58 +a.72 +a.bs + 0.00 +pThis contradicts with E[X2 +2] = Σ22 = 6. Therefore, ˆP∗ /∈ F. +Lastly, we remark that the assumption on the half-space support plays an important role in the +above result. If the half-space support is replaced by the full-space support for both the BCM and +the UCM, then the two models will result in the same optimal values by the projection theorem +in [30]. +5. Conclusion and Further Discussion +To summarize, we study a bivariate distributionally robust optimization problem with the +mean-covariance ambiguity set and half-space support. For a class of widely-adopted objective +functions in inventory management, option pricing, and portfolio selection, we obtain closed- form +tight bounds and optimal distributions of the inner problem. Furthermore, we show that under +the distributionally robust setting, a centralized inventory system does not always reduce the +optimal total inventory. This contradicts with the belief that a centralized inventory system can +reduce the total inventory. In addition, we identify two effects, a conventional pooling effect and +a novel shifting effect. Their combination determines the benefit of incorporating the covariance +information in the ambiguity set. Finally, we demonstrate the importance of keeping the covariance +information in the ambiguity set instead of compressing the information into one dimension through +numerical experiments. +It is worth mentioning that our result of the bivariate moment problem is also useful to derive +a closed-form upper bound of the multivariate moment problem supP∈F EP +� +ℓ +� +wT X +�� +in (A.6) +with ℓ(x) = max {u1x + v1, u2x + v2}. Such multivariate problem is known to be computationally +challenging, and Natarajan et al. provide a mathematically tractable upper bound for the expected +piecewise linear utility function in a numerical way [28]. In consideration of our results, we treat the +n-dimensional random variable X as consisting of a sequence of low-dimensional random variables +Xi with X = (X1, X2, . . . , XI) and Xi ∈ Rni. In turn, we can decompose the ambiguity set +F into corresponding marginal ambiguity sets Fi. Based on such a decomposition, we obtain an +upper bound of the multivariate problem as follows: +sup +P∈F +EP +� +ℓ +� +wT X +�� +≤ +� +i∈I +sup +Pi∈Fi +EPi +� +max{u1wT +i Xi + v1i, u2wT +i Xi + v2i} +� +for all possible v1i and v2i satisfying �I +i=1 v1i = v1 and �I +i=1 v2i = v2. Therefore, if ni ≤ 2 for all i, +we can provide a closed-form upper bound. In terms of numerical algorithms, we can incorporate +�I +i=1 v1i = v1 and �I +i=1 v2i = v2 as two constraints to obtain a better upper bound. 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European Journal of Operational Research, 198(2):557–570. +24 + +Appendix A: Proof of Proposition 1 +Rewrite the loss function +ℓ(x) = max{u1x + v1, u2x + v2} += max{0, (u2 − u1)x + v2 − v1} + (u1x + v1) += +� +(u2 − u1)x + v2 − v1 +� ++ + (u1x + v1) += (u2 − u1) +� +x + v2 − v1 +u2 − u1 +� ++ ++ (u1x + v1). +Let ˜X = w ◦ X, then EP[ ˜X] = w ◦ µ and EP +� ˜X ˜X +T � += wwT ◦ Σ. Thus, the ambiguity set F of +random vector X is equivalent to the ambiguity set ˜F of random vector ˜X. Therefore, +sup +P∈F +EP +� +ℓ +� +wT X +�� += sup +P∈F +EP +� +(u2 − u1) +� +wT X + v2 − v1 +u2 − u1 +� ++ ++ +� +u1wT X + v1 +�� += (u2 − u1) sup +P∈F +EP +�� +wT X + v2 − v1 +u2 − u1 +� ++ +� ++ u1wT µ + v1 += (u2 − u1) sup +P∈ ˜ +F +EP +�� +1T ˜X + v2 − v1 +u2 − u1 +� ++ +� ++ u1wT µ + v1. +Appendix B: Proof of Theorem 1 +As proved in [1], problem supP∈F EP[ℓ(wT X)] is NP-hard for general cases. In this paper, we +study the bivariate case (n = 2). Specifically, we consider the following bivariate moment problem +vP (θ; q) = +sup +P∈F(θ) +EP +� +(X1 + X2 − q)+ +� +, +(B.1) +where θ = (µ1, µ2, Σ11, Σ22, Σ12) and the ambiguity set F(θ) is described as follows +F(θ) = {P ∈ M(R2 ++) : EP[X1] = µ1, EP[X2] = µ2, EP +� +X2 +1 +� += Σ11, EP +� +X2 +2 +� += Σ22, EP [X1X2] = Σ12}. +(B.2) +For simplicity, we denote (Σ11, Σ22, Σ12) := (aµ2 +1, bµ2 +2, cµ1µ2) for the rest of the Appendix. Thus, +the covariance matrix M and correlation ρ can be represented by +M = +� +� +(a − 1)µ2 +1 +(c − 1)µ1µ2 +(c − 1)µ1µ2 +(b − 1)µ2 +2 +� +� +and +ρ = +c − 1 +� +(a − 1)(b − 1) +. +(B.3) +Note that parameters a, b and c follow Assumption 1, which characterizes the feasibility of the +primal problem (B.1). To exclude the trivial cases in which the ambiguity set only contains the +distribution with a fixed single-point marginal distribution, without loss of generality, we assume +that a > 1 and b > 1 in the remaining discussion. +25 + +The dual of problem (B.1) is given as +vD(θ; q) = inf +z +z1 + z2µ1 + z3µ2 + z4aµ2 +1 + z5bµ2 +2 + z6cµ1µ2 +s.t. +h1(x; z) :=z1 + z2x1 + z3x2 + z4x2 +1 + z5x2 +2 + z6x1x2 ≥ 0, +∀(x1, x2) ∈ R2 ++ +h2(x; z) := q + z1 + (z2 − 1)x1 + (z3 − 1)x2 + z4x2 +1 + z5x2 +2 + z6x1x2 ≥ 0, +∀(x1, x2) ∈ R2 ++, +(B.4) +with the dual variable z = (z1, z2, z3, z4, z5, z6) ∈ R6. +Define six conditions of (θ, q) in Table B.1, +Condition 1 +Condition 2 +Condition 3 +Condition 4 +Condition 5 +Condition 6 +Qa ≥ q, +Qb < q, +Qa < q, +Qb < q, +Qa < q, +Qa > |ζa|, +Qb ≥ q +Qb ≤ ζb +Qa ≤ ζa +Qb ≤ −ζb +Qa ≤ −ζa +Qb > |ζb| +Table B.1: Six Conditions +where ζa = aµ1 + cµ2 − q and ζb = cµ1 + bµ2 − q, and +Qa = +� +q2 − 2q a − c +a − 1µ2 + ab − c2 +a − 1 µ2 +2, +Qb = +� +q2 − 2q b − c +b − 1µ1 + ab − c2 +b − 1 µ2 +1, +Qc = +� +q2 − 2q(µ1 + µ2) + aµ2 +1 + bµ2 +2 + 2cµ1µ2. +For simplicity of the later proof, we denote +A = ab − c2 +2(a − c)µ2, A = ab − c2 +2(b − c)µ1, B = A + D · (C − C) +2(a − c)C , B = A + D · (C − C) +2(b − c)C , +(B.5) +where +C = (a − 1)µ1 + (c − 1)µ2, C = (c − 1)µ1 + (b − 1)µ2, +(B.6) +and +D = (a − 1)(aµ1 + cµ2) > 0, D = (b − 1)(cµ1 + bµ2) > 0. +(B.7) +Note that A, B, C, D and A, B, C, D are symmetric if we replace µ1 by µ2 and a by b. +We also present four terms that will be frequently employed for the later analysis: +aµ1 + cµ2 − B = aC +2 + (a − 1)(b − 1) − (c − 1)2 +(a − 1)C +µ2 +2, +(B.8) +cµ1 + bµ2 − B = bC2 + (a − 1)(b − 1) − (c − 1)2 +(b − 1)C +µ2 +1, +(B.9) +A − A = +ab − c2 +(a − c)(b − c)(C − C), +(B.10) +B − B = (C − C) +CC +E, +(B.11) +26 + +where E = (a−1)cµ2 +1 + +� +ab−a−b+c2� +µ1µ2 +(b−1)cµ2 +2. Because of the condition (a−1)(b−1)− +(c−1)2 ≥ 0 in Assumption 1, it is easy to prove that E ≥ (a−1)cµ2 +1 +2(c−1)cµ1µ2 +(b−1)cµ2 +2 = +cVar(X1 + X2) > 0 . +In Appendix B.1, we prove parameters (θ, q) must satisfy one and precisely one of six conditions +in Table B.1. Then we provide the worst-case distributions for every condition and prove the +optimality in Appendix B.2. Through the analyses of the worst-case distribution for each condition, +Theorem 1 can be proved. +Appendix B.1. Proof of Lemma 2 +To show that every feasible input (θ, q) satisfies one and exactly one of six conditions, we first +show that conditions in Table B.1 are equivalent to those in (B.12-B.17). By simple algebra, we +can verify the following equivalency with +Q2 +a ≤ ζ2 +a ⇔ C(q − B) ≤ 0, Q2 +b ≤ ζ2 +b ⇔ C(q − B) ≤ 0, +Qa ≥ q ⇔ 2q(a − c) ≤ (ab − c2)µ2, Qb ≥ q ⇔ 2q(b − c) ≤ (ab − c2)µ1. +Thus, we can rewrite conditions in Table B.1 as follows: +condition 1: 2q(a − c) ≤ (ab − c2)µ2, 2q(b − c) ≤ (ab − c2)µ1; +(B.12) +condition 2: 2q(b − c) > (ab − c2)µ1, C(q − B) ≤ 0, ζb ≥ 0; +(B.13) +condition 3: 2q(a − c) > (ab − c2)µ2, C(q − B) ≤ 0, ζa ≥ 0; +(B.14) +condition 4: 2q(b − c) > (ab − c2)µ1, C(q − B) ≤ 0, ζb ≤ 0; +(B.15) +condition 5: 2q(a − c) > (ab − c2)µ2, C(q − B) ≤ 0, ζa ≤ 0; +(B.16) +condition 6: C(q − B) > 0, C(q − B) > 0. +(B.17) +Furthermore, according to the conditions in (B.12-B.17), we will discuss the feasible region of q +under each condition based on the input θ in different cases. The results are summarized in Table +B.2. Through this table, we can clearly see that the intersection of feasible regions of q in each +column is empty, and the union of those is R+. In addition, as the six cases of input θ are also +mutually exclusive, every feasible input (θ, q) satisfies one and exactly one of six conditions. Now, +we are ready to see how to obtain feasible regions of q in each case. +Case 1 (a > c > b): In this case, we have a > c > b > 1 based on b > 1 in Assumption 1, and +we can first verify that +C > 0, C > 0, and C − C = (a − c)µ1 + (c − b)µ2 > 0. +(B.18) +27 + +Cond +Cases +a > c > b +b > c > a +a > c, b > c +C ≥ C, C > 0 +C ≥ C, C < 0 +C ≤ C, C > 0 +C ≤ C, C < 0 +condition 1 +0 ≤ q ≤ A +0 ≤ q ≤ A +0 ≤ q ≤ A +0 ≤ q ≤ A +0 ≤ q ≤ A +0 ≤ q ≤ A +condition 2 +- +A < q ≤ B +- +- +A < q ≤ B +A < q ≤ B +condition 3 +A < q ≤ B +- +A < q ≤ B +A < q ≤ B +- +- +condition 4 +- +- +- +q ≥ B +- +- +condition 5 +- +- +- +- +- +q ≥ B +condition 6 +q > B +q > B +q > B +B < q < B +q > B +B < q < B +Table B.2: Feasible Regions of q in Different Cases +Condition 1 in (B.12) can be rewritten as +q ≤ ab − c2 +2(a − c)µ2 = A and q ≥ ab − c2 +2(b − c)µ1 = A. +According to H¨older’s inequality, we have (E[X1X2])2 ≤ E[X2 +1]E[X2 +2], i.e., +c2 ≤ ab. +(B.19) +Thus, A ≥ 0 and A ≤ 0 due to (B.19) and b < c. +In all, condition 1 can be represented as +0 ≤ q ≤ A, as q ∈ R+. +Condition 3 in (B.14) can be rewritten as +q > A, q ≤ B, and q ≤ aµ1 + cµ2, +because of C > 0 in (B.18) and a > c. According to (B.8), we have +aµ1 + cµ2 − B = aC +2 + (a − 1)(b − 1) − (c − 1)2 +(a − 1)C +µ2 +2 > 0, +(B.20) +due to C > 0 in (B.18) and (a − 1)(b − 1) ≥ (c − 1)2 in Assumption 1. In addition, we can verify +that B − A = D·(C−C) +2(a−c)C > 0 due to (B.18) and D > 0 in (B.7). In all, condition 3 can be simplified +as A < q ≤ B. +Condition 6 in (B.17) can be written as +q > B and q > B, +due to inequality (B.18). +Moreover, we have B > B, as B − B > 0 by equation (B.11) and +inequality (B.18). In all, condition 6 can be represented as q > max{B, B} = B. +28 + +Similar to the previous analysis, the feasible region of q under condition 2 in (B.13) and that +under condition 4 in (B.15) are empty with q < A < 0. Due to aµ1 + cµ2 − B > 0 in (B.20), +we know that q ≤ B and q ≥ aµ1 + cµ2 together are incompatible, which means that the feasible +region of q under condition 5 in (B.16) is also empty. +In summary, the space q ≥ 0 is fulfilled by condition 1, condition 3, and condition 6 in this +case, and all feasible regions are mutually exclusive, which are summarized in the first column of +Table B.2. +Case 2 (b > c > a): Note that the condition on the input θ under case 2 (b > c > a) is +symmetric to that under case 1 (a > c > b) in the sense of swapping a and b. In fact, condition +2 and condition 3 are symmetric; condition 4 and condition 5 are symmetric; condition 1 and +condition 6 are self-symmetric. Therefore, by the same analysis in case 1, we summarize the result +in the second column of Table B.2. +Case 3 (a > c, b > c, C ≥ C, C > 0): In this case, we can first verify that +A ≥ 0, A ≥ 0, A − A = +ab − c2 +(a − c)(b − c)(C − C) ≤ 0. +(B.21) +due to (B.19). +Condition 1 in (B.12) can be rewritten as +q ≤ A and q ≤ A, +due to a > c, b > c. As A ≤ A holds by (B.21) and q ∈ R+, condition 1 can be represented as +0 ≤ q ≤ A. +Condition 3 in (B.14) can be written as +q > A, q ≤ B, and q ≤ aµ1 + cµ2, +because of C > 0 and a > c. According to (B.8), we have aµ1 + cµ2 − B > 0 due to C ≥ C > 0 +in this case and (a − 1)(b − 1) ≥ (c − 1)2 in Assumption 1. +In addition, we can verify that +B − A = D·(C−C) +2(a−c)C ≥ 0 due to D > 0 in (B.7) and C ≥ C in this case. Thus, aµ1 + cµ2 > B ≥ A. +In all, the condition can be simplified as A < q ≤ B. +Condition 6 in (B.17) can be written as +q > B and q > B, +due to C ≥ C > 0. Moreover, we have B ≥ B, as B − B ≥ 0 by equation (B.11) and C ≥ C. In +all, condition 6 can be simplified as q > B. +29 + +Similar to the previous analysis, we can verify that B − A = D·(C−C) +2(b−c)C ≤ 0. Thus we know +that q ≤ B and q > A together are incompatible, which means that the feasible region of q under +condition 2 in (B.13) and condition 4 in (B.15) are empty. In addition, due to aµ1 + cµ2 − B > 0, +we know that q ≥ aµ1 + cµ2 and q ≤ B together are incompatible, which means that the feasible +region of q under condition 5 in (B.16) is also empty. +In summary, the space q ≥ 0 is fulfilled by condition 1, condition 3, and condition 6 in this +case, and all feasible regions are mutually exclusive, which are summarized in the third column of +Table B.2. +Case 4 (a > c, b > c, C ≥ C, C < 0): In this case, we first prove that C > 0 by contradiction. +Assume C ≤ 0 and C < 0, i.e., (a − 1)µ1 ≤ (1 − c)µ2 and (b − 1)µ2 < (1 − c)µ1. If we cancel +µ1 and µ2, it leads to (a − 1)(b − 1) < (1 − c)2, which contradicts the feasibility requirement in +Assumption 1. +Furthermore, we can verify that +A ≥ 0, A ≥ 0, A − A ≤ 0. +By the same analysis in case 3, the feasible regions of q under condition 1 and condition 3 are the +same as those in case 3. +Condition 4 in (B.15) can be written as +q > A, q ≥ B and q ≥ cµ1 + bµ2, +because of b > c and C < 0. According to (B.9), we have that cµ1 + bµ2 − B < 0 due to C < 0 +in this case and (a − 1)(b − 1) − (c − 1)2 ≥ 0 in Assumption 1. +In addition, we verify that +B − A = D·(C−C) +2(b−c)C ≥ 0 due to D > 0 in (B.7) and C < 0 together with C ≤ C in this case. In all, +condition 4 can be simplified as q ≥ B. +Condition 6 in (B.17) can be written as +q < B and q > B, +because of C > 0 and C < 0. Moreover, we have B ≤ B, as B − B ≤ 0 by equation (B.11) and +C ≥ C. In all, condition 6 can be simplified as B < q < B. +Similar to the previous analysis, due to cµ1+bµ2−B < 0, we know that q ≥ B and q ≤ cµ1+bµ2 +together are incompatible, which means that the feasible region of q under condition 2 in (B.13) +is empty. In addition, due to aµ1 + cµ2 − B > 0, we know that q ≤ B and q ≥ aµ1 + cµ2 together +are incompatible, which means that the feasible region of q under condition 5 in (B.16) is empty. +30 + +In summary, the space q ≥ 0 is fulfilled by condition 1, condition 3, condition 4, and condition +6 in this case, and all feasible regions are mutually exclusive, which are summarized in the fourth +column of Table B.2. +Case 5 (a > c, b > c, C < C, C > 0): Note that the condition about the input θ under case 5 +(a > c, b > c, C < C, C > 0) is symmetric to that under case 3 (a > c, b > c, C ≥ C, C > 0). As +the symmetry of condition 2 and condition 3 and the self-symmetry of condition 1 and condition +6, by the same analysis in case 3, we summarize the result in the fifth column of Table B.2. +Case 6 (a > c, b > c, C < C, C < 0): Note that the condition about the input θ under case +6 (a > c, b > c, C < C, C < 0) is symmetric to that under case 4 (a > c, b > c, C ≥ C, C < 0). +Note that condition 2 and condition 3 are symmetric; condition 4 and condition 5 are symmetric; +condition 1 and condition 6 are self-symmetric. Therefore, by the same analysis in case 4, we +summarize the result in the last column of Table B.2. +Appendix B.2. Proof of Theorem 1 +In this subsection, we provide the optimal values and optimal distributions under each condition +in Lemma B.2.1-B.2.6, and Theorem 1 follows immediately from these lemmas. +Before diving into details, we first present our intuitions on how to derive the optimal distri- +bution under each condition. Let z∗ be the dual optimal solution, so the dual feasibility implies +h1(x; z∗) ≥ 0 and h2(x; z∗) ≥ 0 for all x ∈ R2 ++ in (B.4). Moreover, the complementary slackness +conditions indicate that h1(x∗; z∗) = 0 or h2(x∗; z∗) = 0 for every x∗ in optimal support. Instead +of focusing on h1 and h2 directly, geometrically, we consider a curved surface S with +S = +� +(x1, x2, t) ∈ R3 : z∗ +1 + x1z∗ +2 + x2z∗ +3 + x2 +1z∗ +4 + x2 +2z∗ +5 + x1x2z∗ +6 = t +� +, +and a folded hyperplane T with +T = +� +(x1, x2, t) ∈ R3 ++ : max{x1 + x2 − q, 0} = t +� +. +Accordingly, h1, h2 ≥ 0 is equivalent to S above T, and h1 = 0 or h2 = 0 is equivalent to +S touching T. Through this interpretation, we can quickly analyze the characteristics of optimal +support. Based on these characteristics, we successfully identify an optimal distribution under each +condition. In this derivation, although we are motivated by the framework for solving univariate +moment problems introduced in [19], it also requires a lot of trial and error to obtain an optimal +distribution of our bivariate moment problem. As an illustration, we plot S and T with parameters +µ1 = µ2 = 1, a = 1.1, b = 1.015, ρ = 0.7, q = 1 in Figure B.1. In this figure, the points of the solid +31 + +Figure B.1: An illustration of S and T +line together with (0, q − Qa) are touching points of S and T. In fact, the curved surface S is a +paraboloid. +Now, we are ready to present six lemmas corresponding to each condition. In each proof, we +first verify the primal feasibility. Then, we provide a dual solution and show its feasibility. Lastly, +we verify the zero duality gap and thus the optimality. +Lemma B.2.1. Suppose that Qa ≥ q, Qb ≥ q, and Assumption 1 holds. The optimal distribution +for problem B.1 can be characterized as follows +(i) if b > c, +� +� +� +� +� +� +� +� +� +� +� +� +� +(0, 0), +w.p. p1 = (a−1)(b−1)−(c−1)2 +(ab−c2) +( ab−c2 +b−c µ1, 0), +w.p. p2 = +(b−c)2 +(ab−c2)b +(cµ1, bµ2), +w.p. p3 = 1 +b +(B.22) +(ii) if a > c, +� +� +� +� +� +� +� +� +� +� +� +� +� +(0, 0), +w.p. p1 = (a−1)(b−1)−(c−1)2 +(ab−c2) +(0, ab−c2 +a−c µ2), +w.p. p2 = +(a−c)2 +(ab−c2)a +(aµ1, cµ2), +w.p. p3 = 1 +a. +(B.23) +For both cases, the optimal values are all vP (q; θ) = µ1 + µ2 − q · a+b−2c +ab−c2 . +Proof of Lemma B.2.1. Recall the inequality ab − c2 ≥ 0 in (B.19). We now further assume b > c. +First, we shall verify the feasibility of the primal solution (B.22). Due to b > c and (a − 1)(b − +1)−(c−1)2 ≥ 0, we can verify the non-negativity of the primal supporting points and p1, p2, p3 > 0. +32 + +S +3 +2.5 +2 +O+b +B +t 1.5 +1 +O-b +0.5 +2 +0 +1.5 +0 +0.5 +0.5 +1.5 ++1 +2 +0Besides, it is easy to verify that p1 +p2 +p3 = 1. In addition, we can confirm the following moment +constraints hold, EP[X1] = µ1, EP[X2] = µ2, EP[X2 +1] = aµ2 +1, EP[X1X2] = cµ1µ2, EP[X2 +2] = bµ2 +2. In +all, the primal solution (B.22) is feasible for problem (B.1). +Second, we shall verify the dual feasibility of the following dual solution z∗ with +z∗ = +� +0, 1 − +2q(b − c) +µ1(ab − c2), 1 − 2q(a − c) +µ2(ab − c2), +q(b − c)2 +(ab − c2)2 µ2 +1 +, +q(a − c)2 +(ab − c2)2 µ2 +2 +, 2q(a − c)(b − c) +(ab − c2)2 µ1µ2 +� +. +(B.24) +We consider two dual constraint functions h1(x; z∗) and h2(x; z∗). Since z∗ +4 ≥ 0 and 4z∗ +4z∗ +5 −z∗ +6 +2 = +0, then the Hessian matrix +� +�2z∗ +4 +z∗ +6 +z∗ +6 +2z∗ +5 +� +� is positive semidefinite, which means that both functions +are convex. Next, we take the derivative of functions h1(x; z∗) and h2(x; z∗), +∇xh1(x; z∗) = (z∗ +2 + 2z∗ +4x1 + z∗ +6x2, z∗ +3 + 2z∗ +5x2 + z∗ +6x1)T +(B.25) +and +∇xh2(x; z∗) = (z∗ +2 − 1 + 2z∗ +4x1 + z∗ +6x2, z∗ +3 − 1 + 2z∗ +5x2 + z∗ +6x1)T . +(B.26) +Define x = (x1, x2)T , x∗ = (0, 0)T , x∗ = ( ab−c2 +b−c µ1, 0)T . Substitute x∗ into equation (B.25), and +we have +∇xh1(x∗; z∗)T (x − x∗) = (b − 1)(Q2 +b − q2) +(ab − c2)µ2 +1 +x1 + (a − 1)(Q2 +a − q2) +(ab − c2)µ2 +2 +x2. +Since a > 1 and b > 1 in Assumption 1, Qa ≥ q, Qb ≥ q in condition 1, and inequality (B.19), then +we have +∇xh1(x∗; z∗)T (x − x∗) ≥ 0, ∀x ∈ R2 ++. +Note that h1(x; z∗) is a convex function, which implies for any x ∈ R2 ++, +h1(x; z∗) ≥ h1(x∗; z∗) + ∇xh1(x∗; z∗)T (x − x∗) ≥ h1(x∗; z∗) = 0. +For function h2(x; z∗), substitute x∗ into equation (B.26), and we have +∇xh2(x∗; z∗) = +� +� − 2q(b−c) +µ1(ab−c2) + +2q(b−c)2 +(ab−c2)2µ2 +1 +ab−c2 +b−c µ1 +− 2q(a−c) +µ2(ab−c2) + 2q(a−c)(b−c) +(ab−c2)2µ1µ2 +ab−c2 +b−c µ1 +� +� = +� +�0 +0 +� +�. +Note that the function h2(x; z∗) is convex, which implies for any x ∈ R2 ++, +h2(x; z∗) ≥ h2(x∗; z∗) + ∇xh2(x∗; z∗)T (x − x∗) = h2(x∗; z∗) = 0. +In all, we prove that the dual solution z∗ is feasible for problem (B.4). +33 + +Lastly, it is easy to verify the objective values for the feasible primal-dual pair are all equal to +µ1 + µ2 − q · a+b−2c +ab−c2 . Hence, the zero duality gap implies the optimality of the feasible primal-dual +pair. +When a > c, the dual solution is the same as that of b > c in equation (B.24). By the similar +analysis of the b > c case, we can verify the distribution (B.23) is an optimal solution. The last +case is when a = b = c, in which X1 and X2 are perfectly correlated and Scarf’s bound in Lemma +1 will solve the problem. +Lemma B.2.2. Suppose that Qb < q, Qb ≤ ζb and Assumption 1 hold. Then an optimal distri- +bution for primal distribution for problem B.1 can be characterized as +� +� +� +� +� +� +� +� +� +� +� +� +� +(q − Qb, 0) +w.p. p1 = b−1 +2b + q(b−1)+µ1(c−b) +2bQb +(q + Qb, 0) +w.p. p2 = b−1 +2b − q(b−1)+µ1(c−b) +2bQb +(cµ1, bµ2) +w.p. p3 = 1 +b +, +(B.27) +where Qb = +� +q2 + 2q · c−b +b−1µ1 + ab−c2 +b−1 µ2 +1. The optimal value is vP (q; θ) = b−1 +2b ((q + Qb) − b−c +b−1µ1) + +µ1 + µ2 − q. +Proof of Lemma B.2.2. First, we shall verify the feasibility of the primal solution (B.27). Since +(a − 1)(b − 1) ≥ (c − 1)2 in Assumption 1, we have (ab − c2)(b − 1) − (b − c)2 ≥ (a + b − 2c)(b − +1) − (b − c)2 = (a − 1)(b − 1) − (c − 1)2 ≥ 0, and it further leads to +Q2 +b − (b − c +b − 1µ1 − q)2 = (ab − c2)(b − 1) − (b − c)2 +(b − 1)2 +µ2 +1 ≥ 0. +With Qb ≥ 0, we get Qb ≥ b−c +b−1µ1 − q and Qb ≥ q − b−c +b−1µ1. Now we rewrite p1 and p2 to be +p1 = b − 1 +2bQb +(Qb + q − µ1 +b − c +b − 1) and p2 = b − 1 +2bQb +(Qb − q + µ1 +b − c +b − 1). +Thus, p1 ≥ 0 is due to Qb ≥ b−c +b−1µ1 − q, and p2 ≥ 0 is due to Qb ≥ q − b−c +b−1µ1. Besides, p3 ≥ 0 +holds obviously. Furthermore, by straightforward calculations, we can also verify that the following +constraints hold, EP[1] = p1 + p2 + p3 = 1, EP[X1] = µ1, EP[X2] = µ2, EP[X2 +1] = aµ2 +1, EP[X1X2] = +cµ1µ2, EP[X2 +2] = bµ2 +2. In all, the primal solution (B.27) is feasible for problem (B.1). +Second, we shall verify the dual feasibility of the following dual solution +z∗ = +�(Qb − q)2 +4Qb +, Qb − q +2Qb +, 1 − Qb + q +2Qb +Qb + q − cµ1 +bµ2 +, +1 +4Qb +, (Qb + q − cµ1)2 +4b2µ2 +2Qb +, Qb + q − cµ1 +2bµ2Qb +� +. +We consider two dual constraint functions h1(x; z∗) and h2(x; z∗). Since z∗ +4 ≥ 0 and 4z∗ +4z∗ +5 −z∗ +6 +2 = +0, then the Hessian matrix +� +�2z∗ +4 +z∗ +6 +z∗ +6 +2z∗ +5 +� +� is positive semidefinite, which means that both functions +34 + +are convex. Next, we define x = (x1, x2)T , x∗ = (q − Qb, 0)T and x∗ = (q + Qb, 0)T . Then, we +have +∇xh1(x∗; z∗)T (x − x∗) = ζb − Qb +bµ2 +x2 ≥ 0, ∀x ∈ R2 ++. +Since h1(x; z∗) is a convex function, for any x ∈ R2 ++, we have h1(x; z∗) ≥ h1(x∗; z∗) = 0. For +function h2(x; z∗), substitute x∗ into equation (B.26), and we have ∇xh2(x∗; z∗) = (0, 0)T . Thus +h2(x; z∗) ≥ h2(x∗; z∗) = 0. In all, we prove that the dual solution z∗ is feasible for problem (B.4). +Lastly, it is easy to verify the objective values for the feasible primal-dual pair are all equal to +b−1 +2b +� +(q + Qb) − b−c +b−1µ1 +� ++ µ1 + µ2 − q. Hence, the zero duality gap implies the optimality of the +feasible primal-dual pair. +Lemma B.2.3. Suppose that Qa < q, Qa ≤ ζa and Assumption 1 hold. Then an optimal distri- +bution for primal distribution for problem B.1 can be characterized as +� +� +� +� +� +� +� +� +� +� +� +� +� +(0, q − Qa) +w.p. p1 = a−1 +2a + q(a−1)+µ2(c−a) +2aQa +(0, q + Qa) +w.p. p2 = a−1 +2a − q(a−1)+µ2(c−a) +2aQa +(aµ1, cµ2) +w.p. p3 = 1 +a +, +(B.28) +where Qa = +� +q2 + 2q · c−a +a−1µ2 + ab−c2 +a−1 µ2 +2. The optimal value is vP (q; θ) = a−1 +2a +� +(q + Qa) − a−c +2a µ2 +� ++ +µ1 + µ2 − q. +Proof of Lemma B.2.3. The proof is similar to that in Lemma B.2.2. Note that the conditions in +Lemma B.2.2 and B.2.3 are symmetric, in some sense, through replacing b by a and µ2 by µ1. +Lemma B.2.4. Suppose that Qb < q, Qb ≤ −ζb and Assumption 1 hold. +Then an optimal +distribution for primal distribution for problem B.1 can be characterized as +� +� +� +� +� +� +� +� +� +� +� +� +� +(q − Qb, 0) +w.p. p1 = b−1 +2b + q(b−1)+µ1(c−b) +2bQb +(q + Qb, 0) +w.p. p2 = b−1 +2b − q(b−1)+µ1(c−b) +2bQb +(cµ1, bµ2) +w.p. p3 = 1 +b +, +(B.29) +where Qb = +� +q2 + 2q · c−b +b−1µ1 + ab−c2 +b−1 µ2 +1. The optimal value is vP (q; θ) = b−1 +2b +� +b−c +b−1µ1 − (q − Qb) +� +. +Proof of Lemma B.2.4. First, we observe that the optimal primal solution is the same as that in +Lemma B.2.2, so the primal feasibility of this solution holds. +Second, we shall verify the dual feasibility of the following dual solution +z∗ = +�(Qb − q)2 +4Qb +, Qb − q +2Qb +, (Qb − q)(q − Qb − cµ1) +2bQbµ2 +, +1 +4Qb +, (q − Qb − cµ1)2 +4Qbb2 +, q − Qb − cµ1 +2bQbµ2 +� +. +35 + +We consider two dual constraint functions h1(x; z∗) and h2(x; z∗). Note that z∗ +4 ≥ 0 and 4z∗ +4z∗ +5 − +z∗ +6 +2 = 0, then the Hessian matrix +� +�2z∗ +4 +z∗ +6 +z∗ +6 +2z∗ +5 +� +� is positive semidefinite, which means that both +functions are convex. Next, we construct x∗ = (q − Qb, 0)T and x∗ = (q + Qb, 0)T . Then we have +∇xh1(x∗; z∗)T (x − x∗) = −Qb − ζb +bµ2 +x2 ≥ 0, ∀x ∈ R2 ++. +Since h1(x; z∗) is a convex function, for any x ∈ R2 ++, we have h1(x; z∗) ≥ h1(x∗; z∗) = 0. For +function h2(x; z∗), substitute x∗ into equation (B.26), and we have ∇xh2(x∗; z∗) = (0, 0)T . Thus, +h2(x; z∗) ≥ h2(x∗; z∗) = 0. In all, h1(x; z∗) ≥ 0 and h2(x; z∗) ≥ 0 hold for all x ∈ R2 ++. +Lastly, it is easy to verify the objective values for the feasible primal-dual pair equal to +b−1 +2b +� +b−c +b−1µ1 − (q − Qb) +� +. Hence, the zero duality gap implies the optimality of the feasible primal- +dual pair. +Lemma B.2.5. Suppose that 2(a − c)q > (ab − c2)µ2, Qa ≤ −cµ2 − aµ1 + q, and Assumption 1 +hold. Then an optimal distribution for primal distribution for problem B.1 can be characterized +as +� +� +� +� +� +� +� +� +� +� +� +� +� +(0, q − Qa) +w.p. p1 = a−1 +2a + q(a−1)+µ2(c−a) +2aQa +(0, q + Qa) +w.p. p2 = a−1 +2a − q(a−1)+µ2(c−a) +2aQa +(aµ1, cµ2) +w.p. p3 = 1 +a +, +(B.30) +where Qa = +� +q2 + 2q · c−a +a−1µ2 + ab−c2 +a−1 µ2 +2. The optimal value is vP (q; θ) = a−1 +2a +� +a−c +a−1µ2 − (q − Qa) +� +. +Proof of Lemma B.2.5. The proof is similar to that in Lemma B.2.4. Note that the conditions in +Lemma B.2.4 and B.2.5 are symmetric, in some sense, through replacing b by a and µ2 by µ1. +Lemma B.2.6. Suppose that Qa > |ζa|, Qb > |ζb| and Assumption 1 hold. Then an optimal +distribution of problem B.1 can be characterized as +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +x(1) = ((1 − t0) Ua +Uc µ1, t1) +w.p. p1 = Ubµ2 +2Qct1 +x(2) = (q − Qc, 0) +w.p. p2 = Uaµ1 +2Qct1 t0 +x(3) = (0, q + Qc) +w.p. p3 = Vb u2 +2Qct2 t0 +x(4) = (t2, (1 − t0) Vb +Vc µ2) +w.p. p4 = Vaµ1 +2Qct2 +(B.31) +where Ua = (q + Qc) − (aµ1 + cµ2), Va = (aµ1 + cµ2) − (q − Qc), Ub = (q + Qc) − (bµ2 + +cµ1), Vb = (bµ2 + cµ1) − (q − Qc), Uc = (q + Qc) − (µ1 + µ2), Vc = (µ1 + µ2) − (q − Qc), +Qc = +� +q2 − 2q(µ1 + µ2) + aµ2 +1 + bµ2 +2 + 2cµ1µ2, and +t1 = Ub +Uc +µ2 + t0 +Ua +Uc +µ1, +t2 = Va +Vc +µ1 + t0 +Vb +Vc +µ2, +t0 = +Det(M) +Det(M) + Σ12UcVc +, +36 + +with covariance matrix M defined in (B.3). The optimal value is vP (q; θ) = 1 +2Vc. +Proof of Lemma B.2.6. First, we shall verify the feasibility of the primal solution (B.31). Because +of the condition 6 with Qa > |q − aµ1 − cµ2|, Qb > |q − cµ1 − bµ2|, we have Va, Vb, Ua, Ub > 0 +immediately. Moreover, we also have Vc, Uc > 0 due to Q2 +c − (q − µ1 − µ2)2 = (a − 1)µ2 +1 + 2(c − +1)µ1µ2 + (b − 1)µ2 +2 = Var(X1 + X2) > 0. Thus, the distribution (B.31) has positive support and +probability. As +(p1 + p2) + (p3 + p4) = Ubµ2 + t0Uaµ1 +2Qct1 ++ t0Vbµ2 + Vaµ1 +2Qct2 += Uc +2Qc ++ Vc +2Qc += 1, +we conclude the primal solution (B.31) is a well-defined distribution. To verify this distribution +satisfies corresponding constraints, we state two useful equations q − Qc = Ua +Uc µ1 + Ub +Uc µ2, q + Qc = +Vb +Vc µ2 + Va +Vc µ1 that will be frequently employed. Specially, we have +E[X1] = +4 +� +i=1 +pix(i) +1 += µ1 +�(1 − t0)UaUbµ2 + t0U 2 +aµ1 + t0UaUbµ2 +2QcUct1 ++ Va +2Qc +� += µ1( Ua +2Qc ++ Va +2Qc +) = µ1, +E[X2] = +4 +� +i=1 +pix(i) +2 += µ2 +� Vb +2Qc ++ t0V 2 +b µ2 + t0VaVbµ1 + (1 − t0)VaVbµ1 +2QcVct2 +� += µ2( Ub +2Qc ++ Vb +2Qc +) = µ2, +E[X1X2] = +4 +� +i=1 +pix(i) +1 x(i) +2 += (1 − t0)µ1µ2 +2Qc +�UaUb +Uc ++ VaVb +Vc +� += (1 − t0)µ1µ2 +2Qc +� 2Qc +UcVc +Det(M) +µ1µ2 ++ 2cQc +� += cµ1µ2, +E[X1(X1 + X2)] = +4 +� +i=1 +pix(i) +1 (x(i) +1 + x(i) +2 ) = (q − Qc)[p1(q − Qc − t1)) + p2(q − Qc)] + p4(q + Qc)t2, += (q − Qc)(µ1 − p3x(3) +1 +− p4x(4) +1 ) + p4(q + Qc)t2 = (q − Qc)µ1 + 2Qcp4t2 = aµ2 +1 + cµ1µ2 +E[(X1 + X2)X2] = +4 +� +i=1 +pi(x(i) +1 + x(i) +2 )x(i) +2 += p1(q − Qc)t1 + (q + Qc)[p3(q + Qc) + p4(q + Qc − t2)], += p1(q − Qc)t1 + (q + Qc)(µ2 − p1x(1) +2 +− p2x(2) +2 ) = (q + Qc)µ2 − 2Qcp1t1 = bµ2 +2 + cµ1µ2. +In all, the primal feasibility holds. +Second, we shall verify the dual feasibility of the following dual solution +z∗ = +�(Qb − q)2 +4Qc +, Qc − q +2Qc +, Qc − q +2Qc +, +1 +4QC +, +1 +4QC +, +1 +2QC +� +. +By straightforward calculations, we have h1(x; z∗) = +1 +4Qc (x1 + x2 − q + Qc)2 ≥ 0 and h2(x; z∗) = +1 +4Qc (x1 + x2 − q − Qc)2 ≥ 0. Thus, z∗ is a dual feasible solution. +Lastly, it is easy to verify the objective values for the feasible primal-dual pair equal to 1 +2Vc. +Hence, the zero duality gap implies the optimality of the feasible primal-dual pair. +37 + +Appendix C: Proof of Proposition 2 +From Theorem 2.1 in [35], the bivariate moment problem +sup +P∈F(θ) +EP +� +max +k=1,...,K {ℓk(X)} +� +(C.1) +is equivalent to its dual problem +inf +z +z1 + µ1z2 + µ2z3 + Σ11z4 + Σ22z5 + Σ12z6 +s.t. +z1 + z2x1 + z3x2+z4x2 +1 + z5x2 +2 + z6x1x2 ≥ +max +k=1,...,m +� +w1k + w2kx1 + w3kx2 + w4kx2 +1 + w5kx2 +2 + w6kx1x2 +� +, +for all x ∈ R2 ++. Let y2 +1 = x1 and y2 +2 = x2, and then we have +z1 + z2x1 + z3x2 + z4x2 +1 + z5x2 +2 + z6x1x2 ≥ w1k + w2kx1 + w3kx2 + w4kx2 +1 + w5kx2 +2 + w6kx1x2, ∀x ∈ R2 ++ +⇔ +z1 − w1k + (z2 − w2k)y2 +1 + (z3 − w3k)y2 +2 + (z4 − w4k)y4 +1 + (z5 − w5k)y4 +2 + (z6 − w6k)y2 +1y2 +2 ≥ 0, ∀y ∈ R2. +The left-hand-side of the inequality can be represented by the sum of squares of polynomials, that +is, the above inequality is equivalent to +� +� +� +� +� +� +� +� +� +� +� +� +� +� +y2 +1 +y1y2 +y2 +2 +y1 +y2 +1 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +T +M +� +z − w(k), g(k), h(k)� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +y2 +1 +y1y2 +y2 +2 +y1 +y2 +1 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +≥ 0, +∀y ∈ R2 +⇔ +M +� +z − w(k), g(k), h(k)� +⪰ 0 +for some g(k) and h(k). Therefore, the original problem (C.1) can be reformulated by +inf +z,G,H +z1 + µ1z2 + µ2z3 + Σ11z4 + Σ22z5 + Σ12z6 +s.t. +M +� +z − w(k), g(k), h(k)� +⪰ 0, +k = 1, ..., m +with z ∈ R6, G ∈ R3×K, H ∈ R3×K and the matrix M(˜z, g, h) defined as follows: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +˜z4 +0 +−g1 +0 +−h1 +−g2 +0 +˜z6 + 2g1 +0 +h1 +−h2 +−h3 +−g1 +0 +˜z5 +h2 +0 +−g3 +0 +h1 +h2 +˜z2 + 2g2 +h3 +0 +−h1 +−h2 +0 +h3 +˜z3 + 2g3 +0 +−g2 +−h3 +−g3 +0 +0 +˜z1 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +where w(k), g(k), h(k) are the kth column vectors of matrices W, G, H respectively. +38 + +Appendix D: Proof of Proposition 3 +From Theorem 4.5.7 in book [6], we know that |ρ| = 1 if and only if there exists numbers +ϕ1 ̸= 0 and ϕ2 such that P(X1 = ϕ1X2 + ϕ2) = 1. If X1 and X2 are perfectly correlated with +ρ = 1, we have the following moment constraints due to X1 = ϕ1X2 + ϕ2 and ϕ1 > 0: +E[ϕ1X2 + ϕ2] = µ1, E[X2] = µ2, E[(ϕ1X2 + ϕ2)2] = aµ2 +1, E[X2 +2] = bµ2 +2. +From these equations, ϕ1 and ϕ2 should satisfy ϕ1µ2 + ϕ2 = µ1 and bϕ2 +1µ2 +2 + ϕ2 +2 + 2ϕ1ϕ2µ2 = aµ2 +1. +Then we can obtain ϕ1 = +� +a−1 +b−1 +µ1 +µ2 and ϕ2 = +� +1 − +� +a−1 +b−1 +� +µ1 and thus X1 = +� +a−1 +b−1 +µ1 +µ2 X2 + +� +1 − +� +a−1 +b−1 +� +µ1. Moreover, note that we assume 1 ≤ a ≤ b, then +� +1 − +� +a−1 +b−1 +� +µ1 ≥ 0. From the +nonnegativity of X2, we know that X1 ≥ +� +1 − +� +a−1 +b−1 +� +µ1. +Appendix E: Closed-Form Solution of the DRO Newsvendor Problem +We study the DRO newsvendor problem as follows: +inf +q≥0 +� +sup +P∈F(θ) +EP +� +(X1 + X2 − q)+ +� ++ (1 − η)q +� +, +where F(θ) is the mean-covariance ambiguity set in (B.2), and the critical ratio η is a given constant +in (0, 1). +We explain the procedure of solving the closed-form solution q∗ introduced in Section 4.1. +Firstly, each q∗ +i locates at either a stationary point of vP (q; θ) + (1 − η)q or a boundary point of +the interval Ai. The formulation of each interval Ai is discussed in Table B.2. Thus, we study the +stationary point of +f(q) = vP (q; θ) + (1 − η)q +in each interval Ai. For simplicity, we denote +Sa(η) = +� +(a − 1)(ab − c2) − (c − a)2 +4aη(a − aη − 1) +, +Sb(η) = +� +(b − 1)(ab − c2) − (c − b)2 +4bη(b − bη − 1) +, +Sc(η) = +� +(a − 1)µ2 +1 + (b − 1)µ2 +2 + 2(c − 1)µ1µ2 +4η(1 − η) +. +For condition 2 as an example, we have +f(q) = b − 1 +2b +� +q2 − 2q b − c +b − 1µ1 + ab − c2 +b − 1 µ2 +1 + +�b − 1 +2b +− η +� +q + b + c +2b µ1 + µ2 +39 + +with its derivative +f′(q) = b − 1 +2b +q − b−c +b−1µ1 +� +q2 − 2q b−c +b−1µ1 + ab−c2 +b−1 µ2 +1 ++ b − 1 +2b +− η. +Thus, the stationary point is obtained as follows by setting f′(q) = 0: +q = +� +(2bη − b + 1) +� +(b − 1)(ab − c2) − (c − b)2 +4bη(b − bη − 1) +− (c − b) +� +µ1 +b − 1 += [(2bη − b + 1)Sb(η) − (c − b)] +µ1 +b − 1, +which exists only if 0 < η < 1 − 1 +b. +The stationary points under other conditions can be derived by the same approach shown under +condition 2. We provide Table E.1 to summarize the closed-form stationary points in all cases. +Lastly, we remark that the stationary point shown in this table for each condition may be out of +the interval Ai. In all, given this table, it is not difficult to obtain the closed-form expressions of +the optimal solutions. +Conditions +Stationary Points +Feasible η +condition 1 +f(q) is linear in q ∈ A1 +condition 2 +[(2bη − b + 1)Sb(η) − (c − b)] µ1 +b−1, +0 < η < 1 − 1 +b +condition 3 +[(2aη − a + 1)Sa(η) − (c − a)] µ2 +a−1, +0 < η < 1 − 1 +a +condition 4 +[(2bη − b + 1)Sb(1 − η) − (c − b)] µ1 +b−1, +1 +b < η < 1 +condition 5 +[(2aη − a + 1)Sa(1 − η) − (c − a)] µ2 +a−1, +1 +a < η < 1 +condition 6 +(2η − 1)Sc(η) + µ1 + µ2, +0 < η < 1 +Table E.1: Stationary Points +40 + diff --git a/vdE2T4oBgHgl3EQfgQfX/content/tmp_files/load_file.txt b/vdE2T4oBgHgl3EQfgQfX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0baf63a28c085bcd9be377becd10a1da10915c2b --- /dev/null +++ b/vdE2T4oBgHgl3EQfgQfX/content/tmp_files/load_file.txt @@ -0,0 +1,1741 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf,len=1740 +page_content='Distributionally Robust Optimization under Mean-Covariance Ambiguity Set and Half-Space Support for Bivariate Problems Jiayi Guoa, Hao Qiua, Zhen Wangb,c, Zizhuo Wangb,∗, Xinxin Zhanga aSchool of Information Management and Engineering, Shanghai University of Finance and Economics, Shanghai, 200433, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' China bSchool of Data Science, The Chinese University of Hong Kong, Shen Zhen, Shenzhen, Guangdong, 518172, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' China cUniversity of Science and Technology of China, Hefei, Anhui, 230026, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' China Abstract In this paper, we study a bivariate distributionally robust optimization problem with mean- covariance ambiguity set and half-space support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Under a conventional type of objective function widely adopted in inventory management, option pricing, and portfolio selection, we obtain closed- form tight bounds of the inner problem in six different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Through a primal-dual approach, we identify the optimal distributions in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' As an application in inventory control, we first derive the optimal order quantity and the corresponding worst-case distribution, extending the existing results in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Moreover, we show that under the distributionally robust set- ting, a centralized inventory system does not necessarily reduce the optimal total inventory, which contradicts conventional wisdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Furthermore, we identify two effects, a conventional pooling effect, and a novel shifting effect, the combination of which determines the benefit of incorporat- ing the covariance information in the ambiguity set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Finally, we demonstrate through numerical experiments the importance of keeping the covariance information in the ambiguity set instead of compressing the information into one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Keywords: robustness and sensitivity analysis, distributionally robust optimization, bivariate moment problem, mean-covariance ambiguity set, newsvendor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Introduction In recent years, distributionally robust optimization (DRO) has become a popular approach to address optimization problems affected by uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Its applications have received considerable attention in various fields including economics, management science, and mathematical finance [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' ∗Corresponding author Email addresses: guo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='jiayi@sufe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='cn (Jiayi Guo), qiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='hao@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='sufe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='cn (Hao Qiu), wangzhen@cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='cn (Zhen Wang), wangzizhuo@cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='cn (Zizhuo Wang ), xinxin_zhang@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='sufe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='cn (Xinxin Zhang) Preprint submitted to January 11, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='03936v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='OC] 10 Jan 2023 The appeal of distributionally robust optimization lies in its flexibility in specifying uncertainty beyond a fixed probability distribution, as well as in its ability to produce computationally tractable models [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We refer the readers to [34, 12, 17, 5, 35] for some comprehensive reviews for DRO problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The problem studied in distributionally robust optimization is as follows: inf y∈Y sup P∈F EP [f (y, X)] (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1) where X ∈ Rn is the realization of the uncertainty, y ∈ Y ⊆ Rm is the decision taken before the realization, and f (y, X) : Rm × Rn → R is the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Given the decision y, the inner problem of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1) evaluates the worse-case objective: sup P∈F EP [f (y, X)] , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2) where the ambiguity set, denoted by F, characterizes a set of possible distributions X may follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In the literature, various characterizations of ambiguity set have been considered, such as support [14, 28], moments [34, 2], shape [29], and dispersions [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Among them, one of the most classical ambiguity sets is the mean-covariance ambiguity set given as follows: F = � P ∈ M(S) : EP[X] = µ, EP � XXT � = Σ � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='3) where µ, Σ are the mean and covariance of the distribution and S is the support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' If S = Rn + or S = Rn, then we call the support of F half-space or full-space respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The key advantage of the mean-covariance ambiguity set lies in its simplicity and computational tractability, especially under the full-space support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Generally speaking, the computational complexity of DRO problem depends on the complexity of solving the inner problem, which is a semi-infinite linear program on its probability measure, and NP-hard in general [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In this paper, we consider a class of widely-adopted loss functions for the inner problem as follows: ℓ(X) = max{u1wT X + v1, u2wT X + v2}, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='4) with X ∈ Rn, w ∈ Rn + and u1, u2, v1, v2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Such a loss function is adopted in a wide range of applications, including inventory management [34, 11], option pricing [28, 38, 41, 1], and portfolio selection [10, 27, 26, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Regarding the ambiguity set, we focus on the mean-covariance ambi- guity set, which is a common choice and has been studied in every application listed above (see, for instance, [28, 7, 34, 1, 38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Furthermore, we consider the problem under the half-space sup- port, which is of relevance under many situations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=', nonnegative demand in inventory control 2 [18], nonnegative losses in a stop-loss contract [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Moreover, by incorporating the half-space information, the ambiguity sets shrink, resulting in a less conservative optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' However, for problems with a mean-covariance ambiguity set, the methodology for solving the DRO problem under half-space support is less well-studied than that under full-space support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Specifically, when considering the problems with objectives of a linear reward function under the full-space support, Popescu demonstrates a powerful projection property that can translate an n−dimensional multivariate inner problem into an equivalent univariate problem [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Unfortu- nately, it is not easy to incorporate half-space support information in such an approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In fact, as shown in [1], even determining the feasibility of such a problem under half-space support is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Therefore, given mean and covariance information, much research on incorporating half-space support into DRO focus on approximation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' For example, Rujeerapaiboon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' [33] study the upper Chebyshev bound of a product of non-negative random variables given their first two moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' They show that in order to obtain a tractable numerical procedure, the first two moments should follow a permutation-symmetric structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' [22] formulate a semidefinite programming relaxation for the appointment scheduling problem given the mean and covariance information for the nonnegative random service time variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Natarajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' [28] provide a mathematically tractable lower bound for the expected piecewise linear utility function by taking a convolution of two problems that are computationally simple through relaxing either the half-space support to the full-space or the mean-covariance ambiguity set to the one with only mean information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In this paper, we further shed light on the DRO problem under mean-covariance ambiguity set and half-space support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Particularly, we are able to derive an analytical solution to such a problem for the two-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Two-dimensional problem has been extensively studied in the literature, as it is simple and useful to illustrate the relation between the optimal decisions and the correlations of two random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' For example, two-dimensional problems are considered in the variations of newsvendor model including dual sourcing [39], two markets [23], and two products [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Moreover, the stop-loss problem in option pricing typically involves two types of losses (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=', property losses and liability losses) [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' As far as we know, it is computationally challenging even to solve two-dimensional DRO problems with mean-covariance ambiguity set under half- space support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Specifically, a typical approach to solving those problems under half-space support is to relax the support to the full-space [18, 9, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Moreover, as the projection theorem [30] cannot be applied to those problems, both works [9, 1] obtain numerical solutions to the relaxed problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Govindarajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' achieve analytical solutions to the relaxed problem under strong 3 assumptions [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Unlike the relaxation of the support, Tian relaxes the two-dimensional problem into a univariate one and applies the moment approach [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' To sum up, the analytical result of a two-dimensional problem with mean-covariance ambiguity set under half-space support is largely missing in the literature, and our work fills in the gap with a conclusive answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We summarize the contributions of this paper as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We propose an analytical solution for the two-dimensional DRO problem with mean-covariance ambiguity set and loss function ℓ under half-space support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Specifically, the optimal value of this problem can be characterized by six different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' To obtain the optimal solution in each case, we extend the primal-dual approach in [19], designed for the univariate moment problems, to our two-dimensional problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We extend the loss function ℓ defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='4) to a generalized multi-piece quadratic objective function and provide a semi-definite programming reformulation for the DRO problem with mean-covariance ambiguity set and the generalized loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We apply our analytical result to inventory control problems, providing the optimal order quantity and the worst-case distribution, which is an extension of the result in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' More- over, we find that under the DRO setting with mean-covariance ambiguity set, inventory pooling does not necessarily reduce the optimal total inventory, which contradicts with the conventional wisdom that pooling can reduce the total inventory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Furthermore, we identify two effects, a conventional pooling effect and a novel shifting effect, which together deter- mine the benefit of incorporating the covariance information in the ambiguity set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Finally, we demonstrate through numerical experiments the importance of keeping the covariance information in the ambiguity set, instead of compressing the information in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In Section 2, we introduce our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In Section 3, we formally present the bivariate moment problem and its closed-form solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We also demonstrate some properties of the worst-case distribution and provide a numerical approach for the generalized objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In Section 4, we study an inventory control problem as the outer problem and provide some managerial insights for adopting the mean-covariance DRO model for such problems compared to other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We conclude the paper and point out some future research directions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Model Consider the following distributionally robust optimization problem inf y∈Y sup P∈F EP [f (X, y)] (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5) with feasible region Y ⊆ Rm, ambiguity set F, and loss function f : Rn × Rm → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The loss function depends both on the decision vector y ∈ Rm and the random vector X ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We first investigate the inner worst-case expectation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' For ease of notation, we sup- press the dependence on the decision variable y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Specifically, we consider the inner worst-case expectation problem of the form sup P∈F EP � ℓ � wT X �� (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6) where the loss function ℓ : R → R is a convex piecewise linear function with the following form ℓ(x) = max {u1x + v1, u2x + v2} , and w ∈ Rn + is a nonnegative coefficient of the random vector X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The ambiguity set F of the distribution P consists of nonnegative random vector X with given mean µ and second-order moment Σ, that is F = � P ∈ M(Rn +) : EP[X] = µ, EP � XXT � = Σ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='7) The problem of form (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6) is widely adopted in DRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We provide the following two applications as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (Multi-dimensional Newsvendor Problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We consider a multi-dimensional distri- butionally robust newsvendor problem in a centralized inventory management setting (see [4]) as follows: inf q≥0 � sup P∈F EP �� n � i=1 Xi − q � + � + (1 − η)q � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='8) where we denote (·)+ = max{·, 0} in the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Specifically, q and η are the order quantity and the critical ratio respectively, and �n i=1 Xi represents the pooling of uncertain de- mands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Obviously, the worst-case expectation of the inner problem can be considered as a special case of problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6) with u1 = 1, v1 = −q, u2 = v2 = 0 and w = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (Mean-CVaR Portfolio Selection Problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We consider a capital market consisting of n assets whose uncertain returns are captured by an n−dimensional random vector X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' A 5 portfolio is encoded by an n−dimensional vector w which has to satisfy some constraints w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The distributionally robust mean-CVaR portfolio selection problem can be formulated as follows: inf w∈W � sup P∈F � EP � −wT X � + ρ P-CVaRα � −wT X � �� or equivalently, inf w∈W � − wT µ + ρ inf τ∈R � τ + 1 α sup P∈F EP �� −wT X − τ � + � �� where the equivalence is formally proved in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Again, the worst-case expectation of the inner problem can be considered as a special case of problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6) with u1 = −1, v1 = −τ, and u2 = v2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Without loss of generality, we assume u1 < u2 in the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Otherwise, supposing u1 = u2 = 0, the loss function ℓ(x) can be reduced to a linear function ℓ(x) = ux + max{v1, v2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' As a result, problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6) has a trivial solution as EP[ℓ � wT X � ] = u1EP[wT X] + max{v1, v2} = u1wT µ + max{v1, v2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In fact, with this assumption, we can simplify the formulation of problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6) by the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Suppose u1 < u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We can reformulate the problem supP∈F EP � ℓ � wT X �� in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6) as follows: (u2 − u1) sup P∈ ˜ F EP �� 1T ˜X + v2 − v1 u2 − u1 � + � + u1wT µ + v1 with the transformed random variable ˜X = w ◦ X and the transformed ambiguity set ˜F = � P ∈ M(Rn +) : EP[ ˜X] = w ◦ µ, EP � ˜X ˜X T � = wwT ◦ Σ � where the notation ◦ denotes the Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' This proposition demonstrates that it is sufficient to solve the simplified problem in the form of sup P∈F EP �� n � i=1 Xi − q � + � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='9) so as to solve the original problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6) with q = − v2−v1 u2−u1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We relegate the proof of Proposition 1 to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Generally speaking, even determining the feasibility of problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='9) is NP-hard as shown in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' For univariate case (n = 1), Scarf derives a closed-form solution in [34], which we state below for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 6 Lemma 1 ([34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Let the ambiguity set F† = � P ∈ M(R+) : EP[X] = µ, EP[X2] = Σ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' If 0 ≤ q ≤ Σ 2µ, then the optimal value is sup P∈F† EP[(X − q)+] = µ − q · µ2 Σ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='10) with an optimal distribution X∗ = � � � � � 0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 1 − µ2 Σ Σ µ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' µ2 Σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' If q > Σ 2µ, then the optimal value is sup P∈F† EP[(X − q)+] = 1 2(Q − q + µ) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='11) with an optimal distribution P∗ = � � � � � q − Q w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 1 2 + q−µ 2Q q + Q w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 1 2 − q−µ 2Q , where Q = � q2 − 2µq + Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In this paper, we study the bivariate case (n = 2) of problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='9), which takes the correlation between two random variables into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We provide a closed-form solution for the inner problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='9), based on which an efficient algorithm for the outer DRO problem is also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Bivariate Moment Problem In this section, we study the bivariate case (n = 2) of problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='9) stated as follows: vP (θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' q) = sup P∈F(θ) EP � (X1 + X2 − q)+ � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='12) where θ = (µ1, µ2, Σ11, Σ22, Σ12) represents the moment information of random variables X1 and X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' For simplicity, we denote (Σ11, Σ22, Σ12) := (aµ2 1, bµ2 2, cµ1µ2) for the rest of this paper, and consider the ambiguity set F(θ) = � � � P ∈ M(R2 +) ������ EP[X1] = µ1, EP[X2] = µ2 EP � X2 1 � = aµ2 1, EP � X2 2 � = bµ2 2, EP [X1X2] = cµ1µ2 � � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='13) Thus, we have the correlation ρ = c−1 √ (a−1)(b−1) and the covariance matrix M = � � (a − 1)µ2 1 (c − 1)µ1µ2 (c − 1)µ1µ2 (b − 1)µ2 2 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='14) The nonemptyness of ambiguity set F(θ) requires that the covariance matrix M be positive semi- definite, which means a ≥ 1, b ≥ 1 and (a − 1)(b − 1) ≥ (c − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Note that X1, X2 ∈ R+ and thus c ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' To exclude trivial cases, we focus on a, b > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In all, without loss of generality, we assume that our input parameters satisfy the following assumption in the remaining part of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 7 Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We assume a > 1, b > 1, c ≥ 0 and (a − 1)(b − 1) ≥ (c − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Next, we present the closed-form solution for the bivariate moment problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Closed-Form Solution Before establishing our closed-form solution, we first define three terms Qa, Qb, Qc that will be frequently used in this subsection: Qa = � q2 − 2q a − c a − 1µ2 + ab − c2 a − 1 µ2 2, Qb = � q2 − 2q b − c b − 1µ1 + ab − c2 b − 1 µ2 1, Qc = � q2 − 2q(µ1 + µ2) + aµ2 1 + bµ2 2 + 2cµ1µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In the following lemma, we present the conditions that correspond to six different optimal values of problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Specifically, this lemma shows that such six conditions divide the feasible region into six disjoint sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Every feasible input of (θ, q) satisfies one and only one of the six conditions in the following table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Condition 1 Condition 2 Condition 3 Condition 4 Condition 5 Condition 6 Qa ≥ q, Qb < q, Qa < q, Qb < q, Qa < q, Qa > |ζa|, Qb ≥ q Qb ≤ ζb Qa ≤ ζa Qb ≤ −ζb Qa ≤ −ζa Qb > |ζb| where ζa = aµ1 + cµ2 − q and ζb = cµ1 + bµ2 − q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We relegate the proof of this lemma to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Below we provide a sample plot as a graphical illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Specifically, we plot Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1 with µ1 = 1, µ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5, a+b = 3 and c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='7, when ranging a and q over [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='9] and [0, 10] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Figure 1 shows that the feasible region can be divided into six disjoint subsets, where borderlines are in the form of ¯A, A, ¯B, B, ¯C and C that are functions of input (θ, q) defined in the proof of this lemma, see equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5) in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' For each condition, the following theorem presents a closed-form optimal value of the bivariate moment problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Given a non-empty ambiguity set F(θ) and q > 0, the optimal value vP (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ) can be 8 Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1: The Feasible Region is Divided into Six Conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' characterized as vP (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ) = � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � µ1 + µ2 − q · a + b − 2c ab − c2 if Condition 1 holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' b − 1 2b � (q + Qb) − b − c b − 1µ1 � + µ1 + µ2 − q if Condition 2 holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' a − 1 2a � (q + Qa) − a − c 2a µ2 � + µ1 + µ2 − q if Condition 3 holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' b − 1 2b � b − c b − 1µ1 − (q − Qb) � if Condition 4 holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' a − 1 2a � a − c a − 1µ2 − (q − Qa) � if Condition 5 holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 1 2(Qc − q + µ1 + µ2) if Condition 6 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We relegate the proof of this theorem to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In this theorem, we characterize the optimal value vP (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ) of problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='12) by six different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' To obtain the optimal distributions, we extend the primal-dual approach in [19], designed for the univariate moment problems, to our two-dimensional problems, and successfully identify a corresponding optimal distribution for each case stated in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' There are several interesting observations of the optimal values in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' First of all, the optimal value under condition 6 is the same as that in a univariate pooling problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Specifically, let ¯X = X1 +X2 and we consider the univariate problem supP∈ ¯ F1 EP[( ¯X −q)+] with the ambiguity set ¯F1 = � P ∈ M(R+) : EP[ ¯X] = µ1 + µ2, EP[ ¯X2] = aµ2 1 + bµ2 2 + 2cµ1µ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We can verify that the optimal value under condition 6 is the same as that in this univariate pooling problem via (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='11) in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='9 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='8 A B B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6 6 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='4 B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='3 2 A B 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1 0 1 2 3 4 5 6 7 8 9 10 bSecondly, the optimal value under condition 2 and that under condition 3 are symmetric through replacing b by a and µ1 by µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' This symmetric property also applies to the optimal values under condition 4 and condition 5, while the optimal values under condition 1 and condition 6 are self- symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' With respect to the optimal distributions, from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2 and Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='4 in Appendix B, we can see that the optimal distributions under condition 2 and condition 4 share the same formulation, though their optimal values are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Specifically, both optimal distributions are in the form of � � � � � � � � � � � � � x(1) = (q − Qb, 0) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p1 = b−1 2b + q(b−1)+µ1(c−b) 2bQb ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' x(2) = (q + Qb, 0) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p2 = b−1 2b − q(b−1)+µ1(c−b) 2bQb ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' x(3) = (cµ1, bµ2) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p3 = 1 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' To illustrate, we plot the optimal distribution under condition 2 in Figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The points on the 0 q-Qb q q+Qb q cμ1 bμ2 Demand In Item 1 Demand In Item 2 (a) Condition 2 q-Qb q+Qb q cμ1 bμ2 q 0 Demand In Item 1 Demand In Item 1 (b) Condition 4 Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2: The Three Support Points under Conditions 2 and 4 solid line together with (q − Qb, 0) are potential points in the optimal support, as they satisfy the complementary slackness conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In addition, condition 2 indicates that cµ1+bµ2−q ≥ Qb > 0, so the objective value at point x(3) is strictly positive with � x(3) 1 + x(3) 2 − q � + = cµ1 + bµ2 − q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We also plot the optimal distribution under condition 4 in Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' This condition indicates that cµ1 + bµ2 − q ≤ −Qb < 0, which implies a zero objective value at x(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In short, x(3) affects the objective value under condition 2 but not under condition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In fact, the optimal distributions in other conditions can also be interpreted similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Note that the optimal distributions under conditions 3 and 5 also share the same formulation, as they 10 are symmetric to those under condition 2 and condition 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Furthermore, notice that the optimal distribution under condition 2 becomes invalid under condition 1, because x(1) 1 = q − Qb < 0, which contradicts the non-negativity of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' When q shrinks from above Qb to below Qb, condition 2 is switched to condition 1, which is consistent with the illustration in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1 with q crossing A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Consequently, the optimal distribution under condition 1 always takes (0, 0) in its optimal support from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Lastly, as the optimal value under condition 6 is the same as that in the univariate pooling problem, it is not surprising to see each support point x of optimal distributions lay on either x1 + x2 = q − Qc or x1 + x2 = q + Qc from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6, which is consistent with the supporting points of the univariate pooling problem via (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='11) in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Extension of the Loss Function Beyond the two-piecewise linear objectives, recent literature also considers the multi-piece quadratic loss functions in newsvendor models [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Specifically, quadratic losses are used to measure the cost severity of critical perishable commodities [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Moreover, such multi-piece quadratic functions are also employed in the robust portfolio selection problem to represent the lower partial moment as a measure of risk [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In this subsection, we study the bivariate moment problem under half-space support with a multi-piece quadratic objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' As it is computationally challenging to obtain such a solution in closed form, we propose a numerical approach to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Specifically, we provide a semi-definite programming (SDP) formulation by applying the sum-of-squares technique [31] to our dual problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We relegate the proof to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Given a random vector X ∈ M(R2) with its ambiguity set F(θ) defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='13) and a matrix of coefficients W ∈ R6×K for a K piece-wise quadratic objective, we denote each piece as follows: ℓk(X) = w1k + w2kX1 + w3kX2 + w4kX2 1 + w5kX2 2 + w6kX1X2 for k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=', K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The following bivariate moment problem sup P∈F(θ) EP � max k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=',K {ℓk(X)} � can be solved by an SDP: inf z,G,H z1 + µ1z2 + µ2z3 + Σ11z4 + Σ22z5 + Σ12z6 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' M � z − w(k), g(k), h(k)� ⪰ 0, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=', K 11 with z ∈ R6, G ∈ R3×K, H ∈ R3×K and the matrix M(˜z, g, h) defined as follows: � � � � � � � � � � � � � � ˜z4 0 −g1 0 −h1 −g2 0 ˜z6 + 2g1 0 h1 −h2 −h3 −g1 0 ˜z5 h2 0 −g3 0 h1 h2 ˜z2 + 2g2 h3 0 −h1 −h2 0 h3 ˜z3 + 2g3 0 −g2 −h3 −g3 0 0 ˜z1 � � � � � � � � � � � � � � where w(k), g(k), h(k) are the kth column vectors of matrices W, G, H respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' As far as we know, it is mathematically challenging to obtain an exact reformulation for piece- wise polynomial objectives with either a higher degree or more than two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' This is because the sum-of-squares technique may no longer apply in these settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Specifically, each dual con- straint of our problem is to ensure that the corresponding polynomial is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Indeed, our analysis uses the result by Hilbert in 1888, who showed in [21] that every non-negative polynomial with n variables and degree 2d could be represented as sums of squares of other polynomials if and only if either n = 1 or 2d = 2 or n = 2 and 2d = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Note that the formulation of our proposition is a bivariate polynomial (n = 2) with fourth-order (2d = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Therefore, the sum of squares tech- nique can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Similar challenges for a higher degree or more than two variables are also presented by Bertsimas and Popescu, who briefly review the computational tractability in moment problems [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The Bivariate Newsvendor Problem In the previous section, we provided a closed-form solution for the inner bivariate moment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In this section, we study the outer DRO problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Specifically, we consider the DRO newsvendor problem as follows: inf q≥0 � sup P∈F(θ) EP � (X1 + X2 − q)+ � + (1 − η)q � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='15) where F(θ) is the mean-covariance ambiguity set in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='13), and the critical ratio η is a given constant in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In the following, we first propose an approach to solve this problem in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1, which also reveals the relation between the optimal order q∗ and the moment parameter θ in an analytical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2, we analyze the benefit of incorporating the covariance information into the ambiguity set, as well as the impact on the total inventory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='3, we show that it is 12 important to keep the covariance information instead of compressing such information into one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Optimal Order Quantity We introduce the following procedure in the box below to solve problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In terms of the computational techniques to solve q∗ i in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='16), we first note that each Ai essentially corresponds to an interval of q, which can be solved from a quadratic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' As a result, we obtain each Ai in a closed form shown in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Furthermore, we note that vp(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ)+(1−η)q is the sum of a linear term and a square root of a quadratic term of q, which is strictly convex in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Therefore, it is not difficult to obtain the unique stationary points of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='16) shown in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Since an optimal solution locates at either a stationary point or a boundary point of the interval Ai, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='16) can be solved explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' To summarize, this framework is capable of solving the optimal order q∗ systematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Identify local solutions separately: for each condition i of the six conditions in Theorem 1, solve q∗ i = arg min q∈Ai {vp(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ) + (1 − η)q} (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='16) where Ai = {q ∈ R+ : q satisfies condition i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Obtain global solutions jointly: solve q∗ = arg min q∈{q∗ 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=',q∗ 6} {vp(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ) + (1 − η)q} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The Effects of Inventory Pooling Inventory pooling is frequently employed as an operational strategy to mitigate demand uncer- tainty: combining inventory allows the company to decrease demand variability, cut operational costs, and boost profits, especially if the component market demands are negatively correlated [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The pooling strategy often results in a centralized inventory system [4, 13, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We illustrate the difference between centralized and decentralized structures of inventory systems through Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In Figure 3(a), we show a single supplier serving two demand streams, which we call the centralized system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' in Figure 3(b), we show two suppliers deciding order quantities separately, which we call the decentralized system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' From the perspective of pooling, we consider our model in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='15) as a bivariate centralized model (BCM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In the following, we compare the centralized model with the decentralized model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We first 13 S1 D1 D2 (a) Centralized System S1 S2 D1 D2 (b) Decentralized System Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='3: Two Structures of Inventory Systems denote individual ambiguity sets of demands X1 and X2 by F1(θ) = � P ∈ M(R+) : EP[X1] = µ1, EP[X2 1] = Σ11 � , F2(θ) = � P ∈ M(R+) : EP[X2] = µ2, EP[X2 2] = Σ22 � , as the marginal ambiguity sets of F(θ) in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Note that these marginal ambiguity sets neglect the covariance information in the original ambiguity set F(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Now we are ready to propose the bivariate decentralized model (BDM) as follows: inf q1,q2≥0 {v1(q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ) + v2(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ) + (1 − η)(q1 + q2)} , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='17) where v1(q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ) and v2(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ) are the optimal values of sup P∈F1(θ) EP[(X1 − q1)+] and sup P∈F2(θ) EP[(X2 − q2)+] , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Note that we can decompose the BDM to be the sum of two marginal univariate DRO problems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=', inf q1≥0 {v1(q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ) + (1 − η)q1} + inf q2≥0 {v2(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ) + (1 − η)q2} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Therefore, the optimal solution of the BDM can be obtained through applying Scarf’s result [34] on each individual univariate problem with q∗ i = � � � � � µi + √ Σii−µ2 i 2 2η−1 √ η(1−η), if Σii−µ2 i Σii < η < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 0, if 0 ≤ η ≤ Σii−µ2 i Σii , for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In the rest of this subsection, we compare the optimal values and optimal solutions between the BCM and the BDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' By doing so, we demonstrate the role of the correlation coefficient ρ in the BCM over various critical ratios η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' With regard to the optimal values, the BCM always outperforms the BDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Intuitively, the BDM is inclined to adopt an overly conservative decision, 14 since this model neglects the correlation between demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' As an illustration, we plot the relative gap of optimal values between the BCM and the BDM over correlation coefficient ρ and critical ratio η in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='4, when fixing the other moment information with µ1 = 1, µ2 = 1, a = 2, b = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Specifically, this relative gap is defined as the ratio: κ(ρ, η) = VBDM(θ) − VBCM(θ) VBCM(θ) , where VBCM and VBDM are optimal values of the BCM and the BDM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='4, Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='4: Relative Gap between BCM and BDM it is not surprising to see κ(ρ, η) is relatively large when ρ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' This phenomenon is consistent with the well-known pooling effect, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=', inventory pooling can significantly reduce operational costs when demands are negatively correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Moreover, when η is small, both models tend to accept a zero inventory so that κ(ρ, η) is zero as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' An unexpected observation is that the BCM also enjoys an advantage when ρ is close to 1 and η is in the intermediate range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' This is due to another effect which we call the shifting effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Before introducing the shifting effect, we first illustrate this observation with a concrete example through further fixing η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In such a case, we obtain VBDM = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' For VBCM(ρ), by the method introduced in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1, we have VBCM(ρ) = 1 5 � 5 − 5ρ2 + √ 5 10 ρ + 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' As shown in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5, VBCM(ρ) is a concave function on [− √ 5/5, 1]1 with the optimal value VBCM(ρ∗) = 2 obtained at ρ∗ = √ 5/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The increase of VBCM(ρ) over ρ ∈ [− √ 5/5, √ 5/5], as 1If ρ < − √ 5/5, then Assumption 1 will be violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='35 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='36 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='00 pFigure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5: VBDM and VBCM(ρ) when η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' expected, is because of the pooling effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' On the other hand, the shifting effect dominates the pooling effect over ρ ∈ ( √ 5/5, 1], resulting in a decrease in the objective function of VBCM(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' To explain the shifting effect, we first introduce the following proposition with ρ = 1, whose proof is relegated to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Suppose that two nonnegative random variables X1 and X2 satisfy E[X1] = µ1, E[X2] = µ2, E[X2 1] = aµ2 1, E[X2 2] = bµ2 2, and assume 1 ≤ a ≤ b without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' If X1 and X2 are perfectly correlated with ρ = 1, then it must be X1 ≥ � 1 − � a−1 b−1 � µ1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' From this proposition, we can see that a perfect correlation with ρ = 1 still benefits the BCM by shifting the restriction of uncertain demand X1 from X1 ≥ 0 to X1 ≥ � 1 − � (a − 1)/(b − 1) � µ1, which is considered as the shifting effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Consequently, this effect tightens the ambiguity set and thus results in a less conservative decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' To further illustrate the shifting effect, we note that a large ρ benefits the BCM by reducing the probability of occurrences of a small demand realization, thus tightening the ambiguity set and making the decision less conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' As an illustration, we plot the upper bound of probability P(X1 ≤ ξ) over correlation ρ in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6, when fixing the same parameters with µ1 = 1, µ2 = 1, a = 2, b = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' On the upper left of the red dividing line, the values of maxF∈F(θ) P(X1 ≤ ξ) remain the same as maxF∈F1(θ) P(X1 ≤ ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' On the lower right of this red line, we observe that max F∈F(θ) P(X1 ≤ ξ) < max F∈F1(θ) P(X1 ≤ ξ), 16 21 - VBCM(p) VBDM Objective 2D 15 LB 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='B LD p(a) The Value of maxF ∈F(θ) P(X1 ≤ ξ) (b) The Value of maxF ∈F1(θ) P(X1 ≤ ξ) Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6: Shifting Effect and the shifting effect takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Besides, for ξ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5, the shifting effect keeps getting stronger as the value of ρ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In the extreme case when ρ = 1 and ξ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5 < 1 − √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2, Figure 6(a) shows that maxF∈F(θ) P(X1 ≤ ξ) = 0, which verifies Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Another counter-intuitive observation is that a centralized system does not necessarily reduce the optimal total order quantity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=', the optimal order quantity q∗ BCM may be larger than q∗ BDM Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='7: Gap of Optimal Orders: q∗ BDM − q∗ BCM with q∗ BDM = q∗ 1 BDM + q∗ 2 BDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We illustrate this observation in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='7 with the same set of moment parameters as in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='7, a very large η (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=', η ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='9) results in the worst-case demand distribution taking condition 6 in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In such a case, the conventional 17 138 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='86 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='99 LD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='D ppooling effects dominate [4, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' For an instance with η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='9, ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='3, the centralized optimal order q∗ BCM = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='61, whereas the decentralized orders are q∗ 1 BDM = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='33 and q∗ 2 BDM = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='98 respectively and pooling indeed reduces the optimal total order quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' However, for a medium-large η, the individual optimal order quality could become 0 under sufficiently large demand uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' For an instance with η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='7, ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='3, the centralized optimal order q∗ BCM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='84, whereas the decentralized orders are q∗ 1 BDM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='44 and q∗ 2 BDM = 0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In some sense, demand pooling can reduce demand variability and smooth the change of the optimal orders along with different critical ratios η, and during this process, the optimal total order quantity under the centralized system may be larger than that under the decentralized system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The Effects of Ambiguity Pooling For bivariate moment problems, a straightforward technique of relaxation, often considered in the literature, is to model the problem as a univariate problem [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Such relaxation can simplify the problem definition, reduce the solution complexity, and ease the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' More precisely, this technique of relaxation results in a univariate centralized model (UCM) as follows: inf q≥0 � sup P∈ ¯ F(θ) EP[( ¯X − q)+] + (1 − η)q � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='18) where we treat ¯X = X1 + X2 with ¯µ = µ1 + µ2, ¯Σ = Σ11 + 2Σ12 + Σ22, and define ¯F(θ) = {P ∈ M(R+) : EP[ ¯X] = ¯µ, EP � ¯X2� = ¯Σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Unlike the BDM which neglects the covariance, the UCM compresses the uncertainty of mean and covariance into information in the one-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We call such a maneuver ambiguity pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In the rest of this subsection, we demonstrate the effects of ambiguity pooling by comparing the optimal value and ambiguity sets of the BCM and the UCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='8, we plot the relative gap between the optimal values through fixing the same part of moment parameters with µ1 = 1, µ2 = 1, a = 2, b = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' As shown in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='8, the BCM enjoys a 5% − 15% performance improvement with a medium to large η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' It is not surprising to see that the relative gap is zero when η is small or very large: when η is small, both models will adopt a zero inventory, thus the gap is zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' when η is very large, condition 6 in Theorem 1 holds, resulting in the same optimal values for both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' An essential observation is that the ambiguity pooling process may lead to a loss of information on the uncertainty under half-space support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' To be precise, we consider the following ambiguity 18 Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='8: Relative Gap between BCM and UCM set ˆF(θ) = � P ∈ M(R2) : EP[X1 + X2] = ¯µ, EP � (X1 + X2)2� = ¯Σ, X1 + X2 ≥ 0 � , and the UCM can be reformulated as follows: inf q≥0 � sup P∈ ˆ F(θ) EP � (X1 + X2 − q)+ � + (1 − η)q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' It is obvious that F(θ) ⊂ ˆF(θ), which means the ambiguity pooling process potentially results in a more conservative decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Next, we shall provide an example to demonstrate that the optimal distribution of problem supP∈ ˆ F EP[(X1 + X2 − q)+] does not belong to F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' in other words, F(θ) is a proper subset of ˆF(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We take µ1 = 1, µ2 = 1, a = 2, b = 6, c = 1 as an example, in which ¯µ = 2 and ¯Σ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Based on the Scarf bound in Lemma 1, a class of optimal distributions for problem supP∈ ˆ F EP[(X1 + X2 − q)+] with total K + 1 mass points can be described as follows: ˆP∗ = � � � � � (0, 0), w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p1 = 1 − ¯µ2 ¯Σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6 (x(k) 1 , x(k) 2 ), w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p(k) 2 , for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=', K with �K k=1 p(k) 2 = ¯µ2 ¯Σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='4 and x(k) 1 + x(k) 2 = ¯Σ ¯µ = 5 for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' To show ˆP∗ /∈ F by contradiction, we first assume ˆP∗ ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Thus, the following constraints must hold for X1: E[X1] = K � k=1 x(k) 1 p(k) 2 = µ1 = 1, and E[X2 1] = 10 − 10 + 2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' As a result, the second moment of X2 must satisfy E[X2 2] = K � k=1 � ¯Σ ¯µ − x(k) 1 �2 p(k) 2 = ¯Σ2 ¯µ2 K � k=1 p(k) 2 − 2 ¯Σ ¯µ E[X1] + E[X2 1] = � 1 − 2µ1 ¯µ � ¯Σ + Σ11 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 19 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='00 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='72 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='bs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='00 pThis contradicts with E[X2 2] = Σ22 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Therefore, ˆP∗ /∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Lastly, we remark that the assumption on the half-space support plays an important role in the above result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' If the half-space support is replaced by the full-space support for both the BCM and the UCM, then the two models will result in the same optimal values by the projection theorem in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Conclusion and Further Discussion To summarize, we study a bivariate distributionally robust optimization problem with the mean-covariance ambiguity set and half-space support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' For a class of widely-adopted objective functions in inventory management, option pricing, and portfolio selection, we obtain closed- form tight bounds and optimal distributions of the inner problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Furthermore, we show that under the distributionally robust setting, a centralized inventory system does not always reduce the optimal total inventory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' This contradicts with the belief that a centralized inventory system can reduce the total inventory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In addition, we identify two effects, a conventional pooling effect and a novel shifting effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Their combination determines the benefit of incorporating the covariance information in the ambiguity set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Finally, we demonstrate the importance of keeping the covariance information in the ambiguity set instead of compressing the information into one dimension through numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' It is worth mentioning that our result of the bivariate moment problem is also useful to derive a closed-form upper bound of the multivariate moment problem supP∈F EP � ℓ � wT X �� in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6) with ℓ(x) = max {u1x + v1, u2x + v2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Such multivariate problem is known to be computationally challenging, and Natarajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' provide a mathematically tractable upper bound for the expected piecewise linear utility function in a numerical way [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In consideration of our results, we treat the n-dimensional random variable X as consisting of a sequence of low-dimensional random variables Xi with X = (X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' , XI) and Xi ∈ Rni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In turn, we can decompose the ambiguity set F into corresponding marginal ambiguity sets Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Based on such a decomposition, we obtain an upper bound of the multivariate problem as follows: sup P∈F EP � ℓ � wT X �� ≤ � i∈I sup Pi∈Fi EPi � max{u1wT i Xi + v1i, u2wT i Xi + v2i} � for all possible v1i and v2i satisfying �I i=1 v1i = v1 and �I i=1 v2i = v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Therefore, if ni ≤ 2 for all i, we can provide a closed-form upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In terms of numerical algorithms, we can incorporate �I i=1 v1i = v1 and �I i=1 v2i = v2 as two constraints to obtain a better upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Note that the estimation of the n × n covariance matrix is usually unreliable in practice if n is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 20 Our approach relies on only a number of marginal well-estimated 2 × 2 covariance submatrices, which is considered to be robust against data perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' On the other hand, the weakness of our approach is that we only incorporate ⌊n/2⌋ submatrices at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The open question is how to make more use of the covariance information so as to provide a better bound for this multivariate moment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Another possible research direction is to incorporate the shape of distributions including sym- metry, unimodality, or convexity into the ambiguity set in order to provide less conservative deci- sions appropriate to their respective scenarios, which we also leave for further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' References [1] Bertsimas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' and Popescu, I.' 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+page_content=' and Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' On the value of mix flexibility and dual sourcing in unreliable newsvendor networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Manufacturing & Service Operations Management, 7(1):37–57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' [40] Zhu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' and Fukushima, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Worst-case conditional value-at-risk with application to robust portfolio management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Operations Research, 57(5):1155–1168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' [41] Zuluaga, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=', Pe˜na, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=', and Du, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Third-order extensions of Lo’s semiparametric bound for european call options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' European Journal of Operational Research, 198(2):557–570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 24 Appendix A: Proof of Proposition 1 Rewrite the loss function ℓ(x) = max{u1x + v1, u2x + v2} = max{0, (u2 − u1)x + v2 − v1} + (u1x + v1) = � (u2 − u1)x + v2 − v1 � + + (u1x + v1) = (u2 − u1) � x + v2 − v1 u2 − u1 � + + (u1x + v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Let ˜X = w ◦ X, then EP[ ˜X] = w ◦ µ and EP � ˜X ˜X T � = wwT ◦ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Thus, the ambiguity set F of random vector X is equivalent to the ambiguity set ˜F of random vector ˜X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Therefore, sup P∈F EP � ℓ � wT X �� = sup P∈F EP � (u2 − u1) � wT X + v2 − v1 u2 − u1 � + + � u1wT X + v1 �� = (u2 − u1) sup P∈F EP �� wT X + v2 − v1 u2 − u1 � + � + u1wT µ + v1 = (u2 − u1) sup P∈ ˜ F EP �� 1T ˜X + v2 − v1 u2 − u1 � + � + u1wT µ + v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Appendix B: Proof of Theorem 1 As proved in [1], problem supP∈F EP[ℓ(wT X)] is NP-hard for general cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In this paper, we study the bivariate case (n = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Specifically, we consider the following bivariate moment problem vP (θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' q) = sup P∈F(θ) EP � (X1 + X2 − q)+ � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1) where θ = (µ1, µ2, Σ11, Σ22, Σ12) and the ambiguity set F(θ) is described as follows F(θ) = {P ∈ M(R2 +) : EP[X1] = µ1, EP[X2] = µ2, EP � X2 1 � = Σ11, EP � X2 2 � = Σ22, EP [X1X2] = Σ12}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2) For simplicity, we denote (Σ11, Σ22, Σ12) := (aµ2 1, bµ2 2, cµ1µ2) for the rest of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Thus, the covariance matrix M and correlation ρ can be represented by M = � � (a − 1)µ2 1 (c − 1)µ1µ2 (c − 1)µ1µ2 (b − 1)µ2 2 � � and ρ = c − 1 � (a − 1)(b − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='3) Note that parameters a, b and c follow Assumption 1, which characterizes the feasibility of the primal problem (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' To exclude the trivial cases in which the ambiguity set only contains the distribution with a fixed single-point marginal distribution, without loss of generality, we assume that a > 1 and b > 1 in the remaining discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 25 The dual of problem (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1) is given as vD(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' q) = inf z z1 + z2µ1 + z3µ2 + z4aµ2 1 + z5bµ2 2 + z6cµ1µ2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' h1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z) :=z1 + z2x1 + z3x2 + z4x2 1 + z5x2 2 + z6x1x2 ≥ 0, ∀(x1, x2) ∈ R2 + h2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z) := q + z1 + (z2 − 1)x1 + (z3 − 1)x2 + z4x2 1 + z5x2 2 + z6x1x2 ≥ 0, ∀(x1, x2) ∈ R2 +, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='4) with the dual variable z = (z1, z2, z3, z4, z5, z6) ∈ R6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Define six conditions of (θ, q) in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1, Condition 1 Condition 2 Condition 3 Condition 4 Condition 5 Condition 6 Qa ≥ q, Qb < q, Qa < q, Qb < q, Qa < q, Qa > |ζa|, Qb ≥ q Qb ≤ ζb Qa ≤ ζa Qb ≤ −ζb Qa ≤ −ζa Qb > |ζb| Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1: Six Conditions where ζa = aµ1 + cµ2 − q and ζb = cµ1 + bµ2 − q, and Qa = � q2 − 2q a − c a − 1µ2 + ab − c2 a − 1 µ2 2, Qb = � q2 − 2q b − c b − 1µ1 + ab − c2 b − 1 µ2 1, Qc = � q2 − 2q(µ1 + µ2) + aµ2 1 + bµ2 2 + 2cµ1µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' For simplicity of the later proof, we denote A = ab − c2 2(a − c)µ2, A = ab − c2 2(b − c)µ1, B = A + D · (C − C) 2(a − c)C , B = A + D · (C − C) 2(b − c)C , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5) where C = (a − 1)µ1 + (c − 1)µ2, C = (c − 1)µ1 + (b − 1)µ2, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6) and D = (a − 1)(aµ1 + cµ2) > 0, D = (b − 1)(cµ1 + bµ2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='7) Note that A, B, C, D and A, B, C, D are symmetric if we replace µ1 by µ2 and a by b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We also present four terms that will be frequently employed for the later analysis: aµ1 + cµ2 − B = aC 2 + (a − 1)(b − 1) − (c − 1)2 (a − 1)C µ2 2, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='8) cµ1 + bµ2 − B = bC2 + (a − 1)(b − 1) − (c − 1)2 (b − 1)C µ2 1, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='9) A − A = ab − c2 (a − c)(b − c)(C − C), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='10) B − B = (C − C) CC E, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='11) 26 where E = (a−1)cµ2 1 + � ab−a−b+c2� µ1µ2 +(b−1)cµ2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Because of the condition (a−1)(b−1)− (c−1)2 ≥ 0 in Assumption 1, it is easy to prove that E ≥ (a−1)cµ2 1 +2(c−1)cµ1µ2 +(b−1)cµ2 2 = cVar(X1 + X2) > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1, we prove parameters (θ, q) must satisfy one and precisely one of six conditions in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Then we provide the worst-case distributions for every condition and prove the optimality in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Through the analyses of the worst-case distribution for each condition, Theorem 1 can be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Proof of Lemma 2 To show that every feasible input (θ, q) satisfies one and exactly one of six conditions, we first show that conditions in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1 are equivalent to those in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='12-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' By simple algebra, we can verify the following equivalency with Q2 a ≤ ζ2 a ⇔ C(q − B) ≤ 0, Q2 b ≤ ζ2 b ⇔ C(q − B) ≤ 0, Qa ≥ q ⇔ 2q(a − c) ≤ (ab − c2)µ2, Qb ≥ q ⇔ 2q(b − c) ≤ (ab − c2)µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Thus, we can rewrite conditions in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1 as follows: condition 1: 2q(a − c) ≤ (ab − c2)µ2, 2q(b − c) ≤ (ab − c2)µ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='12) condition 2: 2q(b − c) > (ab − c2)µ1, C(q − B) ≤ 0, ζb ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='13) condition 3: 2q(a − c) > (ab − c2)µ2, C(q − B) ≤ 0, ζa ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='14) condition 4: 2q(b − c) > (ab − c2)µ1, C(q − B) ≤ 0, ζb ≤ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='15) condition 5: 2q(a − c) > (ab − c2)µ2, C(q − B) ≤ 0, ζa ≤ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='16) condition 6: C(q − B) > 0, C(q − B) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='17) Furthermore, according to the conditions in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='12-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='17), we will discuss the feasible region of q under each condition based on the input θ in different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The results are summarized in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Through this table, we can clearly see that the intersection of feasible regions of q in each column is empty, and the union of those is R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In addition, as the six cases of input θ are also mutually exclusive, every feasible input (θ, q) satisfies one and exactly one of six conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Now, we are ready to see how to obtain feasible regions of q in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Case 1 (a > c > b): In this case, we have a > c > b > 1 based on b > 1 in Assumption 1, and we can first verify that C > 0, C > 0, and C − C = (a − c)µ1 + (c − b)µ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='18) 27 Cond Cases a > c > b b > c > a a > c, b > c C ≥ C, C > 0 C ≥ C, C < 0 C ≤ C, C > 0 C ≤ C, C < 0 condition 1 0 ≤ q ≤ A 0 ≤ q ≤ A 0 ≤ q ≤ A 0 ≤ q ≤ A 0 ≤ q ≤ A 0 ≤ q ≤ A condition 2 A < q ≤ B A < q ≤ B A < q ≤ B condition 3 A < q ≤ B A < q ≤ B A < q ≤ B condition 4 q ≥ B condition 5 q ≥ B condition 6 q > B q > B q > B B < q < B q > B B < q < B Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2: Feasible Regions of q in Different Cases Condition 1 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='12) can be rewritten as q ≤ ab − c2 2(a − c)µ2 = A and q ≥ ab − c2 2(b − c)µ1 = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' According to H¨older’s inequality, we have (E[X1X2])2 ≤ E[X2 1]E[X2 2], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=', c2 ≤ ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='19) Thus, A ≥ 0 and A ≤ 0 due to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='19) and b < c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In all, condition 1 can be represented as 0 ≤ q ≤ A, as q ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Condition 3 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='14) can be rewritten as q > A, q ≤ B, and q ≤ aµ1 + cµ2, because of C > 0 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='18) and a > c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' According to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='8), we have aµ1 + cµ2 − B = aC 2 + (a − 1)(b − 1) − (c − 1)2 (a − 1)C µ2 2 > 0, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='20) due to C > 0 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='18) and (a − 1)(b − 1) ≥ (c − 1)2 in Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In addition, we can verify that B − A = D·(C−C) 2(a−c)C > 0 due to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='18) and D > 0 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In all, condition 3 can be simplified as A < q ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Condition 6 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='17) can be written as q > B and q > B, due to inequality (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Moreover, we have B > B, as B − B > 0 by equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='11) and inequality (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In all, condition 6 can be represented as q > max{B, B} = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 28 Similar to the previous analysis, the feasible region of q under condition 2 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='13) and that under condition 4 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='15) are empty with q < A < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Due to aµ1 + cµ2 − B > 0 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='20), we know that q ≤ B and q ≥ aµ1 + cµ2 together are incompatible, which means that the feasible region of q under condition 5 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='16) is also empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In summary, the space q ≥ 0 is fulfilled by condition 1, condition 3, and condition 6 in this case, and all feasible regions are mutually exclusive, which are summarized in the first column of Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Case 2 (b > c > a): Note that the condition on the input θ under case 2 (b > c > a) is symmetric to that under case 1 (a > c > b) in the sense of swapping a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In fact, condition 2 and condition 3 are symmetric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' condition 4 and condition 5 are symmetric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' condition 1 and condition 6 are self-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Therefore, by the same analysis in case 1, we summarize the result in the second column of Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Case 3 (a > c, b > c, C ≥ C, C > 0): In this case, we can first verify that A ≥ 0, A ≥ 0, A − A = ab − c2 (a − c)(b − c)(C − C) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='21) due to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Condition 1 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='12) can be rewritten as q ≤ A and q ≤ A, due to a > c, b > c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' As A ≤ A holds by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='21) and q ∈ R+, condition 1 can be represented as 0 ≤ q ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Condition 3 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='14) can be written as q > A, q ≤ B, and q ≤ aµ1 + cµ2, because of C > 0 and a > c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' According to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='8), we have aµ1 + cµ2 − B > 0 due to C ≥ C > 0 in this case and (a − 1)(b − 1) ≥ (c − 1)2 in Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In addition, we can verify that B − A = D·(C−C) 2(a−c)C ≥ 0 due to D > 0 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='7) and C ≥ C in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Thus, aµ1 + cµ2 > B ≥ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In all, the condition can be simplified as A < q ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Condition 6 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='17) can be written as q > B and q > B, due to C ≥ C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Moreover, we have B ≥ B, as B − B ≥ 0 by equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='11) and C ≥ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In all, condition 6 can be simplified as q > B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 29 Similar to the previous analysis, we can verify that B − A = D·(C−C) 2(b−c)C ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Thus we know that q ≤ B and q > A together are incompatible, which means that the feasible region of q under condition 2 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='13) and condition 4 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='15) are empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In addition, due to aµ1 + cµ2 − B > 0, we know that q ≥ aµ1 + cµ2 and q ≤ B together are incompatible, which means that the feasible region of q under condition 5 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='16) is also empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In summary, the space q ≥ 0 is fulfilled by condition 1, condition 3, and condition 6 in this case, and all feasible regions are mutually exclusive, which are summarized in the third column of Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Case 4 (a > c, b > c, C ≥ C, C < 0): In this case, we first prove that C > 0 by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Assume C ≤ 0 and C < 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=', (a − 1)µ1 ≤ (1 − c)µ2 and (b − 1)µ2 < (1 − c)µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' If we cancel µ1 and µ2, it leads to (a − 1)(b − 1) < (1 − c)2, which contradicts the feasibility requirement in Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Furthermore, we can verify that A ≥ 0, A ≥ 0, A − A ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' By the same analysis in case 3, the feasible regions of q under condition 1 and condition 3 are the same as those in case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Condition 4 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='15) can be written as q > A, q ≥ B and q ≥ cµ1 + bµ2, because of b > c and C < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' According to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='9), we have that cµ1 + bµ2 − B < 0 due to C < 0 in this case and (a − 1)(b − 1) − (c − 1)2 ≥ 0 in Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In addition, we verify that B − A = D·(C−C) 2(b−c)C ≥ 0 due to D > 0 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='7) and C < 0 together with C ≤ C in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In all, condition 4 can be simplified as q ≥ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Condition 6 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='17) can be written as q < B and q > B, because of C > 0 and C < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Moreover, we have B ≤ B, as B − B ≤ 0 by equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='11) and C ≥ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In all, condition 6 can be simplified as B < q < B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Similar to the previous analysis, due to cµ1+bµ2−B < 0, we know that q ≥ B and q ≤ cµ1+bµ2 together are incompatible, which means that the feasible region of q under condition 2 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='13) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In addition, due to aµ1 + cµ2 − B > 0, we know that q ≤ B and q ≥ aµ1 + cµ2 together are incompatible, which means that the feasible region of q under condition 5 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='16) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 30 In summary, the space q ≥ 0 is fulfilled by condition 1, condition 3, condition 4, and condition 6 in this case, and all feasible regions are mutually exclusive, which are summarized in the fourth column of Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Case 5 (a > c, b > c, C < C, C > 0): Note that the condition about the input θ under case 5 (a > c, b > c, C < C, C > 0) is symmetric to that under case 3 (a > c, b > c, C ≥ C, C > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' As the symmetry of condition 2 and condition 3 and the self-symmetry of condition 1 and condition 6, by the same analysis in case 3, we summarize the result in the fifth column of Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Case 6 (a > c, b > c, C < C, C < 0): Note that the condition about the input θ under case 6 (a > c, b > c, C < C, C < 0) is symmetric to that under case 4 (a > c, b > c, C ≥ C, C < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Note that condition 2 and condition 3 are symmetric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' condition 4 and condition 5 are symmetric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' condition 1 and condition 6 are self-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Therefore, by the same analysis in case 4, we summarize the result in the last column of Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Proof of Theorem 1 In this subsection, we provide the optimal values and optimal distributions under each condition in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6, and Theorem 1 follows immediately from these lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Before diving into details, we first present our intuitions on how to derive the optimal distri- bution under each condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Let z∗ be the dual optimal solution, so the dual feasibility implies h1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) ≥ 0 and h2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) ≥ 0 for all x ∈ R2 + in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Moreover, the complementary slackness conditions indicate that h1(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) = 0 or h2(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) = 0 for every x∗ in optimal support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Instead of focusing on h1 and h2 directly, geometrically, we consider a curved surface S with S = � (x1, x2, t) ∈ R3 : z∗ 1 + x1z∗ 2 + x2z∗ 3 + x2 1z∗ 4 + x2 2z∗ 5 + x1x2z∗ 6 = t � , and a folded hyperplane T with T = � (x1, x2, t) ∈ R3 + : max{x1 + x2 − q, 0} = t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Accordingly, h1, h2 ≥ 0 is equivalent to S above T, and h1 = 0 or h2 = 0 is equivalent to S touching T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Through this interpretation, we can quickly analyze the characteristics of optimal support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Based on these characteristics, we successfully identify an optimal distribution under each condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In this derivation, although we are motivated by the framework for solving univariate moment problems introduced in [19], it also requires a lot of trial and error to obtain an optimal distribution of our bivariate moment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' As an illustration, we plot S and T with parameters µ1 = µ2 = 1, a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1, b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='015, ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='7, q = 1 in Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In this figure, the points of the solid 31 Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1: An illustration of S and T line together with (0, q − Qa) are touching points of S and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In fact, the curved surface S is a paraboloid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Now, we are ready to present six lemmas corresponding to each condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In each proof, we first verify the primal feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Then, we provide a dual solution and show its feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Lastly, we verify the zero duality gap and thus the optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Suppose that Qa ≥ q, Qb ≥ q, and Assumption 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The optimal distribution for problem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1 can be characterized as follows (i) if b > c, � � � � � � � � � � � � � (0, 0), w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p1 = (a−1)(b−1)−(c−1)2 (ab−c2) ( ab−c2 b−c µ1, 0), w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p2 = (b−c)2 (ab−c2)b (cµ1, bµ2), w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p3 = 1 b (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='22) (ii) if a > c, � � � � � � � � � � � � � (0, 0), w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p1 = (a−1)(b−1)−(c−1)2 (ab−c2) (0, ab−c2 a−c µ2), w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p2 = (a−c)2 (ab−c2)a (aµ1, cµ2), w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p3 = 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='23) For both cases, the optimal values are all vP (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ) = µ1 + µ2 − q · a+b−2c ab−c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Recall the inequality ab − c2 ≥ 0 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We now further assume b > c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' First, we shall verify the feasibility of the primal solution (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Due to b > c and (a − 1)(b − 1)−(c−1)2 ≥ 0, we can verify the non-negativity of the primal supporting points and p1, p2, p3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 32 S 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5 2 O+b B t 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5 1 O-b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5 2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5 +1 2 0Besides, it is easy to verify that p1 +p2 +p3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In addition, we can confirm the following moment constraints hold, EP[X1] = µ1, EP[X2] = µ2, EP[X2 1] = aµ2 1, EP[X1X2] = cµ1µ2, EP[X2 2] = bµ2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In all, the primal solution (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='22) is feasible for problem (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Second, we shall verify the dual feasibility of the following dual solution z∗ with z∗ = � 0, 1 − 2q(b − c) µ1(ab − c2), 1 − 2q(a − c) µ2(ab − c2), q(b − c)2 (ab − c2)2 µ2 1 , q(a − c)2 (ab − c2)2 µ2 2 , 2q(a − c)(b − c) (ab − c2)2 µ1µ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='24) We consider two dual constraint functions h1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) and h2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Since z∗ 4 ≥ 0 and 4z∗ 4z∗ 5 −z∗ 6 2 = 0, then the Hessian matrix � �2z∗ 4 z∗ 6 z∗ 6 2z∗ 5 � � is positive semidefinite, which means that both functions are convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Next, we take the derivative of functions h1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) and h2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗), ∇xh1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) = (z∗ 2 + 2z∗ 4x1 + z∗ 6x2, z∗ 3 + 2z∗ 5x2 + z∗ 6x1)T (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='25) and ∇xh2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) = (z∗ 2 − 1 + 2z∗ 4x1 + z∗ 6x2, z∗ 3 − 1 + 2z∗ 5x2 + z∗ 6x1)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='26) Define x = (x1, x2)T , x∗ = (0, 0)T , x∗ = ( ab−c2 b−c µ1, 0)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Substitute x∗ into equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='25), and we have ∇xh1(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗)T (x − x∗) = (b − 1)(Q2 b − q2) (ab − c2)µ2 1 x1 + (a − 1)(Q2 a − q2) (ab − c2)µ2 2 x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Since a > 1 and b > 1 in Assumption 1, Qa ≥ q, Qb ≥ q in condition 1, and inequality (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='19), then we have ∇xh1(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗)T (x − x∗) ≥ 0, ∀x ∈ R2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Note that h1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) is a convex function, which implies for any x ∈ R2 +, h1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) ≥ h1(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) + ∇xh1(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗)T (x − x∗) ≥ h1(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' For function h2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗), substitute x∗ into equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='26), and we have ∇xh2(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) = � � − 2q(b−c) µ1(ab−c2) + 2q(b−c)2 (ab−c2)2µ2 1 ab−c2 b−c µ1 − 2q(a−c) µ2(ab−c2) + 2q(a−c)(b−c) (ab−c2)2µ1µ2 ab−c2 b−c µ1 � � = � �0 0 � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Note that the function h2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) is convex, which implies for any x ∈ R2 +, h2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) ≥ h2(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) + ∇xh2(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗)T (x − x∗) = h2(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In all, we prove that the dual solution z∗ is feasible for problem (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 33 Lastly, it is easy to verify the objective values for the feasible primal-dual pair are all equal to µ1 + µ2 − q · a+b−2c ab−c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Hence, the zero duality gap implies the optimality of the feasible primal-dual pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' When a > c, the dual solution is the same as that of b > c in equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' By the similar analysis of the b > c case, we can verify the distribution (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='23) is an optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The last case is when a = b = c, in which X1 and X2 are perfectly correlated and Scarf’s bound in Lemma 1 will solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Suppose that Qb < q, Qb ≤ ζb and Assumption 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Then an optimal distri- bution for primal distribution for problem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1 can be characterized as � � � � � � � � � � � � � (q − Qb, 0) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p1 = b−1 2b + q(b−1)+µ1(c−b) 2bQb (q + Qb, 0) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p2 = b−1 2b − q(b−1)+µ1(c−b) 2bQb (cµ1, bµ2) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p3 = 1 b , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='27) where Qb = � q2 + 2q · c−b b−1µ1 + ab−c2 b−1 µ2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The optimal value is vP (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ) = b−1 2b ((q + Qb) − b−c b−1µ1) + µ1 + µ2 − q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' First, we shall verify the feasibility of the primal solution (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Since (a − 1)(b − 1) ≥ (c − 1)2 in Assumption 1, we have (ab − c2)(b − 1) − (b − c)2 ≥ (a + b − 2c)(b − 1) − (b − c)2 = (a − 1)(b − 1) − (c − 1)2 ≥ 0, and it further leads to Q2 b − (b − c b − 1µ1 − q)2 = (ab − c2)(b − 1) − (b − c)2 (b − 1)2 µ2 1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' With Qb ≥ 0, we get Qb ≥ b−c b−1µ1 − q and Qb ≥ q − b−c b−1µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Now we rewrite p1 and p2 to be p1 = b − 1 2bQb (Qb + q − µ1 b − c b − 1) and p2 = b − 1 2bQb (Qb − q + µ1 b − c b − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Thus, p1 ≥ 0 is due to Qb ≥ b−c b−1µ1 − q, and p2 ≥ 0 is due to Qb ≥ q − b−c b−1µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Besides, p3 ≥ 0 holds obviously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Furthermore, by straightforward calculations, we can also verify that the following constraints hold, EP[1] = p1 + p2 + p3 = 1, EP[X1] = µ1, EP[X2] = µ2, EP[X2 1] = aµ2 1, EP[X1X2] = cµ1µ2, EP[X2 2] = bµ2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In all, the primal solution (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='27) is feasible for problem (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Second, we shall verify the dual feasibility of the following dual solution z∗ = �(Qb − q)2 4Qb , Qb − q 2Qb , 1 − Qb + q 2Qb Qb + q − cµ1 bµ2 , 1 4Qb , (Qb + q − cµ1)2 4b2µ2 2Qb , Qb + q − cµ1 2bµ2Qb � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We consider two dual constraint functions h1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) and h2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Since z∗ 4 ≥ 0 and 4z∗ 4z∗ 5 −z∗ 6 2 = 0, then the Hessian matrix � �2z∗ 4 z∗ 6 z∗ 6 2z∗ 5 � � is positive semidefinite, which means that both functions 34 are convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Next, we define x = (x1, x2)T , x∗ = (q − Qb, 0)T and x∗ = (q + Qb, 0)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Then, we have ∇xh1(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗)T (x − x∗) = ζb − Qb bµ2 x2 ≥ 0, ∀x ∈ R2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Since h1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) is a convex function, for any x ∈ R2 +, we have h1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) ≥ h1(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' For function h2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗), substitute x∗ into equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='26), and we have ∇xh2(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) = (0, 0)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Thus h2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) ≥ h2(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In all, we prove that the dual solution z∗ is feasible for problem (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Lastly, it is easy to verify the objective values for the feasible primal-dual pair are all equal to b−1 2b � (q + Qb) − b−c b−1µ1 � + µ1 + µ2 − q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Hence, the zero duality gap implies the optimality of the feasible primal-dual pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Suppose that Qa < q, Qa ≤ ζa and Assumption 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Then an optimal distri- bution for primal distribution for problem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1 can be characterized as � � � � � � � � � � � � � (0, q − Qa) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p1 = a−1 2a + q(a−1)+µ2(c−a) 2aQa (0, q + Qa) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p2 = a−1 2a − q(a−1)+µ2(c−a) 2aQa (aµ1, cµ2) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p3 = 1 a , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='28) where Qa = � q2 + 2q · c−a a−1µ2 + ab−c2 a−1 µ2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The optimal value is vP (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ) = a−1 2a � (q + Qa) − a−c 2a µ2 � + µ1 + µ2 − q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The proof is similar to that in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Note that the conditions in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='3 are symmetric, in some sense, through replacing b by a and µ2 by µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Suppose that Qb < q, Qb ≤ −ζb and Assumption 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Then an optimal distribution for primal distribution for problem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1 can be characterized as � � � � � � � � � � � � � (q − Qb, 0) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p1 = b−1 2b + q(b−1)+µ1(c−b) 2bQb (q + Qb, 0) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p2 = b−1 2b − q(b−1)+µ1(c−b) 2bQb (cµ1, bµ2) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p3 = 1 b , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='29) where Qb = � q2 + 2q · c−b b−1µ1 + ab−c2 b−1 µ2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The optimal value is vP (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ) = b−1 2b � b−c b−1µ1 − (q − Qb) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' First, we observe that the optimal primal solution is the same as that in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2, so the primal feasibility of this solution holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Second, we shall verify the dual feasibility of the following dual solution z∗ = �(Qb − q)2 4Qb , Qb − q 2Qb , (Qb − q)(q − Qb − cµ1) 2bQbµ2 , 1 4Qb , (q − Qb − cµ1)2 4Qbb2 , q − Qb − cµ1 2bQbµ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 35 We consider two dual constraint functions h1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) and h2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Note that z∗ 4 ≥ 0 and 4z∗ 4z∗ 5 − z∗ 6 2 = 0, then the Hessian matrix � �2z∗ 4 z∗ 6 z∗ 6 2z∗ 5 � � is positive semidefinite, which means that both functions are convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Next, we construct x∗ = (q − Qb, 0)T and x∗ = (q + Qb, 0)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Then we have ∇xh1(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗)T (x − x∗) = −Qb − ζb bµ2 x2 ≥ 0, ∀x ∈ R2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Since h1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) is a convex function, for any x ∈ R2 +, we have h1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) ≥ h1(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' For function h2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗), substitute x∗ into equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='26), and we have ∇xh2(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) = (0, 0)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Thus, h2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) ≥ h2(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In all, h1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) ≥ 0 and h2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) ≥ 0 hold for all x ∈ R2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Lastly, it is easy to verify the objective values for the feasible primal-dual pair equal to b−1 2b � b−c b−1µ1 − (q − Qb) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Hence, the zero duality gap implies the optimality of the feasible primal- dual pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Suppose that 2(a − c)q > (ab − c2)µ2, Qa ≤ −cµ2 − aµ1 + q, and Assumption 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Then an optimal distribution for primal distribution for problem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1 can be characterized as � � � � � � � � � � � � � (0, q − Qa) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p1 = a−1 2a + q(a−1)+µ2(c−a) 2aQa (0, q + Qa) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p2 = a−1 2a − q(a−1)+µ2(c−a) 2aQa (aµ1, cµ2) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p3 = 1 a , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='30) where Qa = � q2 + 2q · c−a a−1µ2 + ab−c2 a−1 µ2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The optimal value is vP (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ) = a−1 2a � a−c a−1µ2 − (q − Qa) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The proof is similar to that in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Note that the conditions in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='4 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5 are symmetric, in some sense, through replacing b by a and µ2 by µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Suppose that Qa > |ζa|, Qb > |ζb| and Assumption 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Then an optimal distribution of problem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1 can be characterized as � � � � � � � � � � � � � � � � � � � � � x(1) = ((1 − t0) Ua Uc µ1, t1) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p1 = Ubµ2 2Qct1 x(2) = (q − Qc, 0) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p2 = Uaµ1 2Qct1 t0 x(3) = (0, q + Qc) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p3 = Vb u2 2Qct2 t0 x(4) = (t2, (1 − t0) Vb Vc µ2) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' p4 = Vaµ1 2Qct2 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='31) where Ua = (q + Qc) − (aµ1 + cµ2), Va = (aµ1 + cµ2) − (q − Qc), Ub = (q + Qc) − (bµ2 + cµ1), Vb = (bµ2 + cµ1) − (q − Qc), Uc = (q + Qc) − (µ1 + µ2), Vc = (µ1 + µ2) − (q − Qc), Qc = � q2 − 2q(µ1 + µ2) + aµ2 1 + bµ2 2 + 2cµ1µ2, and t1 = Ub Uc µ2 + t0 Ua Uc µ1, t2 = Va Vc µ1 + t0 Vb Vc µ2, t0 = Det(M) Det(M) + Σ12UcVc , 36 with covariance matrix M defined in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The optimal value is vP (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ) = 1 2Vc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' First, we shall verify the feasibility of the primal solution (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Because of the condition 6 with Qa > |q − aµ1 − cµ2|, Qb > |q − cµ1 − bµ2|, we have Va, Vb, Ua, Ub > 0 immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Moreover, we also have Vc, Uc > 0 due to Q2 c − (q − µ1 − µ2)2 = (a − 1)µ2 1 + 2(c − 1)µ1µ2 + (b − 1)µ2 2 = Var(X1 + X2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Thus, the distribution (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='31) has positive support and probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' As (p1 + p2) + (p3 + p4) = Ubµ2 + t0Uaµ1 2Qct1 + t0Vbµ2 + Vaµ1 2Qct2 = Uc 2Qc + Vc 2Qc = 1, we conclude the primal solution (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='31) is a well-defined distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' To verify this distribution satisfies corresponding constraints, we state two useful equations q − Qc = Ua Uc µ1 + Ub Uc µ2, q + Qc = Vb Vc µ2 + Va Vc µ1 that will be frequently employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Specially,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' we have E[X1] = 4 � i=1 pix(i) 1 = µ1 �(1 − t0)UaUbµ2 + t0U 2 aµ1 + t0UaUbµ2 2QcUct1 + Va 2Qc � = µ1( Ua 2Qc + Va 2Qc ) = µ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' E[X2] = 4 � i=1 pix(i) 2 = µ2 � Vb 2Qc + t0V 2 b µ2 + t0VaVbµ1 + (1 − t0)VaVbµ1 2QcVct2 � = µ2( Ub 2Qc + Vb 2Qc ) = µ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' E[X1X2] = 4 � i=1 pix(i) 1 x(i) 2 = (1 − t0)µ1µ2 2Qc �UaUb Uc + VaVb Vc � = (1 − t0)µ1µ2 2Qc � 2Qc UcVc Det(M) µ1µ2 + 2cQc � = cµ1µ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' E[X1(X1 + X2)] = 4 � i=1 pix(i) 1 (x(i) 1 + x(i) 2 ) = (q − Qc)[p1(q − Qc − t1)) + p2(q − Qc)] + p4(q + Qc)t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' = (q − Qc)(µ1 − p3x(3) 1 − p4x(4) 1 ) + p4(q + Qc)t2 = (q − Qc)µ1 + 2Qcp4t2 = aµ2 1 + cµ1µ2 E[(X1 + X2)X2] = 4 � i=1 pi(x(i) 1 + x(i) 2 )x(i) 2 = p1(q − Qc)t1 + (q + Qc)[p3(q + Qc) + p4(q + Qc − t2)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' = p1(q − Qc)t1 + (q + Qc)(µ2 − p1x(1) 2 − p2x(2) 2 ) = (q + Qc)µ2 − 2Qcp1t1 = bµ2 2 + cµ1µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In all, the primal feasibility holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Second, we shall verify the dual feasibility of the following dual solution z∗ = �(Qb − q)2 4Qc , Qc − q 2Qc , Qc − q 2Qc , 1 4QC , 1 4QC , 1 2QC � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' By straightforward calculations, we have h1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) = 1 4Qc (x1 + x2 − q + Qc)2 ≥ 0 and h2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z∗) = 1 4Qc (x1 + x2 − q − Qc)2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Thus, z∗ is a dual feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Lastly, it is easy to verify the objective values for the feasible primal-dual pair equal to 1 2Vc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Hence, the zero duality gap implies the optimality of the feasible primal-dual pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 37 Appendix C: Proof of Proposition 2 From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1 in [35], the bivariate moment problem sup P∈F(θ) EP � max k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=',K {ℓk(X)} � (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1) is equivalent to its dual problem inf z z1 + µ1z2 + µ2z3 + Σ11z4 + Σ22z5 + Σ12z6 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' z1 + z2x1 + z3x2+z4x2 1 + z5x2 2 + z6x1x2 ≥ max k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=',m � w1k + w2kx1 + w3kx2 + w4kx2 1 + w5kx2 2 + w6kx1x2 � , for all x ∈ R2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Let y2 1 = x1 and y2 2 = x2, and then we have z1 + z2x1 + z3x2 + z4x2 1 + z5x2 2 + z6x1x2 ≥ w1k + w2kx1 + w3kx2 + w4kx2 1 + w5kx2 2 + w6kx1x2, ∀x ∈ R2 + ⇔ z1 − w1k + (z2 − w2k)y2 1 + (z3 − w3k)y2 2 + (z4 − w4k)y4 1 + (z5 − w5k)y4 2 + (z6 − w6k)y2 1y2 2 ≥ 0, ∀y ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The left-hand-side of the inequality can be represented by the sum of squares of polynomials, that is, the above inequality is equivalent to � � � � � � � � � � � � � � y2 1 y1y2 y2 2 y1 y2 1 � � � � � � � � � � � � � � T M � z − w(k), g(k), h(k)� � � � � � � � � � � � � � � y2 1 y1y2 y2 2 y1 y2 1 � � � � � � � � � � � � � � ≥ 0, ∀y ∈ R2 ⇔ M � z − w(k), g(k), h(k)� ⪰ 0 for some g(k) and h(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Therefore, the original problem (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1) can be reformulated by inf z,G,H z1 + µ1z2 + µ2z3 + Σ11z4 + Σ22z5 + Σ12z6 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' M � z − w(k), g(k), h(k)� ⪰ 0, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=', m with z ∈ R6, G ∈ R3×K, H ∈ R3×K and the matrix M(˜z, g, h) defined as follows: � � � � � � � � � � � � � � ˜z4 0 −g1 0 −h1 −g2 0 ˜z6 + 2g1 0 h1 −h2 −h3 −g1 0 ˜z5 h2 0 −g3 0 h1 h2 ˜z2 + 2g2 h3 0 −h1 −h2 0 h3 ˜z3 + 2g3 0 −g2 −h3 −g3 0 0 ˜z1 � � � � � � � � � � � � � � where w(k), g(k), h(k) are the kth column vectors of matrices W, G, H respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' 38 Appendix D: Proof of Proposition 3 From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='7 in book [6], we know that |ρ| = 1 if and only if there exists numbers ϕ1 ̸= 0 and ϕ2 such that P(X1 = ϕ1X2 + ϕ2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' If X1 and X2 are perfectly correlated with ρ = 1, we have the following moment constraints due to X1 = ϕ1X2 + ϕ2 and ϕ1 > 0: E[ϕ1X2 + ϕ2] = µ1, E[X2] = µ2, E[(ϕ1X2 + ϕ2)2] = aµ2 1, E[X2 2] = bµ2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' From these equations, ϕ1 and ϕ2 should satisfy ϕ1µ2 + ϕ2 = µ1 and bϕ2 1µ2 2 + ϕ2 2 + 2ϕ1ϕ2µ2 = aµ2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Then we can obtain ϕ1 = � a−1 b−1 µ1 µ2 and ϕ2 = � 1 − � a−1 b−1 � µ1 and thus X1 = � a−1 b−1 µ1 µ2 X2 + � 1 − � a−1 b−1 � µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Moreover, note that we assume 1 ≤ a ≤ b, then � 1 − � a−1 b−1 � µ1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' From the nonnegativity of X2, we know that X1 ≥ � 1 − � a−1 b−1 � µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Appendix E: Closed-Form Solution of the DRO Newsvendor Problem We study the DRO newsvendor problem as follows: inf q≥0 � sup P∈F(θ) EP � (X1 + X2 − q)+ � + (1 − η)q � , where F(θ) is the mean-covariance ambiguity set in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2), and the critical ratio η is a given constant in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We explain the procedure of solving the closed-form solution q∗ introduced in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Firstly, each q∗ i locates at either a stationary point of vP (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ) + (1 − η)q or a boundary point of the interval Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The formulation of each interval Ai is discussed in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Thus, we study the stationary point of f(q) = vP (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' θ) + (1 − η)q in each interval Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' For simplicity, we denote Sa(η) = � (a − 1)(ab − c2) − (c − a)2 4aη(a − aη − 1) , Sb(η) = � (b − 1)(ab − c2) − (c − b)2 4bη(b − bη − 1) , Sc(η) = � (a − 1)µ2 1 + (b − 1)µ2 2 + 2(c − 1)µ1µ2 4η(1 − η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' For condition 2 as an example, we have f(q) = b − 1 2b � q2 − 2q b − c b − 1µ1 + ab − c2 b − 1 µ2 1 + �b − 1 2b − η � q + b + c 2b µ1 + µ2 39 with its derivative f′(q) = b − 1 2b q − b−c b−1µ1 � q2 − 2q b−c b−1µ1 + ab−c2 b−1 µ2 1 + b − 1 2b − η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Thus, the stationary point is obtained as follows by setting f′(q) = 0: q = � (2bη − b + 1) � (b − 1)(ab − c2) − (c − b)2 4bη(b − bη − 1) − (c − b) � µ1 b − 1 = [(2bη − b + 1)Sb(η) − (c − b)] µ1 b − 1, which exists only if 0 < η < 1 − 1 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' The stationary points under other conditions can be derived by the same approach shown under condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' We provide Table E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1 to summarize the closed-form stationary points in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Lastly, we remark that the stationary point shown in this table for each condition may be out of the interval Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' In all, given this table, it is not difficult to obtain the closed-form expressions of the optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content=' Conditions Stationary Points Feasible η condition 1 f(q) is linear in q ∈ A1 condition 2 [(2bη − b + 1)Sb(η) − (c − b)] µ1 b−1, 0 < η < 1 − 1 b condition 3 [(2aη − a + 1)Sa(η) − (c − a)] µ2 a−1, 0 < η < 1 − 1 a condition 4 [(2bη − b + 1)Sb(1 − η) − (c − b)] µ1 b−1, 1 b < η < 1 condition 5 [(2aη − a + 1)Sa(1 − η) − (c − a)] µ2 a−1, 1 a < η < 1 condition 6 (2η − 1)Sc(η) + µ1 + µ2, 0 < η < 1 Table E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} +page_content='1: Stationary Points 40' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE2T4oBgHgl3EQfgQfX/content/2301.03936v1.pdf'} diff --git a/x9FKT4oBgHgl3EQf6i75/content/tmp_files/2301.11942v1.pdf.txt b/x9FKT4oBgHgl3EQf6i75/content/tmp_files/2301.11942v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9b40894a36c0021e581d9b6014534287557e1944 --- /dev/null +++ b/x9FKT4oBgHgl3EQf6i75/content/tmp_files/2301.11942v1.pdf.txt @@ -0,0 +1,4082 @@ +MNRAS 000, 1–34 (2022) +Preprint 31 January 2023 +Compiled using MNRAS LATEX style file v3.0 +The Origin of Stars in the Inner 500 Parsecs in TNG50 Galaxies +Alina Boecker1,2★, Nadine Neumayer1, Annalisa Pillepich1, Neige Frankel1,3,4, Rahul Ramesh5, +Ryan Leaman6, Lars Hernquist7 +1Max-Planck Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany +2Instituto de Astrofísica de Canarias, C/ Vía Láctea s/n, E-38205 La Laguna, Spain +3Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, ON M5S 3H8, Canada +4Department of Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada +5Universität Heidelberg, Zentrum für Astronomie, Institut für theoretische Astrophysik, Albert-Ueberle-Str. 2, D-69120 Heidelberg, Germany +6Department of Astrophysics, University of Vienna, Türkenschanzstrasse 17, 1180 Wien, Austria +7Harvard–Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA +Accepted 2022 December 16. Received 2022 November 25; in original form 2022 April 7 +ABSTRACT +We investigate the origin of stars in the innermost 500 pc of galaxies spanning stellar masses of 5 × 108−12 M⊙ at z = 0 using the +cosmological magnetohydrodynamical TNG50 simulation. Three different origins of stars comprise galactic centers: 1) in-situ +(born in the center), 2) migrated (born elsewhere in the galaxy and ultimately moved to the center), 3) ex-situ (accreted from other +galaxies). In-situ and migrated stars dominate the central stellar mass budget on average with 73% and 23% respectively. The +ex-situ fraction rises above 1% for galaxies ≳ 1011 M⊙. Yet, only 9% of all galaxies exhibit no ex-situ stars in their centers and +the scatter of ex-situ mass is significant (4−6 dex). Migrated stars predominantly originate closely from the center (1−2 kpc), but +if they travelled together in clumps distances reach ∼ 10 kpc. Central and satellite galaxies possess similar amounts and origins +of central stars. Star forming galaxies (≳ 1010 M⊙) have on average more ex-situ mass in their centers than quenched ones. +We predict readily observable stellar population and dynamical properties: 1) migrated stars are distinctly young (∼ 2 Gyr) and +rotationally supported, especially for Milky Way mass galaxies, 2) in-situ stars are most metal-rich and older than migrated stars, +3) ex-situ stars are on random motion dominated orbits and typically the oldest, most metal-poor and 𝛼-enhanced population. We +demonstrate that the interaction history with other galaxies leads to diverse pathways of building up galaxy centers in a ΛCDM +universe. Our work highlights the necessity for cosmological context in formation scenarios of central galactic components and +the potential to use galaxy centers as tracers of overall galaxy assembly. +Key words: methods: numerical – galaxies: formation – galaxies: evolution – galaxies: stellar content – galaxies: structure – +galaxies: nuclei – galaxies: bulges +1 INTRODUCTION +The center of a galaxy depicts its brightest and densest region. Thus +observations of galaxy centers provide us with the highest data qual- +ity, which should enable us to make the most precise predictions +about their formation. On the other hand, being also the deepest +point of the potential well, the center witnessed the galaxy’s overall +stellar assembly from the earliest cosmic times onward, as understood +from the inside-out formation scenario of galaxies within a ΛCDM +(Lambda-Cold-Dark-Matter) Universe. Therefore, many transforma- +tive processes of galaxy evolution influence a galaxy’s center until +the present day, which need to be taken into account to uniquely +interpret even the highest quality observations. +As a consequence, a variety of central stellar structures are found +in galaxies. Decreasing in size from the order of one kpc to sub +parsec scales, these range from bars and (pseudo)bulges (see e.g. +Laurikainen et al. 2016; Kormendy & Kennicutt 2004, for a sum- +mary), which can include other structures such as nuclear rings and +★ E-mail: aboecker@iac.de +disks, to nuclear star clusters (NSCs; see e.g. Neumayer et al. 2020, +for a summary) and supermassive black holes (SMBHs; see e.g. Ko- +rmendy & Ho 2013, for a summary). Some galaxies may exhibit +more than one of these components or none at all. Many of these +components possess scaling relations of their structural parameters, +such as the Sérsic (1968) index and effective radius of bulges (e.g. +Gadotti 2009; Fisher & Drory 2010) and the luminosity/mass-size +relation of NSCs (e.g. Böker et al. 2004; Côté et al. 2006; Georgiev +& Böker 2014), as well as scaling relations with each other, such as +the bulge-SMBH-mass (e.g. Häring & Rix 2004; Sani et al. 2011; +Läsker et al. 2016) and NSC-SMBH-mass relations (e.g. Ferrarese +et al. 2006; Georgiev et al. 2016), which also scale with the stellar +mass of their underlying host galaxy (e.g. Scott & Graham 2013; +Reines & Volonteri 2015; Sánchez-Janssen et al. 2019). Some of +these scaling relations can differ for early-type and late-type galax- +ies, or depend on the bulge type or the presence of a bar (e.g. Gadotti +& Kauffmann 2009; Georgiev et al. 2016; Davis et al. 2019; Sahu +et al. 2019). +As diverse as the structural properties of central components are, so +are the formation scenarios trying to explain them. Broadly speaking, +© 2022 The Authors +arXiv:2301.11942v1 [astro-ph.GA] 27 Jan 2023 + +2 +Boecker et al. +all of these formation scenarios can be divided into internal and +external processes. For example, bulges are thought to form from +merger events (e.g. Hopkins et al. 2009, 2010), from rapid early-on +star formation (e.g. Guedes et al. 2013; Okamoto 2013), from secular +evolution (e.g. Kormendy & Kennicutt 2004; Athanassoula 2005) or +from the migration of clumps formed in the disk at high redshift (e.g. +Elmegreen et al. 2009; Dekel et al. 2009); bars form through disk +instabilities either in isolation (e.g. Bottema 2003; Athanassoula et al. +2013) or in a cosmological context (e.g. Romano-Díaz et al. 2008; +Kraljic et al. 2012; Peschken & Łokas 2019); nuclear star clusters +are thought to form through either star formation (e.g. Maciejewski +2004; Aharon & Perets 2015) or through the migration and successive +merging of globular clusters in the center (e.g. Hartmann et al. 2011; +Agarwal & Milosavljević 2011); SMBHs can grow by accreting gas +and by merging with other SMBHs (e.g. Croton et al. 2006; Malbon +et al. 2007; Fanidakis et al. 2011; Lapiner et al. 2021). In many +cases, the formation of any one component will also influence the +others. For example, once a bar is formed it can re-arrange the orbits +of stars causing radial migration, or it can efficiently funnel gas to +the center, which can trigger star formation in the center and also +feed the SMBH. In turn, the AGN (active galactic nucleus) feedback +caused by the SMBH will then influence the gas supply and hence +truncate the formation of stars. Thus, it is important to also understand +the interplay between the presence and formation of several central +components. +Observationally, we can only indirectly deduce constraints on any +of these formation scenarios from the stellar population and dynam- +ical properties of a galaxy’s central structure(s). For external galax- +ies, such necessary measurements are only possible with integral +field units (IFUs) that provide spatially resolved stellar population +and kinematical maps (e.g. Gadotti et al. 2020; Bittner et al. 2020). +While major progress has been made in producing these maps with +increasing quality, it is still difficult to disentangle stars from centrally +overlapping galaxy components due to the line-of-sight integration +- let alone identify stars of different origins within a given central +component. This is possibly further complicated by the fact that +stars with properties characteristic of one formation scenario might +be subdominant in luminosity or mass compared to the bulk stellar +population. +Even in the Milky Way, it has only become evident fairly recently +that all major central components contain metal-poor subpopula- +tions of stars that also exhibit different kinematics. For the Galactic +bulge (see e.g. Barbuy et al. 2018, for a summary) there is a smooth +transition from rotation to dispersion dominated kinematics for stars +decreasing from (super-)solar metallicity all the way to the lowest +metallicities ([Fe/H] < −2.0 dex) (Ness et al. 2013; Zoccali et al. +2017; Arentsen et al. 2020). To a lesser extent this decrease is also +seen for the nuclear stellar disk (Schultheis et al. 2021) with addi- +tional evidence of recent star formation activity (< 1 Gyr) on top of +the overall old bulk population (> 8 Gyr) (Nogueras-Lara et al. 2020, +2021). The nuclear star cluster, which hosts the most metal-rich stars +in the Milky Way, also has a subpopulation of sub-solar metallicity +stars, which show an asymmetric spatial distribution and a higher +degree of rotation (Feldmeier-Krause et al. 2020; Do et al. 2020). +Generally, signs of young, metal-rich and kinematically cold stars +in these central structures such as bulges and NSCs, are associated +with being formed in-situ from gas infall, while old, metal-poor +and dispersion dominated systems are thought to originate from +merger processes. However, stars formed in-situ at the beginning +of a galaxy’s lifetime are also metal-poor and might as well become +dispersion dominated over time through various processes, such as +resonances created by the bar. Therefore, even though observed prop- +erties of stars in the centers of galaxies act as a fossil record of their +origin, we need simulations to disentangle which (combinations of) +formation scenarios are able to predict those observations. +Cosmological, hydrodynamical galaxy simulations (see e.g. +Somerville & Davé 2015; Vogelsberger et al. 2020, for a summary) +are ideal to study the complex formation pathways of galaxy centers +as they encompass the most complete conglomeration of galaxy for- +mation processes in a ΛCDM framework, thus capturing internal and +external formation processes alike. The most recent simulations are +able to produce a realistic, diverse population of galaxies (see e.g. +Vogelsberger et al. 2014b; Nelson et al. 2019a, and references therein +for Illustris/TNG specifically). Typically, large simulation boxes are +used to study global galaxy properties across an array of different +galaxies (e.g. Illustris: Genel et al. 2014; Vogelsberger et al. 2014a,b; +EAGLE: Schaye et al. 2015; Crain et al. 2015; Horizon-AGN: Dubois +et al. 2014, 2016; Magneticum: Hirschmann et al. 2014; Teklu et al. +2015; Bocquet et al. 2016; IllustrisTNG: Weinberger et al. 2017; +Pillepich et al. 2018a; SIMBA: Davé et al. 2019), while zoom-in (re- +)simulations focus on internal galaxy structures and dynamics (e.g. +ERIS: Guedes et al. 2011; NIHAO: Wang et al. 2015; Latte: Wetzel +et al. 2016; Auriga: Grand et al. 2017; FIRE-2: Hopkins et al. 2018; +NIHAO-UHD: Buck et al. 2020). To understand the mass build-up of +galaxy centers we need the advantages of both: a big enough box to +probe many different assembly histories and thus galaxy demograph- +ics, and a zoom-in like resolution to focus on the center of galaxies +and capture internal dynamical processes. +We therefore focus our analysis on the origin of stars in the central +few hundred parsecs of galaxies in TNG50 (Pillepich et al. 2019; Nel- +son et al. 2019b) from the IllustrisTNG simulations. The 51.73 cMpc3 +volume captures two 1014 M⊙ halos and hundreds of Milky Way like +galaxies, whereas the spatial resolution provides hundreds to tens of +thousands stellar particles inside the central 500 pc for a four dex +range in galaxy stellar mass. Importantly, TNG50 starts to capture +the diversity of central components, such as low and high Sérsic +index bulges in Milky Way like galaxies (Gargiulo et al. 2021), and +performs well in a statistical comparison of simulated and observed +bar properties (Rosas-Guevara et al. 2021; Frankel et al. 2022); both +which were previously not possible with zoom-in simulations. Hence, +TNG50 offers the unique opportunity to study the contribution of +stars with different (internal or external) origins to the formation of +the galaxy center across diverse galaxy formation pathways and de- +mographics, while predicting the observable imprint that the different +formation scenarios impose on the stars in a galaxy’s center. +The goal of this study is to appeal to different scientific commu- +nities that focus on various central stellar structures of the Milky +Way and external galaxies to provide an understanding where the +most central stars of galaxies originate across a wide range of galaxy +masses inside the TNG modelling framework. Specifically, we also +study, for the first time, stars that have migrated towards the center +to address formation scenarios of central structures that include the +necessity for these processes such as NSC formation. Even though +NSCs are not explicitly resolved in TNG50, we hope to offer new in- +centives for simulations (Antonini et al. 2012; Perets & Mastrobuono- +Battisti 2014; Guillard et al. 2016) and (semi-)analytical models +(Antonini 2013; Antonini et al. 2015; Leaman & van de Ven 2021) +that are tailored towards NSC formation channels. Lastly, we aim +to demonstrate that there are possibilities to use the bright centers +of galaxies as a tracer of the galaxy’s overall assembly history with +readily available observables from current surveys such as SDSS +(e.g. Gallazzi et al. 2021). +This paper is organized as follows. In Section 2 we briefly describe +the TNG50 simulation and the definition of properties of galaxies +MNRAS 000, 1–34 (2022) + +Centers of TNG50 Galaxies +3 +and stars (i.e. stellar particles) that we will analyze at z = 0. We +also provide a detailed description and verification of selecting stars +belonging to a galaxy’s center and our galaxy sample selection. In +Section 3 we present the three different possible origins for stars re- +siding in a galaxy’s center and discuss their birth locations. In Section +4 we show the results of the different contributions of central stars +of different origins across different galaxy population demographics +and their observable stellar population and dynamical properties at +z = 0. In Section 5, we discuss our findings and implications from +TNG50 on the central mass assembly of galaxies in a cosmological +context. We also provide outlooks in the context of the formation of +central galaxy components as well as the assembly of the overall host +galaxy tailored towards measurements of extragalactic observations. +Finally, we conclude our study in Section 6. +2 TOOLS AND METHODS +We briefly introduce the TNG50 simulation below as well as the +properties of TNG50 galaxies and their stars (Section 2.3). We then +describe in Section 2.4 how we define stellar particles that belong to +a galaxy’s center. +2.1 The TNG50 simulation +In this work we primarily study galaxies in TNG50 (Pillepich et al. +2019; Nelson et al. 2019b), which is the highest resolution installment +of the IllustrisTNG (Illustris The Next Generation) (Pillepich et al. +2018b; Springel et al. 2018; Nelson et al. 2018; Naiman et al. 2018; +Marinacci et al. 2018) suite of cosmological, magnetohydrodynam- +ical simulations1. It provides unprecedented zoom-in like resolution +within a representative cosmological volume with a box of 51.7 cMpc +on each side. +The simulation was performed with the Arepo code (Springel +2010; Pakmor et al. 2011; Pakmor & Springel 2013; Pakmor et al. +2016), which employs a finite-volume method on a moving-mesh +to solve the equations of magnetohydrodynamics coupled with a +tree-particle-mesh method for self-gravity. TNG50(-1) has a mass +resolution of 4.6 × 105 M⊙ for dark matter and 8.4 × 104 M⊙ for +baryonic particles. The softening length is 288 cpc for collisionless +particles for z ≤ 1 and 576 cpc for z > 1, whereas the softening length +of the gas particles is adaptive depending on the local cell size of the +moving mesh with a floor value of 74 cpc. TNG50 is accompanied +by three additional simulation runs (-2,-3,-4) that decrease the spatial +resolution each time by half. The initial conditions are set according +to cosmological parameters measured by Planck Collaboration et al. +(2016). +Additionally, the TNG simulations implement a list of physical +subgrid models, which describe galaxy formation and evolution, +such as stellar formation and feedback, chemical enrichment, galactic +winds, supermassive black hole growth and feedback. Details can be +found in Weinberger et al. (2017); Pillepich et al. (2018a). +Importantly, the TNG framework successfully reproduces key ob- +servational results such as the galaxy stellar mass function up until +z < 4 (Pillepich et al. 2018b), bi-modality in galaxy color distribution +(Nelson et al. 2018), the fraction of quiescent galaxies (Donnari et al. +2019, 2021b), scaling relations, such as the galaxy mass-size relation +(Genel et al. 2018), the gas-phase mass-metallicity relation (Torrey +1 IllustrisTNG also encompasses two larger volume runs, namely TNG100 +and TNG300 with subsequently coarser resolution. +et al. 2019) and certain element abundances (Naiman et al. 2018), +as well as the clustering of galaxies (Springel et al. 2018) and mag- +netic fields of massive halos (Marinacci et al. 2018). Specifically, the +resolution of TNG50 allows for the study of internal dynamics and +structures of galaxies (Pillepich et al. 2019) as well as the influence +of stellar and black-hole driven outflows on galaxy evolution (Nelson +et al. 2019b). +Results from the TNG simulation are output in 100 snapshots +ranging from z = 20 until today with an approximate time step +of 150 Myr since z = 4. For each snapshot dark matter halos are +identified by the friends-of-friends (FoF) algorithm (Davis et al. +1985) with a linking length of 0.2, with baryonic particles being +attached to the same FoF group based on their nearest dark matter +particle. Substructures within these halos, i.e. subhalos, are found +through the Subfind algorithm (Springel et al. 2001), which is run on +both dark matter and baryonic particles. To track the mass assembly of +subhalos/galaxies through cosmic time, merger trees are constructed +based on the Sublink algorithm (Rodriguez-Gomez et al. 2015). The +merger trees were constructed twice, once based on dark matter and +once based on baryonic matter alone. +The entire simulations’ particle information for the 100 snapshots, +the halo and subhalo catalogues, merger trees as well as many more +additional supplementary data catalogues are made publicly available +on the TNG website2 (see also Nelson et al. 2019a, for the public +data release). +2.2 General note on calculations +Unless otherwise stated we employ the following definitions in our +subsequent calculations and plots. To center the coordinate system +on a galaxy of interest we choose the position of the particle (of any +type) with the minimum gravitational potential energy as the galaxy’s +center, as given by SubhaloPos in the subhalo catalogue. For the +systemic velocity of a galaxy we use the median velocity of the +5% most bound stellar particles. For face-on or edge-on projections, +galaxies are oriented such that the z-axis is aligned with the total +angular momentum of stellar particles within twice the stellar half +mass radius. To track back galaxies in time we exclusively use the +merger trees based on following baryonic particles (‘Sublink_gal’). +Plots that display summary statistics of galaxy populations use a +running median with a bin size of 0.25-0.3 dex, which is adapted, +if necessary, to ensure a minimum number of ten galaxies per bin. +Furthermore, all displayed quantities are in physical units and all +provided SubfindIDs refer to galaxies at z = 0. +Throughout this study the terms in-situ, migrated and ex-situ al- +ways refer to stars within the central 500 pc of galaxies unless other- +wise stated. +2.3 Galaxy characteristics and properties of their stars +Throughout this study we are interested in two sets of demographics: +1) How does the central mass assembly of galaxies change as a +function of a galaxy’s overall bulk properties?, 2) How do the intrinsic +properties of stars in the center of galaxies differ for different origins? +To address the first question we do not only study the central 500 pc +of galaxies as a function of the galaxy’s total stellar (dynamical) +mass, but we also divide our galaxy sample into different types of +galaxies characterized at z = 0. To address the second question +we study individual properties of stars (i.e. stellar particles) in the +2 https://www.tng-project.org +MNRAS 000, 1–34 (2022) + +4 +Boecker et al. +Property +Short description +Detailed description +Results +Overall galaxy +Mass +total stellar or dynamical (i.e. stars+gas+dark) mass +Appendix A1 +Section 4.1 +Environment +central or satellite +Star formation activity +star forming or quenched +Morphology +(kinematically) disk or bulge dominated +Bar-like feature +present or not, based on Fourier decomposition +AGN feedback +above or below average AGN feedback based on mass of the SMBH +Physical Size +compact or extended with respect to the mass-size relation +Individual stellar particle +Age [Gyr] +the lookback time when the star was born +Appendix A2 +Section 4.2 +Metallicity [log10 𝑍/𝑍⊙] +the total amount of metals +[Mg/Fe] [dex] +the abundance of magnesium as a proxy for 𝛼-elements +Circularity 𝜖 +indicates the type of orbit the star is on +Table 1. Properties of galaxies and their individual stars (stellar particles) in TNG50 at z = 0 investigated in this study. A detailed description on the exact +calculation of the properties as well as the results with respect to the centers of galaxies are found in the indicated sections. +center of galaxies at z = 0. These investigated characteristics are +briefly summarized in Table 1, whereas a detailed description on +their calculations can be found in Appendix A. +2.4 Defining stars belonging to a galaxy’s center +The most straightforward way to define a galaxy’s center at z = 0 +is to select all stellar particles within a 3D spherical aperture with a +given radius 𝑟cut around its center. This simple selection will give us +knowledge about stellar particles that have an instantaneous radius +smaller than the selected aperture. However, as we are interested in +the mass assembly of the center of galaxies, we want to make sure +that selected particles roughly stay inside the spherical aperture over +their orbital time at z = 0. This ensures that we track particles that +changed their orbit, should they have migrated to the center, and not +particles that are just on more eccentric orbits. +To estimate, whether the particles are on such orbits confined to +the center at z = 0, we calculate the specific energy 𝐸cut a particle +on a circular orbit with guiding radius 𝑟cut would have, i.e.: +𝐸cut = 𝑣circ(𝑟cut)2 +2 ++ Φ(𝑟cut). +(1) +The circular velocity 𝑣circ is calculated from the spherically enclosed +mass (stellar, gas and dark matter particles) 𝑣circ(𝑟)2 = 𝐺𝑀 (< 𝑟) +𝑟 +, +whereas the gravitational potential energy Φ is given by the simu- +lation and interpolated to 𝑟cut. Stellar particles with total energies +𝐸 = |v|2 +2 ++ Φ(x) less than 𝐸cut should roughly be confined on orbits +that are within the spherical volume with radius 𝑟cut, whereas parti- +cles with higher energies are able to move to larger radii and hence +spend less time in the center. +We additionally enforce that the specific angular momentum in the +z-direction 𝐿z of stellar particles in the center lies between 𝐿cut = +±𝑣circ𝑟cut, as we noticed that some lower mass galaxies with stellar +masses ≲ 1010 M⊙ had very large 𝐿z and hence large radii with 𝐸 < +𝐸cut, which probably stems from the fact that they are undergoing +tidal stripping at present time. If particles with large orbital radii +(> 2 kpc) still persisted after this cut, we disregard them as well. +These additional steps do not significantly affect the amount of central +particles selected for galaxies with stellar masses ≳ 1010 M⊙. +In general, the selection based on Equation 1 is a simplification as +it assumes a spherical mass distribution, but it gives a good enough +estimate of which particles are truly confined to the center without +actually integrating their orbits; see Appendix B1 for validation of +this with two galaxies contained in the subbox with higher time +cadence. We visualize the difference between a simple selection in +radius and the one in energy using Equation 1 in Figure B1. +2.4.1 Choice of the central region +The last step in selecting stellar particles belonging to a galaxy’s +center is to set a value for 𝑟cut, with which we can in turn calculate +𝐸cut. +We choose a fixed value of 500 pc3 for 𝑟cut across all galaxies +to avoid running too close into the numerical softening length (see +Section 5.4 for further elaboration on this). We explicitly do not +choose to adopt a mass-dependent size, as already with a 10% scaling +of the mass-size relation of TNG50 galaxies, we are at the softening +length of 1010 M⊙ in stellar mass galaxies, while for the highest mass +galaxies we approach 5 kpc, which we do not deem to be central +anymore. We also refer the reader to Section 5.4 and Appendix D +for a more detailed discussion and investigation about numerical +resolution effects and the choices of 𝑟cut. +2.4.2 Galaxy sample selection +Due to the choice of a fixed central aperture of 500 pc we have to make +some selection for our galaxy sample considered in this analysis. +Generally, sizes of TNG50 galaxies are numerically well con- +verged above a stellar mass of ∼ 3 × 108 M⊙ (see Pillepich et al. +2019) at z = 0, but we employ a slightly higher lower mass cut +of 5 × 108 M⊙ ensuring that the galaxies have a sufficient number +of stellar particles for our analysis. We also only consider subha- +los/galaxies that are of cosmological origin (i.e. SubhaloFlag is +true). Any scaling relations used in this analysis such as the galaxy +mass-size or the stellar-mass-black-hole-mass relation (to determine +for example if galaxies lie above or below the median at fixed stellar +mass) are always computed with respect to this galaxy sample, which +contains 4344 galaxies. +Furthermore, for our main analysis of the centers of TNG galaxies, +3 Due to the selection of stellar particles belonging to the 500 pc center +based on their energies, some particles will have instantaneous radii larger +than 500 pc at z = 0, but typically not larger than 1 kpc. +MNRAS 000, 1–34 (2022) + +Centers of TNG50 Galaxies +5 +1 +10 +5 +20 +Percentage of Stars in Center +109 +1010 +1011 +1012 +Total Stellar Mass [M ] +1 +10 +2 +50 +3D Stellar Half Mass Radius [kpc] +56% +23% +7% +0% +2531/4344 Galaxies +TNG50 +z = 0 +10 +100 +2 +4 +R1/2 in Factors of rcut +Figure 1. Sample selection of TNG50 galaxies at z = 0. Stellar mass-size +relation of TNG50 galaxies at z = 0 considered in this analysis colored coded +according to their percentage of stars in the center relative to their total number +of stellar particles. We employ a lower total stellar mass cut of 5 × 108 M⊙ +leaving 4344 Galaxies. We additionally impose a minimum 3D stellar half +mass radius R1/2 of 2 kpc (i.e., R1/2/rcut ≥ 4 with rcut = 500 pc) resulting in +a sample size of 2531 galaxies. Excluded galaxies are shown with grey points +and percentages show their number fractions in bins of 0.5 dex. The median +stellar mass-size relation of all TNG50 galaxies is shown as the black line. +we enforce that the ratio of the 3D stellar half mass radius R1/2 and the +central aperture (rcut = 500 pc) is greater than four, i.e. R1/2 ≥ 2 kpc. +Otherwise, galaxies are too compact for our selected central aperture +and about half of the entire galaxy will be classified as “central”. +Additionally, we make sure that at least a hundred stellar particles +are within the central 500 pc according to Section 2.4, otherwise the +galaxy is disregarded. +Our final galaxy sample selection yields 2531 TNG50 galaxies +and their masses and sizes are visualized in Figure 1. The data points +are color-coded by the percentage of stars inside the central 500 pc +compared to the total amounts of stars. The color trend is neither +uniform in the direction of increasing stellar mass nor size. This +hints at different density profiles for galaxies across their stellar +mass-size plane. We note that our subsequent results do not show any +strong differential trends, even though our constrain of R1/2 ≥ 2 kpc +disregards half the galaxies with stellar masses < 109.5 M⊙ (see +Section 5.4 for a further discussion). +In Figure 2 we show the cumulative number of stellar particles as a +function of their instantaneous radius at z = 0 for our galaxy sample. +The average number of stellar particles in the center, i.e. within +500 pc, is around 103 for galaxy stellar masses between 5 × 108 M⊙ +and 5 × 109 M⊙ and increases towards 105 for the highest mass +bin4. Hence, our choice for 𝑟cut ensures that we have enough stellar +4 A synonymous measure can be achieved with the StellarHsml field, +which gives an approximation for the spatial extent a single stellar particle +samples from the underlying stellar density field. The spherical radius of +stars within 500 pc ranges between 10 − 100 pc for the highest to lowest mass +galaxies respectively. +5 × 108 5 × 109 +1010 +5 × 1010 5 × 1012 +Total Stellar Mass [M ] +0.001 +0.01 +0.1 +1 +10 +Radius [kpc] +1 +102 +104 +106 +108 +Cumulative Stellar Particle Number +TNG50 +z = 0 +2531 Galaxies +rcut +Figure 2. Number of stellar particles in the central 500 pc of our TNG50 +galaxy sample at z = 0. Cumulative number of stellar particles as a function +of radius per individual galaxy are shown as thin gray lines. The thick colored +lines show the average per galaxy stellar mass bin as depicted by the colorbar. +The dashed black line shows our adopted rcut value of 500 pc. The average +number of stellar particles within rcut lies between 103 and 105 for the lowest +and highest galaxy mass bins respectively. No individual galaxy in our sample +has less than 100 stellar particles in the center. +particles in the center to reliably study their properties. We can also +observe a turn-over in the stellar particle number profile at radii +around 0.5−1 kpc confirming that we are indeed probing the densest +(central) region of TNG50 galaxies. +3 THE DIFFERENT ORIGINS OF STARS IN THE CENTER +OF TNG50 GALAXIES +After selecting stars in the center of TNG50 galaxies at z = 0, we +investigate their different origins. We find three general populations +of stars in the central region of galaxies, which we describe in Section +3.1 in detail. We also present the distribution of their birth origin for +stacks in galaxy stellar mass in Section 3.2. +3.1 Definition of different origins +We define the following different origins of stars in the center of +TNG50 galaxies at z = 0: +• in-situ: stars were born inside the host galaxy’s center and are +still found there at z = 0. +• migrated: stars were born gravitationally bound inside the host +galaxy but outside its center. At z = 0 they reside in the host galaxy’s +center. +• ex-situ: stars were born inside other galaxies, which merged with +the host and are ultimately found inside the host’s center at z = 0. +MNRAS 000, 1–34 (2022) + +6 +Boecker et al. +3.1.1 Born inside or outside the host galaxy +To determine whether a star is born inside a galaxy or was brought +in through merger events, we use the stellar assembly catalogue5 +produced by methods of Rodriguez-Gomez et al. (2016) for TNG50. +This classifies stellar particles that formed along the main progenitor +branch of a galaxy, i.e. the galaxy with the most massive history +behind it, as in-situ (InSitu = 1) and otherwise as ex-situ (InSitu = +0). The ex-situ stars generally have two possible origins: they either +came from galaxies that completely merged with the main galaxy, +i.e. they are present in the merger tree of the host, or were stripped +from galaxies that do not belong to the host’s merger tree, e.g. flybys. +Additionally, we treat subhalos/satellites that directly merged onto +the main progenitor branch of a galaxy but are flagged as not be- +ing of cosmological origin (i.e. SubhaloFlag = 0 in the subhalo +catalogue) differently in this study. These subhalos are often formed +within another galaxy as e.g. a fragment of the baryonic disk, contain +little dark matter and hence are not thought of as galaxies (see also +Nelson et al. 2019a, Section 5.2). Because the construction of the +stellar assembly catalogue involves the use of merger trees, which +only track stellar particles and star-forming gas cells of subhalos, +these spurious galaxies are counted as of ex-situ origin. Here, we +change their labelling back to in-situ (i.e. their InSitu flag in the +stellar assembly catalogue becomes true again) for now (see Section +3.1.3 for the implications of this), because we only consider ex-situ +particles coming from true external galaxies. We verify with Figure +C1 in Appendix C that this change does not alter the overall total +ex-situ stellar mass fraction of TNG50 galaxies significantly. We note +that spurious galaxies brought to the main progenitor branch of the +host galaxy through prior merging with a real galaxy are continued +to be counted as ex-situ. +3.1.2 Born in-situ or migrated to the center +To address whether a stellar particle is born inside the center of the +host galaxy or migrated to the center from elsewhere inside the host +galaxy, we need to determine its birth radius. A stellar particle with +a birth radius smaller than rcut = 500 pc is then consequently born +in-situ and otherwise counts as migrated6. +In TNG, two new fields (BirthPos and BirthVel) for their stellar +particles were added. These represent the spatial position and velocity +of the star-forming gas cell that parented the stellar particle at its +exact time of birth (i.e. GFM_StellarFormationTime). In theory, +this provides us with knowledge of the exact birth condition of a +stellar particle at the original time step resolution of the simulation; +and not only at the output time steps of the snapshots. +Because these quantities are provided in the reference frame of +the simulation box, we need to center them on the reference frame +of the galaxy of interest. This however becomes an impossible task +to do to the precision needed for our analysis, as we only know the +center position of subhalos at the one hundred output snapshots, but +the information of its trajectory in-between is lost. We find that even +interpolating the subhalos’ position with a higher order spline to the +exact birth times of stars can lead to centering offsets of several kpc, +especially when there is a merger in process or a pericenter passage +around another galaxy (see Figure B2). As we are interested in typical +scales of one kpc or less in this study, this problem is severe and will +5 This particular catalogue has not been publicly released. +6 We here apply a simple cut in the birth radius instead of calculating 𝐸cut +(i.e. following Section 2.4), as the potential is not recorded for every snapshot. +result in a strong bias towards stars being classified as migrated even +though they where formed inside our selected spherical aperture. +We therefore define the birth position of stellar particles as the +position they have in the snapshot they first appear in. Practically this +is done by matching particles at z = 0 to their birth snapshot through +their unique ParticleIDs. The caveat of this approach is that the +stellar particles have already moved since their exact formation time, +which can also lead to a wrong classification of migrated and in-situ +stars. However, the error created by this approach is much smaller +than the incorrect centering described above (see Figure B3). +We verify this approach by looking at two subhalos that reside +in the subboxes of TNG50. The subbox has 3600 snapshot outputs, +which makes it possible to track the center position of galaxies across +a much finer time resolution of a few Myr. The reader is referred to +Appendix B2 for details. +3.1.3 Clumpy or smoothly migrated +Because we have changed the InSitu flag from the stellar assembly +catalogue for spurious galaxies, in Section 3.1.1, we now find two +types of migrated stellar particles in the center of galaxies. Stars +either travelled individually (‘smoothly’ migrated) or together in +clumps (‘clumpy’ migrated) to their galaxy’s center. Smoothly mi- +grated stars are genuinely born on the main progenitor branch of the +subhalo/galaxy in question and the clumpy migrated stars originate +from these spurious galaxies, i.e. stellar clumps. +Generally, these clumps are ubiquitous in TNG50 galaxies, about +36% of all galaxies considered in this work have at least one through- +out their life time. In stellar or gas surface mass density maps they +look like massive star cluster like objects (see Figure E1 for an exam- +ple) that form within spiral arms or gaseous (disk) fragments during +galaxy interactions. However, we want to be extremely cautious here, +as it is unclear, if their formation is physical or due to some numerical +artifact, even though measures against artificial fragmentation are in +place. In fact, their sizes (i.e. 3D stellar half mass radii) lie mostly +below the gravitational softening length of TNG50. +Once these clumps are formed, however, their dynamical evolution +within the host galaxy is determined by gravity, which we believe is +well captured in TNG50 (modulo the softening). Hence, depending +on their density and the exerted tidal forces on the clumps, they are +either completely disrupted or travel to the center of their host galaxy +due to dynamical friction and deposit their stellar particles there. +Their typical stellar masses are ∼ 108 M⊙. We point the interested +reader to Appendix E for more statistics on the clumps and their +properties. We provide an extensive discussion on the existence and +formation of stellar clumps in simulations and observation in Section +5.3. +For the rest of the paper, we sometimes make the distinction be- +tween migrated particles coming from the ‘smooth’ or ‘clumpy’ +migration, if it is explicitly stated. Otherwise, all general references +to migrated properties always include both types. +3.2 Birth locations of the central stars +The distributions of birth radii of in-situ, migrated and ex-situ central +stars are illustrated in Figure 3 in stacks of galaxy stellar mass. +The in-situ stars are born (by definition) in the center of the host +galaxy at z = 0. The peak of the birth radii distribution is around +200 − 300 pc for galaxies larger than 1010 M⊙ and shifts slightly +towards larger radii for the lower mass galaxy bins. We also see that +higher mass galaxies birth more in-situ stars at all radii and hence +are more centrally concentrated (see also Figure 2). +MNRAS 000, 1–34 (2022) + +Centers of TNG50 Galaxies +7 +0.001 +0.01 +0.1 +0.5 +Birth Radius [kpc] +106 +108 +1010 +Particle Mass [M ] +TNG50 +z = 0 +Insitu +1 +10 +100 +0.5 +Birth Radius [kpc] +TNG50 +z = 0 +Migrated +'Smooth' +1 +10 +100 +0.5 +Birth Radius [kpc] +106 +108 +1010 +Particle Mass [M ] +TNG50 +z = 0 +Migrated +'Clumpy' +5 × 1010 M +< M , tot +5 × 1012 M +1010 M +< M , tot +5 × 1010 M +5 × 109 M +< M , tot +1010 M +5 × 108 M +< M , tot +5 × 109 M +1 +102 +104 +106 +# Particles +1 +102 +104 +106 +# Particles +106 +108 +1010 +Particle Mass [M ] +TNG50 +z = 0 +Exsitu +106 +108 +1010 +Particle Mass [M ] +TNG50 +z = 0 +Exsitu +1 +102 +104 +106 +# Particles +1 +102 +104 +106 +# Particles +0.001 0.01 +0.1 +1 +10 +100 +Birth Radius [kpc] +w.r.t Host at Birth +0.001 +0.01 +0.1 +1 +10 +100 +Radius [kpc] +w.r.t Primary at Stripping +TNG50 +z = 0 +Exsitu +20% +50% +90% +99% +1 +102 +104 +# Particles +106 +108 +Particle Mass [M ] +Figure 3. Distribution of birth radii of in-situ, migrated and ex-situ stellar populations in the central 500 pc of TNG50 galaxies at z = 0. Left: Histograms +of birth radii of the in-situ, ‘smooth’ migrated and ‘clumpy’ migrated stars colored according to stacks of galaxy stellar mass. The left hand side of the y-axis +shows stacked particle mass, whereas the right hand side shows the number of stellar particles. The migrated stars can originate from large radii (> 10 kpc) +and the smoothly and clumpy migrated stars show different distributions. Right: 2D histogram of the birth radius with respect to the host galaxy at birth and +the radius with respect to the primary galaxy (i.e. final z = 0 host) at the time of stripping for all central ex-situ stars in our galaxy sample. The contours (from +thicker to thinner lines) include 20%, 50%, 90% and 99% of all ex-situ particles. The respective 1D histograms for stacks in galaxy stellar mass are also shown. +Most ex-situ stars are born in the central ∼ 1 kpc of their birth galaxy and stay together until they are deposited also within the central ∼ 1 kpc of their z = 0 host +galaxy. +3.2.1 Individually migrated stars originate close to the galaxy’s +center +Most of the smoothly migrated stellar particles were also born close +to the center with radii between 500 pc and 1 kpc, which is partly +due to how we have defined them (i.e. purely based on their birth +radius) and partly a consequence of the typical density profile of +galaxies (i.e. more stars reside in the center of galaxies). For galaxies +below 1010 M⊙ the distribution of birth radii declines exponentially +reaching the highest values of about 10 kpc. The lowest mass galaxies +in our sample (≤ 5 × 109 M⊙) have 11% of smoothly migrated stars, +which are born in the range of 1 − 10 kpc, whereas this increases +slightly to 17% for the next higher mass bin (≤ 1010 M⊙). +For galaxies above 1010 M⊙ we observe a plateau for the distribu- +tion of birth radii starting at ∼ 10 kpc, which stops at around 30 kpc +and 60 kpc for galaxies with ≤ 5 × 109 M⊙ and > 5 × 109 M⊙ re- +spectively. The migrated stars originating from these large distances +likely come from gas that was stripped during a merger, but which +was already attributed to be gravitationally bound to the primary +galaxy according to the Subfind algorithm and hence was counted +as being born in-situ to the primary host. The percentage of smoothly +migrated stars with birth radii larger than 1 kpc is around 20% and +14% for the two highest stellar mass bins respectively. +3.2.2 Stars migrated in clumps originate from the outskirts of +galaxies +The clumpy migrated stars show a distinctively different distribution +than the smoothly migrated ones. For galaxies below 1010 M⊙ their +contribution is negligible. For galaxies between 1010 M⊙ and 5 × +1010 M⊙ the clumpy migrated stars are only 3% of the total migrated +stars, whereas for galaxies above 5 × 1010 M⊙ the contribution rises +to almost 50%. Therefore, clump migration is only important for high +mass galaxies, where it becomes the dominant driver for contributing +migrated stars in the centers (see also Section E). +Furthermore, the peak of the birth radii distribution of clumpy +migrated stars is above 10 kpc for the high mass galaxies. This is in +agreement with the fact that the gaseous disk of galaxies is much +more extended than the stellar one (e.g. Nelson et al. 2012). Stars +travelling in stellar clumps are therefore able to migrate to the center +of galaxies from much farther distances compared to when they travel +individually. +3.2.3 Central ex-situ stars originate from the nuclei of their birth +galaxies +Regarding the ex-situ stars, we investigate two different locations: 1) +their birth place with respect to their birth host galaxy and 2) the +location they were deposited inside their z = 0 host (primary). The +latter is defined as the radius the stellar particles have with respect to +the primary at stripping time, i.e. the time they last switched galaxies. +We show the distribution of these two quantities also in Figure 3 in +the same stacks of galaxy stellar mass, as well as the 2D distribution +of all ex-situ stars for these two radii. About half of all ex-situ stars +that reside in the center of galaxies at z = 0 exhibit values between +100 pc and 1 kpc for both radii respectively. This means that the ex- +situ stars are also born in the center of their respective birth galaxies +MNRAS 000, 1–34 (2022) + +8 +Boecker et al. +as well as remain in said center until they are deposited right in the +center of the primary galaxy during the merger process. Hence, the +central, most bound cores of galaxies are more likely to stay together +during accretion events until they arrive close to the center of the +primary galaxy and ultimately deposit a large quantity of stars there. +This is a consequence of mergers preserving the rank order of the +particles’ binding energy (Barnes 1988; Hopkins et al. 2009). +We also find two other cases of ex-situ stars, albeit much lower +in number. Firstly, TNG predicts a slight excess of ex-situ stars that +are born at larger radii (1 − 100 kpc), but are still deposited close +to the primary galaxy at stripping time, i.e. within ∼ 1 kpc. These +stars represent a second generation of ‘migrated’ stars; or likely in +the case of ex-situ stars with birth radii of ≥ 10 kpc, stars that were +formed from stripped gas during secondary mergers, which only +appear for the most massive z = 0 hosts. Consequently, these ex-situ +stars were born at large radii in their respective host galaxies (i.e. +which will become the secondary galaxy during the merger process +onto the z = 0 host), then migrated to the center of said galaxy in +order to be deposited close to the center of the primary host during +accretion. We confirm this by explicitly checking that their radii are +indeed central (≲ 1 kpc) with respect to the merging host galaxy one +snapshot before the merger coalesces. +The second case represents ex-situ stars that were deposited at +larger radii from the primary (> 1 kpc), but born within the central +1 kpc of their birth galaxy. Despite being stripped outside the center +of the z = 0 host galaxy, these stars still were able to migrate such that +they are found in the center of their respective galaxy at z = 0. There +is a possibility that these stars were stripped earlier, i.e. before the +merger coalesces, but their dynamics were still following the orbit of +the galaxy undergoing the merger and hence they could arrive at the +center of the final host galaxy. +4 THE CENTRAL IN-SITU, MIGRATED AND EX-SITU +POPULATIONS ACROSS TNG50 GALAXIES +In this section we present our results of in-situ, migrated and ex-situ +populations within the central ∼ 500 pc of TNG50 galaxies. +We study their contributions across different galaxy properties +(Section 4.1) and examine differences in their stellar population and +dynamical properties (Section 4.2). +4.1 Galaxy population demographics +Below we depict the contribution of the central stellar mass of the +different origins as an overall trend with galaxy mass (Section 4.1.1), +in correlation to each other (Sec 4.1.2) and for different galaxy types +(Section 4.1.3). +4.1.1 Galaxy mass trends +In Figure 4 we give an overview of the absolute and relative contri- +bution of the ex-situ, migrated and in-situ population across galaxy +masses (both stellar and dynamical) in TNG50. +For all three populations the central stellar mass increases with +increasing galaxy mass with the in-situ population dominating at all +galaxy masses. Whereas the relation for the in-situ and the smoothly +migrated stars have the same shape, the slope for the ex-situ popu- +lation is steeper. The latter also shows a larger overall scatter due to +the stochasticity of merger events contributing stars to the center. +Even though the fractional mass of the ex-situ population in the +center is negligible for galaxy stellar masses below 1011 M⊙, there +are only 227 (9%) galaxies in our total sample that have no central +ex-situ mass, i.e. they do not possess a single stellar particle of ex- +situ origin in their central 500 pc, or in other words they possess less +than ∼ 8 × 104 M⊙, the mass of a stellar particle, in ex-situ stars. +Above 1011 M⊙ the ex-situ mass becomes of the same order as the +in-situ and migrated population, which is a consequence of mergers +contributing a significant amount stellar mass to build up of these +galaxies. The ex-situ mass reaches about 10% of the total central +stellar mass at the highest galaxy masses, albeit with a large scatter +of up to 60%. +Around galaxy stellar masses of about 5 × 1010 M⊙, the relation +flattens for the in-situ and smoothly migrated stars, with the in-situ +population reaching about 4% of the total galaxy stellar mass. Al- +though we have low number statistics of galaxies in this regime within +the TNG50 volume (there are 18 galaxies with stellar masses above +5×1011 M⊙), it is reasonable that the in-situ mass goes down, because +the ex-situ mass increases in addition to galaxies being quenched +by AGN feedback. The consequential increased stochasticity is also +seen by the larger scatter in the in-situ and migrated population at +the highest galaxy stellar masses7. +The contribution of clumpy migrated stars to the overall central +migrated population only starts to significantly affect galaxies with +stellar masses higher than 5 × 1010 M⊙. For galaxies higher than +2 × 1011 M⊙ the clumps are responsible for roughly quadrupling +the mass of migrated stars, or, in fractional terms, increasing the +contribution of migrated stars to the total central mass from below +10% to slightly above 20%. Hence, the clumps are important driver +to bring in stars from the outskirts of galaxies in TNG50. +Taking +into +account +the +entire +migrated +population +(‘smooth+clumpy’), we find a contribution of around 20% to +the total stellar mass in the center across all TNG50 galaxies. +Interestingly, the total central migrated fraction around galaxy stellar +masses of ∼ 1010 M⊙ slightly increases, with the 84th percentile +reaching almost 40%. We explicitly confirm that this is not due to +mixing galaxies with different sizes and hence different total central +stellar masses (see also Figure 1). +The statements made so far also apply when correlating the central +stellar masses of the three populations with the total dynamical mass +of TNG50 galaxies. The larger scatter in all three relations is due to +the scatter in the stellar-to-halo mass relation. +4.1.2 The diversity of central stellar mass at fixed galaxy mass +In Figure 5 three correlations between the central ex-situ, migrated +and in-situ stellar mass as 2D Gaussian kernel estimates in bins of +total galaxy stellar masses are shown. The bins were specifically +chosen based on the change of the average migrated fraction as a +function of galaxy stellar mass and, in case of the highest mass bin, +to ensure enough galaxies per bin to reliably perform the kernel +density estimate. +In-situ vs. migrated mass (Figure 5, first panel): At all stellar +masses, the mass of migrated stars correlates strongly with the in- +situ stellar mass with three times as much in-situ than migrated mass. +This reflects our previous statements that the shape of the median +7 Another possibility for the large scatter at high galaxy stellar masses for the +in-situ and migrated central stellar mass could be stars formed from accreted +gas, which was brought in by gas-rich mergers. We do not quantify this further +as it is beyond the scope of this study. +MNRAS 000, 1–34 (2022) + +Centers of TNG50 Galaxies +9 +104 +105 +106 +107 +108 +109 +1010 +Central Stellar Mass [M ] +TNG50 +z = 0 +2531 Galaxies +Insitu +Migrated 'Smooth+Clumpy' +Migrated 'Smooth' +Exsitu +TNG50 +z = 0 +2531 Galaxies +109 +1010 +1011 +1012 +Total Stellar Mass [M ] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Central Stellar Mass Fraction +TNG50 +z = 0 +1011 +1012 +1013 +Total Dynamical Mass [M ] +TNG50 +z = 0 +Figure 4. Central (500 pc) stellar mass of the in-situ, migrated and ex-situ populations of TNG50 galaxies at z = 0. Median trends of central stellar mass +(top panels) and central stellar mass fraction (bottom panels) as a function of the galaxies’ total stellar mass (left) and total dynamical mass (right) divided in +the three origins: in-situ (pink), migrated (orange) and ex-situ (blue). The migrated population is shown for both ‘smooth+clumpy’ (solid line) and just ‘smooth’ +(dashed line) migration (see 3.1.3 for details). Shaded areas show the 16th and 84th percentiles. Overall the in-situ population dominates on average across +all galaxy masses with the migrated population contributing around 20% to the total central stellar mass. Only above galaxy masses of 1011 M⊙ the ex-situ +population starts to significantly contribute to the central mass build-up. +107 +108 +109 +1010 +Migrated Mass in Center [M ] +107 +108 +109 +1010 +Insitu Mass in Center [M ] +TNG50 +z = 0 +5 × 108 M +< M , tot +5 × 109 M +5 × 109 M +< M , tot +1010 M +1010 M +< M , tot +5 × 1010 M +5 × 1010 M +< M , tot +5 × 1012 M +1% +20% +50% +90% +107 +108 +109 +1010 +Insitu Mass in Center [M ] +104 +105 +106 +107 +108 +109 +1010 +1011 +Exsitu Mass in Center [M ] +TNG50 +z = 0 +107 +108 +109 +1010 +Migrated Mass in Center [M ] +TNG50 +z = 0 +107 +108 +109 +1010 1011 1012 +Total Exsitu Mass [M ] +TNG50 +z = 0 +Figure 5. 2D distributions of the central (500 pc) in-situ, migrated, ex-situ as well as the total ex-situ stellar mass of TNG50 galaxies at z = 0. Gaussian +kernel density estimates for different combinations of in-situ, migrated and ex-situ mass in the center as well as total ex-situ mass color-coded according to four +different total stellar mass bins. The contours show different percentiles encompassing 1%, 20%, 50% and 90% of all data points (thickest to thinnest line). +The black dashed line shows the one-to-one relation in all panels. The correlation between the central mass of the in-situ and migrated population follow the +one-to-one relation closely with a fixed offset, whereas the central ex-situ population exhibits a large scatter across the other central populations. +MNRAS 000, 1–34 (2022) + +10 +Boecker et al. +relation of the central in-situ and migrated mass versus the total +stellar galaxy mass is similar. +Nevertheless, for some galaxies the migrated mass is larger than +the in-situ mass in the center as seen by the 90% contours for the three +highest galaxy mass bins in the range of 5×109−5×1012 M⊙. Galax- +ies in this regime are dominated by clumpy migration. Hence, the +mass contributed to the center by clumps can be significant enough to +break the otherwise tight one-to-one relation of migrated and in-situ +mass. +Lastly, we find for the 90% contour in the highest mass bin for +galaxies above 5 × 1010 M⊙ that there is a larger tail of galaxies with +lower in-situ and migrated mass in the center. Most galaxies situated +in this space have a high ex-situ central mass fraction of 40% or +higher. +Ex-situ vs. in-situ and migrated mass (Figure 5, second and third +panels): At roughly fixed central in-situ or migrated masses there is a +large variety of ex-situ mass that is deposited in the center of galaxies. +The scatter of the ex-situ mass in the center increases roughly from +four to six dex the from smallest to largest galaxy stellar mass bin. +Compared to that the scatter in the in-situ and migrated mass direction +is rather small, being roughly one dex across all galaxy stellar mass +bins. However, some galaxies in mass bins below 5 × 1010 M⊙ have +lower in-situ masses of up to one dex below the majority of the other +galaxies in their respective bins. Galaxies lying in this region have +above average migrated fractions of 40% or more with some reaching +extreme values of above 80%. +The spread in migrated mass compared to the in-situ mass is larger +for the highest mass galaxies. This is mainly due to the increased +stochasticity in the total central stellar mass for the 18 galaxies above +5 × 1011 M⊙ in galaxy stellar mass, which almost spans one dex +as opposed to only a quarter dex for galaxies between 5 × 1010 and +5×1011 M⊙. These 18 galaxies lie between the 50% and 90% contour +and have ex-situ masses spanning from 107 to 1011 M⊙. Their in- +situ masses are exclusively below the 50% contour, whereas their +respective migrated masses can lie towards lower or higher values. +The peak of the central ex-situ mass distributions begins to rise +for galaxies above 1010 M⊙ in total stellar mass, going from about +three dex below the one-to-one relation to one dex. This break +point roughly translates to 4 × 108 M⊙ in central in-situ mass and +1 − 2 × 108 M⊙ in central migrated mass. The former roughly +coincides with the critical mass needed for the SMBH to be in the +kinetic feedback mode (e.g. Zinger et al. 2020, Figure 1). The total +ex-situ mass also begins to rise for galaxies with total stellar masses +above a few 1010 M⊙ (see Figure C1). +Central ex-situ vs. total ex-situ mass (Figure 5, fourth panel): +Lastly, we also show the correlation between the central ex-situ mass +and the total ex-situ mass, i.e. all stars that were ever accreted onto the +z = 0 host galaxy. For fixed total galaxy stellar mass the slope of the +contours depict that a higher total ex-situ mass generally also implies +a higher central ex-situ mass. The slope of this correlation is rather +steep. While the total ex-situ mass spans approximately two dex per +galaxy stellar mass bin, the ex-situ mass in the center spans four to six +dex from the lowest to highest galaxy stellar masses. Consequently, +it is quite stochastic which merging satellite galaxies deposit stellar +mass in the center. +Furthermore, the central density contours shift closer to the one- +to-one relation with increasing galaxy stellar mass. This means that +more galaxies in the highest mass bin have mergers that are more +effective in bringing a larger fraction of their total ex-situ mass into +their center as opposed to lower mass galaxies. Nevertheless, the +90% contours for galaxy stellar masses above 5 × 109 M⊙ extend +right up to the one-to-one relation, meaning that there some galaxies +that have almost all their ex-situ mass in the central 500 pc. +4.1.3 Trends for different galaxy types +Galaxies with different present-day properties are thought to have +undergone different formation pathways. Is this reflected in different +contributions of in-situ, migrated and ex-situ stars building up the +center of these galaxies? +In Figure 6 we show the running median of the central stellar mass +of the three origins as a function of total galaxy stellar mass split +into six different galaxy properties. The definitions of the different +galaxy properties are summarized in Table 1 and described in detail +in Appendix A1. +All in all, the most significant differences are seen in the central +ex-situ population across various galaxy properties. This significance +manifests in separation between the median relations including +the scatter (which we do not show, however, in favour of clar- +ity). Small differences for the in-situ and migrated populations are +not significant with respect to the scatter around the median relations. +Centrals vs. Satellites (Figure 6, top left panel): On average, +centrals and satellites contain the same amount of central in-situ, +migrated and ex-situ mass, showing that their central 500 pc +is unaffected by their environment at z = 0. This is sensible +considering that galaxy centers likely assemble before the galaxy +becomes a satellites. Additionally, most environmental effects +should first take effect in the outskirts of galaxies. Similarly, we +find no significant difference in the central mass of the three pop- +ulations, when the galaxies are divided by the mass of their host halo. +Quenched vs. Star Forming (Figure 6, top middle panel): +Quenched galaxies between 5 × 109 and 5 × 1010 M⊙ have slightly +higher central in-situ and migrated mass than for star forming ones. +This difference primarily arises because the star forming galaxies +tend to have lower central densities on average than quenched ones. +A larger difference is seen in the ex-situ. For galaxy stellar masses +above 1010 M⊙ the average ex-situ mass starts to rise more rapidly +for star forming galaxies than for quenched ones. For galaxies around +5×1010 M⊙ this difference becomes largest, with the median central +ex-situ mass of star forming galaxies being higher by more than one +dex. +While this trend may seemingly be counter-intuitive for the +current consensus of galaxy evolution, the difference is also true +when considering the total ex-situ stellar mass in TNG50 (and also +TNG100) as seen in Figure C1 in Appendix B2. This could be an +indication that today’s star forming galaxies had more or larger +mass ratio mergers with galaxies with high gas content at later +cosmic times (see Section 5.1 for a further discussion). We obtain a +consistent picture when galaxies are divided according to their 𝑔 − 𝑖 +colour or total gas mass at z = 0. +Bulgey vs. Disky (Figure 6, top right panel): The in-situ and +migrated central mass for disky and bulgey galaxies show a similar +trend as for the star forming and quenched population. However, the +trend for the central ex-situ mass is distinct. Bulgey galaxies below +1010 M⊙ in stellar mass have higher ex-situ masses (by roughly half +a dex) in their centers than their disky counterparts. This difference +disappears for galaxy stellar masses above 1010 M⊙. +We have checked the median relation for the total ex-situ mass and +find that bulgey galaxies have a constant higher offset of about 0.25 +MNRAS 000, 1–34 (2022) + +Centers of TNG50 Galaxies +11 +109 +1010 +1011 +1012 +Total Stellar Mass [M ] +105 +106 +107 +108 +109 +1010 +Central Stellar Mass [M ] +TNG50 +z = 0 +Insitu +Migrated 'Smooth+Clumpy' +Exsitu +Centrals (1678) +Satellites (853) +109 +1010 +1011 +1012 +Total Stellar Mass [M ] +TNG50 +z = 0 +Quenched (615) +Star Forming (1916) +109 +1010 +1011 +1012 +Total Stellar Mass [M ] +TNG50 +z = 0 +Bulgey (1150) +Disky (1381) +109 +1010 +1011 +1012 +Total Stellar Mass [M ] +105 +106 +107 +108 +109 +1010 +Central Stellar Mass [M ] +TNG50 +z = 0 +Insitu +Migrated 'Smooth+Clumpy' +Exsitu +'Barred' (1610) +No 'Bar' (921) +109 +1010 +1011 +1012 +Total Stellar Mass [M ] +TNG50 +z = 0 +Overmassive BH (832) +Undermassive BH (1699) +109 +1010 +1011 +1012 +Total Stellar Mass [M ] +TNG50 +z = 0 +Extended (1888) +Compact (643) +Figure 6. Differences in the central (500 pc) in-situ, migrated and ex-situ populations for different galaxy properties of TNG50 galaxies at z = 0. Each +panel shows median trends of central stellar mass divided in the three origins: in-situ (pink), migrated (orange) and ex-situ (blue) as a function of the galaxies’ +total stellar mass. The dashed and solid lines split the TNG50 galaxy population according to different properties, which are (form left to right): central vs. +satellite, quenched vs. star forming, bulgey vs. disky, ‘barred’ vs. ‘non barred’, overmassive vs. undermassive black holes (BH) and extended vs. compact +galaxies. The bracketed numbers show the total amount of galaxies in each category. +dex compared to disk galaxies across the whole galaxy mass range. +Hence, disky galaxies below 1010 M⊙ have not only lower absolute +central and total ex-situ masses, but also a lower central-to-total ex- +situ fraction of about 0.4% as compared to 1% for bulge dominated +galaxies. Thus the relative amount of ex-situ mass that is deposited +in the center might be an important driver for morphological trans- +formation in these lower galaxy mass regimes. +For galaxies above 1010 M⊙ in stellar mass, the central-to-total +ex-situ fraction decreases strongly as a function of galaxy stellar +mass, with disk galaxies having consequently slightly higher values. +This could be an indication that once a massive rational support +exists in the stellar component it is hard to destroy it through mergers. +Similar relations are found when adopting other definitions for disky +and bulgy galaxies, such as the ratio of the kinetic energy in ordered +motion compared to the total kinetic energy (see Rodriguez-Gomez +et al. 2017). +Barred vs. No Bar (Figure 6, bottom left panel): For galaxies +below 1010 M⊙ TNG50 predicts no difference in the central in-situ +and migrated mass, however the galaxies with bar-like features have +higher ex-situ masses than galaxies with no bar-like features. This +trend is similar for the bulgey vs. disky galaxies. We have explicitly +checked that indeed high ex-situ masses in the center of galaxies +within this mass regime mainly occur in bulgey and barred galaxies, +whereas bulgey and unbarred galaxies as well as disky galaxies, both +barred and unbarred, have lower central ex-situ masses by approxi- +mately one dex. +For galaxies above 1010 M⊙ in total stellar mass, this relation for +the central ex-situ mass swaps. In this regime unbarred galaxies have +higher ex-situ masses in the center regardless whether they are disky +or bulgey. We find that the same statements for barred and unbarred +galaxies across the entire mass range are true when correlating the +total ex-situ mass of galaxies. +Lastly, the in-situ and migrated mass in the center is higher for +barred galaxies between 1010 M⊙ and 1011 M⊙. Hence, barred +galaxies in this mass regime have higher central densities than +unbarred galaxies in TNG50, which is consistent with observations +(see Díaz-García et al. 2016b). +Over- vs. Undermassive Black Holes (Figure 6, bottom middle +panel): We could expect that AGN feedback has an influence on the +stellar mass growth in the center of galaxies. We therefore split our +TNG50 sample according to whether the galaxies have an over- or +undermassive black hole at z = 0. Identical relations are found when +the galaxy population is split according to the cumulative energy +injection of each feedback mode or both. +MNRAS 000, 1–34 (2022) + +12 +Boecker et al. +On average, galaxies between 1010 M⊙ and 1011 M⊙ in stellar +mass with an undermassive black hole have a higher central ex-situ +mass by about one dex than galaxies with an overmassive black hole +at the same stellar masses. For galaxies with total stellar masses in +the range of 5 × 109 − 5 × 1010 M⊙, the ones with an overmassive +black hole have in-situ and migrated masses in the center that are +about half a dex higher than for galaxies with an undermassive black +hole. Consequently, galaxies with overmassive black holes in this +mass regime have higher central densities. +We find that mainly all of these differences in the in-situ, migrated +and ex-situ mass for galaxies with over- and undermassive black +holes emerge because galaxies at fixed stellar mass with overmassive +black holes tend to be more compact in TNG50 and vice versa. +Therefore, a similar behaviour of the central stellar mass in the +three populations with total galaxy stellar mass is found when +the galaxy population is split into compact and extended galaxies +(see below). This connection between black hole masses, central +densities and sizes of galaxies at fixed galaxy stellar mass is also +found in observations (Chen et al. 2020). +Extended vs. Compact (Figure 6, bottom right panel): Extended +galaxies tend to have on average more ex-situ mass in the center than +compact galaxies at the same total stellar mass. The difference is +around one dex for galaxies above 1010 M⊙ in stellar mass. +When we correlate with the total ex-situ mass, we find an opposite +behaviour in TNG50. Galaxies ≲ 5 × 1010 M⊙ and with higher total +ex-situ fractions are on average more extended (see Figure C2 in +Appendix C). +Compact galaxies between 5 × 109 M⊙ and 5 × 1010 M⊙ have +more in-situ and migrated mass in the center, and therefore higher +central densities (and black hole masses, see above). As a matter of +fact, this difference is also seen for quenched vs. star forming and +bulgey vs. disky galaxies, even though to a lesser extent. This stems +from the fact that generally star forming galaxies tend to be more +disky and hence more extended and vice versa. +4.2 Stellar population and dynamical properties +Are there distinguishable features in the stellar population and dy- +namical properties of the in-situ, migrated and ex-situ stars? The +short answer is yes, especially for galaxies below ≲ 1011 M⊙, where +the majority of ex-situ stars originate from lower mass satellites. +4.2.1 Average age, metallicity and [Mg/Fe] of central stars +Metallicities, ages and magnesium-to-iron abundances [Mg/Fe] of +stars encode information about their birth places. Figure 7 illustrates +average quantities of stellar populations belonging to the in-situ, +migrated and ex-situ origin as a function of their galaxy’s stellar +mass. We also show separate relations for migrated stars that have +birth radii larger than 1 kpc to exclude the majority of migrated stars +that were born close to the center, which dominate the average stellar +population properties (see smoothly migrated stars in Figure 3). +Metallicity (Figure 7, left panel): Stars in the central 500 pc follow +a mass-metallicity relation, where galaxies at the lowest mass end +(5 × 108 M⊙) have on average solar metallicity and galaxies at the +highest mass end (∼ 1012 M⊙) have metallicities of around 0.5 dex. +The total mass-metallicity relation of all the stars in the center is very +close to the one for the in-situ population only, as they dominate the +mass in the center of galaxies on average (see Figure 4). +Furthermore, the average metallicity for central stars is consistently +offset by about 0.3 dex towards higher metallicities across the whole +galaxy mass range compared to the mass-metallicity relation which +takes into account all the stars belonging to a given galaxy. This +emphasizes the self-similarity of galactic chemical enrichment. +On top of that, the total central mass-metallicity relation is tight +having a scatter of around 0.1 dex, which also holds when only in-situ +or only migrated stars are considered. Hence, there is little galaxy-to- +galaxy variation at fixed stellar mass regarding in-situ star formation. +The average metallicity of the in-situ population is the highest, +followed by the migrated stars, which is less than a quarter dex +lower across the whole galaxy mass range. This small difference is +expected as most of the migrated stars are born very close to the +center (0.5 − 1 kpc). When including only migrated stars with large +birth radii (> 1 kpc), the difference becomes larger to about half a +dex due to internal metallicity gradients present in galaxies, which is +in turn caused by less efficient star formation in the galactic outskirts. +Above galaxy stellar masses of 2 × 1011 M⊙ the average metallicity +of all migrated stars and only those with birth radii larger than 1 kpc +becomes similar again. This is because migrated stars from clumps +are dominating at these galaxy masses, which originate from larger +distances (median distance is 30 kpc) and have high metallicities +(median metallicity is 0.2 dex). +The mass-metallicity relation for the central ex-situ stars follows a +steeper slope than the one for the in-situ and migrated stars, because +we are showing the mass of the z = 0 host galaxy and not of the galaxy +they were born in. The average metallicity of ex-situ stars is around +0.5 dex lower at the lowest galaxy masses compared to the metallicity +for the in-situ stars. At the highest mass end the average metallicity +of the ex-situ stars becomes close to the one for the migrated stars, +which is around 0.25 dex. This steeper slope emphasizes that ex-situ +stars in the center of low mass galaxies originate from galaxies of +even lower mass, while most of the central ex-situ stars in high mass +galaxies originate from galaxies of more similar stellar mass. +Lastly, the galaxy-to-galaxy variation at fixed galaxy stellar mass +for the average metallicity of ex-situ stars is much larger compared to +the in-situ and migrated population. The scatter varies from around +one dex at the low mass galaxy end to close to a quarter dex for the +highest galaxy masses. This emphasizes that at lower host galaxy +stellar mass, a larger variety of satellite galaxies (i.e. with different +stellar masses) can deposit stars in the center of their respective +z = 0 hosts. +Age (Figure 7, middle panel): The ex-situ stars have a rather con- +stant, old age of around 10 Gyr across the whole galaxy mass range, +albeit with a large scatter of around 2 Gyr. This is not surprising as +most mergers happen before the redshift of one, which corresponds to +a lookback time of around 8 Gyr. The flat relation for the average age +of the ex-situ stars is not in conflict with their corresponding mass- +metallicity relation. Because high mass galaxies are more efficient in +chemical enrichment than low mass galaxies, they will consequently +have higher metallicities at fixed stellar age. +The median relations for the average age for the in-situ and mi- +grated stars are again similar to each other with the in-situ stars being +slightly older by around 1 Gyr or less at fixed galaxy stellar mass. +Overall, in-situ and migrated stars are younger, with average ages +between 3 and 6 Gyr, in the lowest mass galaxies (∼ 109 M⊙), and +become increasingly older with average ages of around 8 − 10 Gyr at +the highest mass end (∼ 1012 M⊙). +The scatter of the average ages for the in-situ and migrated stars is +much larger than their corresponding variations in metallicity. This +could have multiple reasons, for example: different pathways in star +MNRAS 000, 1–34 (2022) + +Centers of TNG50 Galaxies +13 +109 +1010 +1011 +1012 +Total Stellar Mass [M ] +1.5 +1.0 +0.5 +0.0 +0.5 +Central Metallicity [log10(Z/Z )] +TNG50 +z = 0 +Insitu +Migrated 'Smooth+Clumpy' +Migrated Birth Radius +1 kpc +Exsitu +109 +1010 +1011 +1012 +Total Stellar Mass [M ] +2 +4 +6 +8 +10 +12 +14 +Central Age [Gyr] +TNG50 +z = 0 +109 +1010 +1011 +1012 +Total Stellar Mass [M ] +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Central [Mg/Fe] [dex] +TNG50 +z = 0 +Figure 7. Average stellar population properties of in-situ, migrated and ex-situ stars in the central 500 pc of TNG50 galaxies at z = 0. From left to right: +The central mass-weighted average metallicity, age and magnesium-to-iron abundance [Mg/Fe] as a function of total galaxy stellar mass for the in-situ (pink), +migrated (orange) and ex-situ (blue) stars. The shaded bands depict the 16th and 84th percentile. The dashed orange line only shows the quantities for migrated +stars that have birth radii larger than 1 kpc. Overall, the stellar population properties of in-situ and migrated stars are similar, whereas the ex-situ stars are more +metal-poor, older and have higher [Mg/Fe] across the galaxy mass range. +formation histories (i.e. star formation rate as a function of time) +can result in the same metallicity but different average ages, or the +metallicity enrichment starts to saturate once a metallicity above +solar is reached and therefore it does not matter, if star formation +continues for another few Gyr. +Galaxies above 1011 M⊙ in stellar mass exhibit a larger scatter of +the average age of their migrated population compared to their in- +situ stars. This arises because migration to the center in this regime +is dominated by clumps, which have a rather flat formation time +distribution with the majority forming between 4 and 10 Gyr ago +(see Figure E1). +Below galaxy stellar mass of 1011 M⊙, the migrated stars, which +were born at distances larger than 1 kpc, have a running median of +averages ages that are around 1 − 2 Gyr older than the total migrated +population. As these stars need significantly more time to arrive in +the center, their ages are consequently older. +[Mg/Fe] (Figure 7, right panel): In extragalactic studies magne- +sium is the predominant 𝛼-element present in optical spectra (see +e.g. Martín-Navarro et al. 2018a, 2019, 2021; Gallazzi et al. 2021). +We therefore show the running median of the mass-weighted average +magnesium-to-iron abundance as a function of galaxy stellar mass +as a proxy for the total 𝛼-to-iron abundance. This abundance ratio +provides to first hand information about the star formation time scale +before supernovae type Ia significantly enrich the interstellar medium +with iron peak elements8. +The average central [Mg/Fe] is almost constant for the in-situ +population across the whole galaxy mass range with a value of about +0.3 dex. For galaxies between 2 × 109 M⊙ and 6 × 1010 M⊙, the +migrated stars have slightly lower values. The lowest average [Mg/Fe] +of around 0.25 dex is reached for galaxies around 2 × 1010 M⊙. This +directly maps to the increased difference of the average age between +8 Influences on [𝛼/Fe] due to IMF (initial mass function) changes are not +captured in the simulation, as a Chabrier IMF (Chabrier 2003) is assumed for +every stellar particle +in-situ and migrated stars of around 1 Gyr in the same mass regime. +Hence, in-situ stars of these galaxies form on average earlier and +more rapidly as opposed to their migrated stars. +Above 6 × 1010 M⊙, the average [Mg/Fe] for the migrated pop- +ulations rises above the one for the in-situ stars to around 0.35 dex +at the highest mass end. This cross-over is not seen in the average +ages. An explanation for this could be that in the high galaxy mass +regime, an increasing number of migrated stars can originate from +larger distances and possibly formed from stripped gas of merging +lower mass systems (see Section 3.2), which have larger [Mg/Fe] +values due to lesser efficiency in chemical enrichment. +When only including migrated stars originating from distances +farther than 1 kpc away from the center, the average [Mg/Fe] be- +comes larger by around 0.1 dex across all galaxy stellar masses. For +galaxies below ∼ 1011 M⊙ the corresponding ages become older, +which is thus consistent in having formed from true in-situ gas of +their respective host galaxies. +The age for migrated stars with birth radii > 1 kpc in galaxies +above 1010 M⊙ does not increase even though their [Mg/Fe] increase +as well. This could indeed provide evidence for some migrated stars +having formed from stripped gas for galaxies in this mass regime. +The ex-situ stars have overall higher average [Mg/Fe] values of +around 0.45 dex, which decreases to around 0.35 dex for galaxies +above 1011 M⊙ in stellar mass. This is consistent with their old ages +and being formed in lower mass satellite galaxies that produced stars +less efficiently than their respective z = 0 hosts. +The scatter in average [Mg/Fe] for the ex-situ population is signif- +icantly larger than for the in-situ and migrated population across all +galaxy masses, but especially ≲ 1011 M⊙. The onset of type Ia su- +pernovae creates probably more stochasticity in lower mass galaxies +as single supernovae events can significantly enrich the interstellar +medium of the entire host galaxy. +MNRAS 000, 1–34 (2022) + +14 +Boecker et al. +109 +1010 +1011 +1012 +Total Stellar Mass [M ] +1.0 +0.5 +0.0 +0.5 +1.0 +Central Circularity +TNG50 +z = 0 +Insitu +Bulgey +Disky +109 +1010 +1011 +1012 +Total Stellar Mass [M ] +TNG50 +z = 0 +Migrated +Bulgey +Disky +109 +1010 +1011 +1012 +Total Stellar Mass [M ] +TNG50 +z = 0 +Exsitu +Bulgey +Disky +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +Fraction in +per Stellar Mass Bin +Figure 8. Average differences in the dynamical properties of in-situ, migrated and ex-situ stars in the central 500 pc of TNG50 galaxies at z = 0. From +left to right: The central circularity distributions in stacks of total galaxy stellar mass for in-situ (pink), migrated (orange) and ex-situ (blue) stars. Per galaxy, +stellar particles in the center belonging to either the in-situ, migrated or ex-situ population are binned according to their circularity 𝜖 and normalized to unity +respectively. They are then stacked together according to the displayed galaxy stellar mass bins and then normalized again. The lines trace the circularity bin +with the maximum fractional mass across galaxy stellar masses divided by bulgey (solid) and disky (dashed-dotted) galaxies respectively. Clearly the migrated +population has the most rotational support for galaxies around 1010 M⊙ in total stellar mass, regardless of the host being disk or bulge dominated. +4.2.2 Stacked circularity distributions +Different birth origins also leave imprints on the stars’ dynamics, +which can still be visible until the present-day. We investigate such +imprints by quantifying the instantaneous circularity 𝜖 of stars (see +Zhu et al. 2021). Circularities close to one indicate circular orbits, +values around zero indicate random motion dominated orbits and +negative ones show counter-rotating orbits. We then compute the +normalized circularity distribution for each galaxy with a bin size of +0.1 for 𝜖. Circularity distributions are than stacked together according +to the galaxy’s total stellar mass in bins of approximate 0.5 dex and +are re-normalized. +The results for the different in-situ, migrated and ex-situ popula- +tions are displayed in Figure 8. The lines in Figure 8 trace the peak +of the circularity distributions across galaxy stellar mass, separately +for disky and bulgey galaxies. +The circularity distribution of the in-situ population is centered +on random motion dominated orbits for galaxies with stellar masses +smaller than 3 × 109 M⊙ and larger than 1011 M⊙. Galaxies with +stellar masses in between have a circularity distribution with a peak +shifted towards slightly higher circularities of around 0.25. We see +that this shift is caused by galaxies that are overall disky, as the +bulge dominated galaxies have a circularity peak that stays around +zero. Nevertheless, the in-situ stars are in summary on warm to hot +orbits even for disk dominated galaxies, which is not surprising as +the velocity dispersion generally rises towards the center of galaxies. +For galaxies below 1010 M⊙ the stacked circularity distributions +for in-situ stars have a sharper peak, whereas galaxies of higher +masses have an overall broader distribution in 𝜖. This could be an +indication of a smaller galaxy-to-galaxy variation of the circular- +ity distribution in the center of the smallest galaxies, regardless of +whether they are disky or bulgey, as in this mass regime the abso- +lute numbers of those two galaxy types are approximately the same in +TNG50. At the high mass end on the other hand, a broader circularity +distribution could indicate that in-situ stars become redistributed in +their orbits due to the increased influence of mergers and contribution +of ex-situ stars. +For the migrated population the circularity distribution is again +centered on random motion orbits for galaxies < 3 × 109 M⊙ and +> 1011 M⊙, although now the distribution is also overall broader for +the low mass galaxies. Migrated stars in intermediate mass galaxies +are on even higher circularity orbits than their corresponding in- +situ stars reaching a peak of around 0.5 for galaxy stellar masses of +∼ 1010 M⊙. This peak is seen in disk and bulge dominated galaxies +alike. Hence, migrated stars tend to have the most rotational support +for galaxies in the intermediate mass regime, which could be an +indication for migration being caused by different mechanisms across +the galaxy mass range in TNG50. However, migrated stars are also on +average younger than the in-situ stars in these galaxies, which might +be the reason why they are still more on circular orbits (see Figure +7). We also point out that some galaxies above ∼ 3 × 1010 M⊙ have +a very double peaked (i.e. one around zero and around 0.5 or higher) +circularity distribution, which is washed out in Figure 8 due to the +stacking. These stars originate from (recently) migrated clumps. +Ex-situ stars have circularities centered around zero across the en- +tire galaxy mass range in TNG50 and also for both disk and bulge +dominated galaxies. Because they originate from stochastic merger +events, stars are put on average on hot, random motion dominated +orbits. Nevertheless, we see a a large scatter throughout the circu- +larity distributions for the ex-situ stars in the different galaxy stellar +mass bins indicating a lot of individual galaxy-to-galaxy variation. +Depending on the exact time the merger occurred and how the orbits +between the host and merging satellite were configured, ex-situ stars +can very well retain some rotational support and often be on counter +rotating orbits. +4.2.3 2D distributions of ages, metallicity and circularities +In Figures 9 and 10 we show the 2D distributions of age and metallic- +ity and age and circularity of the central in-situ, migrated and ex-situ +stars respectively in stacks of galaxy stellar mass. For each galaxy +we first compute the mass-weighted and normalized 2D histogram +of the respective quantities with bin sizes of 0.5 Gyr for age, 0.25 +dex for metallicity and 0.1 for circularity. We then stack those ac- +cording to the total stellar mass bin of the galaxies, normalize again +MNRAS 000, 1–34 (2022) + +Centers of TNG50 Galaxies +15 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +Central Metallicity +[log10(Z/Z )] +TNG50 +z = 0 +145 Galaxies +Insitu +Migrated 'Smooth+Clumpy' +Exsitu +Insitu +Migrated 'Smooth+Clumpy' +Exsitu +TNG50 +z = 0 +129 Galaxies +TNG50 +z = 0 +92 Galaxies +TNG50 +z = 0 +95 Galaxies +Quenched +TNG50 +z = 0 +145 Galaxies +0 +4 +8 +12 +Central Age [Gyr] +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +Central Metallicity +[log10(Z/Z )] +109 M +TNG50 +z = 0 +587 Galaxies +0 +4 +8 +12 +Central Age [Gyr] +109.5 M +TNG50 +z = 0 +573 Galaxies +0 +4 +8 +12 +Central Age [Gyr] +1010 M +TNG50 +z = 0 +415 Galaxies +0 +4 +8 +12 +Central Age [Gyr] +1010.5 M +TNG50 +z = 0 +222 Galaxies +0 +4 +8 +12 +Central Age [Gyr] +Star Forming +1011.5 M +TNG50 +z = 0 +76 Galaxies +Figure 9. Age-metallicity distributions of central (500 pc) stars of TNG50 galaxies in stacks of stellar mass at z = 0. Gaussian kernel density estimates for +in-situ (pink), migrated (orange) and ex-situ (blue) stars encompassing 1%, 20%, 50% and 90% of all central stellar mass (thickest to thinnest) are shown in the +respective galaxy stellar mass bin increasing from left to right as depicted by the colorbar. The galaxy mass bins are centered on the indicated stellar mass and +are 0.5 dex wide, except for the last panel, which is approximately one dex wide. Prior to stacking the age-metallicity distribution of each galaxy is normalized. +The top row shows quenched and bottom row shows star forming galaxies respectively. In each panel the number of galaxies in the corresponding stellar mass +bin are indicated. The galaxy-averaged age-metallicity distribution of the three origins becomes best separated around galaxies with stellar masses of 1010 M⊙. +and then compute the Gaussian kernel density estimate. The galaxy +stellar mass bins are 0.5 dex wide for galaxies between 108.75 M⊙ +and 1010.75 M⊙. We stack all galaxies with stellar masses between +1010.75 M⊙ and 1012 M⊙ together as a finer binning did not reveal +any mass-dependent trends and also became stochastic due to low +number statistics in this mass regime. The five galaxies with stel- +lar masses above 1012 M⊙ are not included. Additionally, we show +the stacked age-metallicity distributions for quenched and star form- +ing galaxies separately to avoid averaging over too many dissimilar +galaxies in this parameter space. Similarly, we divide between bulge +and disk dominated galaxies for the age-circularity distributions. +With the 2D distributions we can observe a couple of new trends +that are not necessarily apparent from the average stellar population +properties in Figure 7 and the 1D circularity distributions of Figure 8. +Age-metallicity (Figure 9): For star forming galaxies (bottom row +of Figure 9), the average distribution of the migrated stars changes +very little in shape and position, apart from shifting towards higher +metallicities, from the lowest mass galaxies until the 1010.5 M⊙ +galaxy stellar mass bin. They are centered between 2−4 Gyr. The ex- +situ stars behave similarly and are centered around 12 Gyr. However, +the average distribution of the in-situ stars shows an entirely different +mass trend. While the in-situ stars are almost entirely coinciding with +the migrated stars in age-metallicity space for the lowest mass bin, +the peak of the in-situ distribution gradually shifts towards older ages +from 2 Gyr to 8 Gyr for increasing galaxy mass. In the process the +in-situ average age-metallicity distribution becomes more elongated +around 109.5 M⊙ in the age direction when focusing on the 20% +contour. +For 1010 M⊙ galaxies the in-situ distribution becomes more cen- +trally concentrated again. Furthermore, the average age-metallicity +distributions of the three origins are maximally separated in this mass +regime, with the migrated stars being the youngest (1 − 6 Gyr for the +20% contour), followed by the in-situ stars (6 − 10 Gyr for the 20% +contour) at similar metallicity and the ex-situ stars populating the +oldest (10 − 13 Gyr for the 20% contour) and most metal-poor tail. +Thus, there must be a mechanism for these galaxies that halts in-situ +star formation in their centers, while it continues outside of it in order +to be able to produce young migrated stars. It is likely that this is +connected to the (kinetic) AGN feedback implement in TNG, which +quenches galaxies from inside-out (Nelson et al. 2021, see also 5.2 +and Figure 11). +Starting at 1010.5 M⊙ the average age-metallicity distribution of +the migrated stars also becomes more elongated towards older ages +and above 1010.5 M⊙ coincides again with the one of the in-situ stars. +The peak of the ex-situ distribution increases towards metallicities +similar to those of the in-situ and migrated stars. For galaxies between +1011 M⊙ and 1012 M⊙ in stellar mass the average distributions for the +in-situ, migrated and ex-situ stars become almost indistinguishable +in age-metallicity space. +For quenched galaxies (top row of Figure 9) the behaviour for the +age-metallicity distributions of the in-situ and migrated stars across +galaxy stellar mass is different. They are not clearly separated in any +galaxy stellar mass bin as was the case for the star forming galaxies. +Both the in-situ and migrated average age-metallicity distributions +are more centrally concentrated than for star forming galaxies, their +shapes are very similar to each other and their peaks are both at old +ages (around 8 Gyr) exhibiting little galaxy mass dependence. The +peak of the age-metallicity distribution for the migrated stars seem to +be slightly younger for galaxies in mass bins between 109.5 M⊙ and +1010 M⊙ and slightly older otherwise. Interestingly at both the low +and high mass end, the separation in metallicity between the in-situ +and migrated stars is larger for the quenched galaxies as for the star +forming ones. For galaxies between 1011 M⊙ and 1012 M⊙ the three +distributions are again indistinguishable. +Age-circularity (Figure 10): Stars with higher circularities are +MNRAS 000, 1–34 (2022) + +16 +Boecker et al. +1.0 +0.5 +0.0 +0.5 +1.0 +Central Circularity +TNG50 +z = 0 +441 Galaxies +Insitu +Migrated 'Smooth+Clumpy' +Exsitu +Insitu +Migrated 'Smooth+Clumpy' +Exsitu +TNG50 +z = 0 +297 Galaxies +TNG50 +z = 0 +129 Galaxies +TNG50 +z = 0 +96 Galaxies +Bulgey +TNG50 +z = 0 +145 Galaxies +0 +4 +8 +12 +Central Age [Gyr] +1.0 +0.5 +0.0 +0.5 +1.0 +Central Circularity +109 M +TNG50 +z = 0 +291 Galaxies +0 +4 +8 +12 +Central Age [Gyr] +109.5 M +TNG50 +z = 0 +405 Galaxies +0 +4 +8 +12 +Central Age [Gyr] +1010 M +TNG50 +z = 0 +378 Galaxies +0 +4 +8 +12 +Central Age [Gyr] +1010.5 M +TNG50 +z = 0 +221 Galaxies +0 +4 +8 +12 +Central Age [Gyr] +Disky +1011.5 M +TNG50 +z = 0 +76 Galaxies +Figure 10. Age-circularity distributions of central (500 pc) stars of TNG50 galaxies in stacks of stellar mass at z = 0. Gaussian kernel density estimates +for in-situ (pink), migrated (orange) and ex-situ (blue) stars encompassing 1%, 20%, 50% and 90% of all stellar mass (thickest to thinnest) are shown in the +respective galaxy stellar mass bin increasing from left to right as depicted by the colorbar. The galaxy mass bins are centered on the indicated stellar mass and +are 0.5 dex wide, except for the last panel, which is approximately one dex wide. Prior to stacking the age-circularity distribution of each galaxy is normalized. +The top row shows bulge dominated and bottom row shows disk dominated galaxies respectively. In each panel the number of galaxies in the corresponding +stellar mass bin are indicated. The galaxy-averaged age-circularity distribution of the three origins becomes best separated around galaxies with stellar masses +of 1010 M⊙. +usually younger. The distribution for migrated stars of disky galax- +ies (bottom row of Figure 10) in the lowest galaxy stellar peaks at +around 2 Gyr with high circularity values of around 0.5, whereas the +distributions for the in-situ stars peaks at older ages (6 Gyr) centered +on circularity values of zero. Nevertheless, the 20% contour for the +in-situ stars still has a tails towards younger ages and slightly above +zero circularities. In the next higher galaxy stellar mass bin the 20% +contour of the distribution for the migrated stars looses its tail of +older ages (4−8 Gyr) and zero circularities. The 20% contour for the +in-situ stars in now centered on even older ages (8 Gyr). Beginning +around galaxy stellar masses of 1010 M⊙ the 20% contour for the +migrated stars elongates towards older ages spanning now 1 − 8 Gyr, +while roughly maintaining the high circularity. The distribution for +the in-situ stars becomes broader and slightly shifts towards above +zero circularities. In the 1010.5 M⊙ galaxy stellar mass bin the peak of +the migrated stars shifts from young (2 Gyr) to old (8 Gyr) ages with +just a slight decrease in circularity. Above galaxies with 1010.5 M⊙ in +stellar mass the migrated stars switch from a rotationally supported +distribution to random motion dominated one until they coincide +with the age-circularity distributions of the in-situ and ex-situ stars +at the highest galaxies. +The peak of the age-circularity distribution for the in-situ stars, +albeit having the same young age as for the migrated stars in the lowest +stellar mass bin, is near zero circularity. With increasing galaxy mass +the age-circularity distribution for in-situ stars shifts towards old ages +and becomes broader in the circularity direction, but stays mostly +centered around zero circularity with perhaps a slight shift towards +higher circularities around the 1010 M⊙ galaxy stellar mass bin as +already observed in Figure 8. The age-circularity distributions for the +ex-situ stars show practically no galaxy mass dependence; they are +centered on random motion dominated orbits and the oldest ages. +The centers of bulge dominated galaxies (top row of Figure 10) +above 109 M⊙ have overall similar age-circularity distributions as +disk dominated galaxies. However, the absolute values of the mi- +grated distribution do not reach the same high circularities as for the +disky galaxies and its peak transitions quicker to old ages (8 Gyr) be- +tween mass bins of 9.5 and 10 dex. Below 109 M⊙ both the migrated +and in-situ distrbition are centered on zero circularities and old ages; +distinct to the disky galaxies. +For both bulge and disk dominated galaxies the average age- +circularity distribution of the in-situ, migrated and ex-situ stars are +well separated in mass ranges between 109.5 M⊙ and 1010 M⊙. This +dependence of increasing circularity for younger ages, especially +prominent for the migrated stars, gives an indication that recently +(i.e. young) migrated stars travel to the center of their host galaxies +by loosing their angular momentum (“churning”; see e.g. Frankel +et al. 2020, for the Milky Way disk) and then, once they have arrived +in the center, become dynamically heated over time. +5 DISCUSSION, IMPLICATIONS AND OUTLOOKS +In this section we discuss the implications of the studied mass as- +sembly of the central 500 pc in TNG50 galaxies on the formation +scenarios of central galaxy components. We also discuss the clumps +found in TNG50 as well as the robustness of our results within the +TNG modelling framework. In addition, we assess how our results +on the stellar population and dynamical properties can be compared +to observations and used to understand the mass build-up of galaxies +in general. +MNRAS 000, 1–34 (2022) + +Centers of TNG50 Galaxies +17 +5.1 The build-up of galaxy centers in a 𝚲CDM cosmology +Throughout this paper, we have unravelled a set of relations between +the properties of the stellar centers of galaxies at z = 0. Galaxy +centers are dominated by in-situ stars (see Figure 4) and follow well +established relations (e.g. Gallazzi et al. 2005) that correlate their +increasing stellar masses with increasing average ages and metallici- +ties (see Figure 7). Stars that migrated to the center are second most +abundant. They follow the trends for the in-situ stars, however are +often distinctively younger and on more rotation supported orbits. +Ex-situ stars in the center become only significant (in mass) at high +galaxy stellar masses (> 1011 M⊙) (see Figure 4). The majority of +ex-situ stars originate from the centers of the accreted galaxies (see +also Gao et al. 2004). Moreover, they are amongst the oldest and most +metal-poor and random motion dominated stars (see Figures 7 and 8 +as well as), which is in agreement with El-Badry et al. (e.g. 2018), +who studied three MW-like galaxies from the Latte project (Wetzel +et al. 2016). +While these trends are consistent with our general understanding +of galaxy formation in a ΛCDM cosmology, we find others that +may be more surprising. For example, there seems to be no average +difference between the central mass assembly of central and satellite +galaxies (see Figure 6). Generally, central galaxies are thought to have +accreted more satellite galaxies. We have checked thisrelation also for +the total accreted mass within TNG50 and also found no significant +difference between centrals and satellites on average. Thus, perhaps +TNG50 is not probing enough very high mass central galaxies around +stellar masses of 1012 M⊙, where this trend might become apparent. +Another, rather unexpected result compared to usual assumptions, +is that star forming galaxies above 1010 M⊙ possess on average more +ex-situ mass in their centers compared to quenched ones (see Figure +6). Again, this difference, even though much less significant remains +when considering the total amount of ex-situ mass (see Figure C1). +This trend also exists for the larger box of TNG100, thus eliminating +the fact for low number statistics at the higher mass end, and is in +contrast to the original Illustris simulation (see Rodriguez-Gomez +et al. 2016, Figure 5). We have checked the median mass growth of +the central ex-situ stars for star forming and quenched galaxies alike +between stellar masses of 1010 − 1011 M⊙ and found that quenched +galaxies stop acquiring ex-situ mass in their centers after z ∼ 1.7 +(lookback time ∼ 10 Gyr). Only if we split quenched galaxies further +into bulgey and disky as well as barred and non barred do we see that +quenched, bulgey and non barred galaxies have a similarly high ex- +situ mass in their centers as their star forming counterparts. Together, +this is an indication that the time of accretion and consequently the +absolute amount of stellar and gas mass of the secondary galaxy (the +former will be higher at later cosmic times and the latter will influence +the amount of newly formed stars during the merger process) will +matter in the build-up of ex-situ mass in the center of the primary +and ultimately dictate what properties it has today. +On top of that, the fraction of in-situ, migrated and ex-situ stars +in the center of galaxies has a significant scatter at fixed galaxy +stellar mass regardless of the galaxy’s bulk properties at z = 0 (see +Figure 5). Hence, median trends for different galaxy populations only +reveal half of the picture, as the stochasticity of galaxy mergers and +interactions in a ΛCDM cosmology leads to diverse pathways in the +build-up of stellar mass in the centers of galaxies. Thus, characteristic +properties of galaxies at z = 0 are only a limited indicator of the exact +formation history of an individual galaxy. For example, there are +perfectly regular MW-like spiral galaxies in TNG50 z = 0, of which +some have experienced (multiple) major mergers and of which some +had a more quiet assembly (see also Sotillo-Ramos et al. 2022). +This diversity in the central 500 pc of TNG50 galaxies potentially +reflects the variety of central galaxy components seen in observations +(see Section 1). Even though nuclear rings, disks and star clusters +are at or below the resolution limit of TNG50, the stellar population +and dynamical properties that we find for central stars of different +origins might be a first indication that this would also manifest in +structurally distinct components. For example, the distinctly high +circularities of migrated stars in 109 − 1010 M⊙ galaxies (see Figure +10) reflect that nuclear disk-like configurations are able to arise. Even +more intriguing are their predominantly younger ages of 1 − 2 Gyr +compared to the underlying old (∼ 8 Gyr) in-situ population, which +is in line with observational findings of nuclear disks/rings (Bittner +et al. 2020). Typically, the formation of nuclear rings in disk galaxies +is associated with bars funnelling gas towards the center (see e.g. +Seo et al. 2019; Tress et al. 2020; Sormani et al. 2020, for dedicated +simulations). Even though we did not explicitly investigate the inflow +of gas in this study, we see that the migration of stars to the center is +likely connected to temporarily induced non-axisymmetries during +galaxy interactions (see Section 5.2). Hence, this shows that mech- +anisms that are associated with producing distinct nuclear galaxy +components are captured in TNG50. Follow-up zoom-in simulations +of TNG50 galaxies would show if indeed nuclear components such +as disks and rings form from these mechanisms (see Section 5.4). +5.2 Mechanisms for the formation and deposit of stars in the +center of galaxies +The cosmological framework of TNG50 produces diverse properties +of galaxies and their centers. Consequently, the mechanisms that are +responsible for the formation and deposit of stars in the centers of +galaxies also have to be diverse. +To visualize possible mechanisms for the formation and deposit of +stars in the center of galaxies we walk through the central assembly +history of an individual galaxy as seen in Figure 11. We picked this +particular galaxy, which has a stellar mass of 1010.8 M⊙ at z = 0, +as it shows many of the possible mechanisms that can be present +in the formation of galaxy centers. This however does not mean +that all galaxies show the same amount of complexity. Most galaxies +will only exhibit one or two of these mechanisms with varying impact +depending on their individual formation pathway. The galaxy’s center +at z = 0 consists of around 50% in-situ, 30% migrated and 20% ex- +situ stars. +The main summary of the subsequent sections and Figure 11 is +the following: galaxy mergers and other interactions are probably +the most important driver in central stellar mass assembly, as they +also strongly influence the formation of in-situ stars. Due to the +diverse statistics of galaxy interactions, many of the proposed for- +mation scenarios of central galaxy components arise naturally and in +conjunction to each other, when hierarchical galaxy formation is con- +sidered. Thus, TNG50 highlights the necessity to study the central +mass assembly of galaxies in a cosmological context. +5.2.1 In-situ stars +Galaxy mergers can trigger bursts of star formation as the tidal forces +compress and shock gas efficiently (e.g Mihos & Hernquist 1996; Di +Matteo et al. 2007; Cox et al. 2008; Di Matteo et al. 2008), even in +its nuclear region (Powell et al. 2013). While the relative enhance- +ment of star formation rates depend on the specific configuration +of the merging galaxies, e.g. merger mass ratio, gas content, orbital +infall parameters, the times of intense star formation coincide with +MNRAS 000, 1–34 (2022) + +18 +Boecker et al. +Figure 11. Central (500 pc) assembly history of an individual galaxy (SubfindID 184937) in TNG50 with a total stellar mass of 1010.8 M⊙. This galaxy +encompasses many mechanisms that can shape the stellar mass build-up in the center of galaxies in a 𝚲CDM cosmology. Top panel: Points show all +individual stellar particles that belong to that galaxy at z = 0. Their distance at the time of birth is shown with respect to their current host in the case of in-situ +formed stars (light gray: all in-situ particles, pink: central in-situ stars only, orange: central migrated stars only). In the case of the ex-situ formed stars the +distance is shown with respect to their future host galaxy at the time of birth (color-coded according to the colorbar: all ex-situ stars, blue: central ex-situ stars +only). The distance to individual satellite galaxies (only with maximum stellar masses above 106 M⊙) that will merge with the primary at some point are shown +with thinner solid lines. Their coloring also follows the colorbar, which visualizes the merger mass ratio taken at the time tmax, when the secondary galaxy +reaches maximum stellar mass. The thick black solid line shows the radius of the FoF Group the galaxy belongs to at a given lookback time (represented as R200, +where the group’s density is 200 times the critical density of the Universe). The thick gray dashed line shows the distance between the individual galaxy and the +central galaxy of the FoF group it belongs to. Approximately 7 Gyr ago the galaxy fell into another group and became a satellite galaxy. Before that it was the +central of its own FoF group. The vertical black dotted line represents the time the kinetic AGN feedback starts to take effect, which quenches the center. This +galaxy has 50% in-situ, 30% migrated (of which only 9% are ’smoothly’ migrated and the rest comes from migrated clumps) and 20% ex-situ stars in its center. +Bottom panel: Histograms of formation times of in-situ (top), formation (solid) and arrival (dashed-dotted) times at the center for ‘smoothly’ migrated (middle) +as well as ex-situ and ‘clumpy’ migrated (bottom) stars. Additionally, in the panel for the in-situ stars, we mark the time of coalescence for the six most massive +mergers of this galaxy with thin blue colored solid lines. According to the colorbar of the top panel, a darker blue means a higher merger mass ratio. Pericenter +passages for two mergers are shown by thin dashed lines following the same colorcode. The approximate time of the galaxy falling into its z = 0 FoF group is +shown by the thick black solid line and the onset of the kinetic AGN feedback is shown as the black dotted line. In the panel for the ‘smoothly’ migrated stars +we also show the A2 mode of the stars for a given lookback time (see Appendix A1 for a definition). In the panel for the ex-situ and ‘clumpy’ migrated stars, we +show the time of coalescence of the two mergers that deposited ex-situ stars in the galaxy’s center (blue solid lines) as well as the three pericenter passages of +the galaxy around its central galaxy after it became a satellite (gray dashed lines). +MNRAS 000, 1–34 (2022) + +SubfindID 184937 +1000 + Ratio at tmax +100= + R200 of FoF Group +10# + Distance to Central +[odx] +-0.1 +Clumps +Merger Mass +Birth Radius +of Stellar Particles +w.r.t Primary +Insitu: 50 % +0.1 +TT +Migrated: 30 % +Onset of +Kinetic AGN Feedback +Exsitu: 20 % +0.01 +0.01 +14 +12 +10 +6 +4 +0 +Lookback Time [Gyr] +Merger Pericenter +Merger Coalescence +Infall into Group +Kinetic AGN Feedback +Pericenters around Central +0.8 +Insitu +# +0.6 +At Formation +0.4 +0.2 +0.0 +0.8 +Migrated +# +Smooth' +0.6. +0.4 +At Formation +At Arrival in Center +0.2 +A2 of Stars +0.0 +Exsitu +Migrated +Normalized +'Clumpy' +2 +At Formation +At Arrival in Center +0- +14 +12 +10 +8 +6 +Lookback Time [Gyr]Centers of TNG50 Galaxies +19 +pericenter passages and coalescence (see also Sotillo-Ramos et al. +2022). +Peaks in the formation time of central in-situ stars in Figure 11 +coincide with times of pericenters and coalescence of mergers that +this galaxy has experienced. Thus the formation history of the central +in-situ stars is directly connected with the merger history of a galaxy. +However, it has to be further quantified whether also the bulk of +in-situ stars is formed during such events or if that actually happens +in-between galaxy interactions. Nevertheless, it is clear that a variety +of different mergers are able to produce peaks in the formation of +central in-situ stars. +For example, the peak between 10 Gyr and 12 Gyr ago was induced +by a very minor merger9 with a stellar merger mass ratio of around +0.02. At these high redshifts the primary still had a high gas fraction +(∼ 80%) and thus the minor merger was enough to trigger a peak +of star formation in the center. Evidently, the formation of in-situ +stars in the center decreased between 10 Gyr and 8 Gyr ago, as the +amount of available gas decreased. Thus, in order to trigger another +significant peak in in-situ star formation later on, the merger between +8 Gyr and 7 Gyr had to bring in a large amount of gas. While the ratio +of the stellar mass between the secondary and primary was around +one (and therefore a major merger) at the time when the secondary +reached its maximum stellar mass, the secondary still had around 20 +times more gas than the primary. +Furthermore, this major merger, as well as two smaller ones that +coalesced around 6 Gyr ago, happened while the primary galaxy was +in the process of falling into another FoF Group, i.e. transitioning +from being a central galaxy of its own FoF group to being a satellite +galaxy of another FoF group. This is seen by the two sharp jumps in +R200 between 9 Gyr and 7 Gyr ago. Evidently, this process produced +another peak of in-situ star formation at around 7.5 Gyr ago, which +could stem from the new, higher density environment. We have also +seen in other galaxies, that were able to retain enough gas in their +centers after infalling into a group, that in-situ star formation was +triggered during the pericenter passages around the central until the +galaxy became quenched. In such occasions, again tidal forces are +able to compress the gas efficiently. +Within TNG50, there are two main processes that can quench the +in-situ star formation in the center of galaxies. The first one is the +onset of the kinetic AGN feedback mode implemented in TNG. Often +this feedback mode switches on after a merger has been completed, +as is the case for our galaxy in Figure 11 at around 7 Gyr, shortly +after the major merger coalesces. We see that only the central 1 kpc +becomes quenched, while the outskirts of the galaxy continue to form +stars. This is because in TNG, AGN driven quenching proceeds from +inside out (see Weinberger et al. 2017; Nelson et al. 2019b, 2021, +for details). After this mode is switched on only occasional gas-rich +mergers or migrated clumps are able to bring in new gas to the center +to cause new in-situ star formation, as seen at 5.5 Gyr in Figure 11. +Lastly, the thermal feedback mode, which is often active prior to the +kinetic mode switches on and also injects relatively more energy, +is not responsible for quenching the centers of galaxies in TNG50 +(Zinger et al. 2020). +The second process that will shut down star formation in the center +of TNG50 galaxies is when the galaxy as a whole becomes quenched, +either through environmental processes, e.g. after a few pericenters +after infall into a group (as is the case for the galaxy in Figure 11 +around 3 Gyr ago) or through AGN feedback, which is primarily +9 Weadopt the definition ofRodriguez-Gomez etal.(2016)for thecalculation +of merger ratios. +important for the highest mass galaxies (see also Donnari et al. +2021a). +5.2.2 Migrated stars +The formation times of ‘smoothly’ migrated stars in Figure 11 is +closely related to the formation times of the in-situ stars, which is not +the case for the ‘clumpy’ migrated stars. This is reasonable, because +the majority of ‘smoothly’ migrated stars are born already close +to the center (≲ 2 kpc), while the ‘clumpy’ migrated stars formed +predominantly in the outer disk. The ‘clumpy’ migrated stars make +up 91% of the total mass of migrated stars in the center of this galaxy. +However, the star formation in the central 2 kpc is not a guarantee +to produce a significant amount of ‘smoothly’ migrated stars, as seen +between lookback times of 9 − 11 Gyr and also between 7 − 7.5 Gyr +in Figure 11. Thus, specific conditions must be met that transport +stars from around 1 − 2 kpc to the center. +Non-axisymmetric features, such as spiral arms and bars, are well +known to be able to diffuse the angular momentum of stars and cause +radial migration (e.g. Sellwood & Binney 2002; Minchev & Famaey +2010). While this effect is mainly studied in the (outer) disk of +galaxies, we show here in Figure 11 that similar non-axisymmetries +are likely responsible for the inward migration of stars to center +of galaxies. We see that peaks in the A2 mode (see Appendix A1 +for a definition) of the stellar mass distribution occur before peaks +of migrated stars arriving in the galaxy center. We detect similar +enhancements for the Fourier modes of the gas mass distribution +(see also Di Matteo et al. 2007). +These temporary enhancements of non-axisymmetric features are +clearly induced during galaxy interactions and the exerted torques +on the gas and stars can produce these ‘smoothly’ migrated stars. +This also indicates that it is possible for a galaxy to have experienced +migration events of stars, even if the galaxy itself does not exhibit +any signs of bar- or spiral-like features today. +The ‘clumpy’ migrated stars form after the first pericenter passage +of the galaxy around its central around 5.5 Gyr ago and arrive at +the center shortly before the second pericenter passage around 2 Gyr +later. Similarly, we have seen qualitatively for other galaxies that +clumps formed rather recently (z < 1) are mainly induced by fly-bys, +as these are still able to destabilize the disk significantly after the +predominant merger phase of the Universe is over. However, clumps +are also able to form without any significant galaxy interactions and +a follow-up study is needed to characterize this further as well as +establish overall the credibility of the formation of the clumps (see +Section 5.3 for a further discussion). +5.2.3 Ex-situ stars +The two mergers that are responsible for the majority of the ex-situ +stars in the center of the galaxy in Figure 11, are the 1:1 and 1:10 +merger that coalesced around 7 Gyr and 5.5 Gyr ago respectively. +Both mergers brought in a comparable total amount of stellar mass +of around 1.2 × 1010 M⊙ and 8.1 × 109 M⊙ respectively. However, +the major merger deposited around 10 times less stars in the central +500 pc compared to the minor merger, i.e. 0.6% and 5% of their +respective total stellar mass arrived in the center. This highlights +that the merger mass ratio cannot be the only parameter determining +the amount of ex-situ stellar mass that is deposited in the center of +galaxies. We expect that the spin-orbit coupling of the primary and +secondary galaxy as well as other orbital parameters play a role in +this, as the exerted tidal forces and the influence of dynamical friction +MNRAS 000, 1–34 (2022) + +20 +Boecker et al. +differ for different configurations (see e.g. Renaud et al. 2009, for a +study). +Around 67% and 93% of the ex-situ stars that arrived from the +major and minor merger respectively were formed after both satellite +galaxies entered R200 of the primary’s FoF halo around 8.75 Gyr ago. +As also all central ex-situ stars were born in the center (∼ 500 pc) of +their respective birth galaxies, this confirms that significant nuclear +star formation is also triggered in the secondary galaxy after infall. +Most of the ex-situ stars in the center arrive there immediately +after the merger coalesces. This is the case if their distance to the +center of the primary was less than 500 pc at the time of stripping. +Otherwise it can take up to 2 Gyr. Interestingly, the arrival of the +‘clumpy’ migrated stars at the center around 3.5 Gyr ago induced +a second peak in the arrival of ex-situ stars from the minor merger +into the center, albeit being ten times lower and hence not visible in +Figure 11. +5.3 The case of stellar clumps +Disk fragmentation can occur due to gravitational instabilities in a +galaxy’s gas-rich and turbulent disk (Toomre 1964; Springel et al. +2005; Hopkins 2013). This fragmentation can form highly star form- +ing clumps, which have been reproduced in several studies using +hydrodynamical galaxy simulations, either isolated or fully cosmo- +logical ones (e.g. Bournaud et al. 2007; Genel et al. 2012; Bournaud +et al. 2014; Mandelker et al. 2014, 2017; Buck et al. 2017). The +execution of these simulations was motivated by the discovery of +the clumpy morphology in the rest-frame UV light of high redshift, +star forming galaxies (e.g. Elmegreen et al. 2007; Guo et al. 2015). +Therefore, these simulations are tailored to focus on clump formation +in massive disk galaxies 1010−11 M⊙ at z ≥ 1. +In observations, clumps have masses between 107 M⊙ and +109 M⊙, as well as sizes of 1 kpc or less. Clumps in the simula- +tions are usually identified via regions of enhanced gaseous surface +mass density or from mock stellar light images. In TNG50, the iden- +tification of clumps is (so far) a passive byproduct of the Subfind +algorithm, nevertheless the extracted baryonic mass distribution of +the clumps peaks at 108 M⊙ exhibiting overall high gas fractions +(see Figure E1) are in agreement with the other studies. However, all +clumps in TNG50 have 3D baryonic half mass radii below 300 pc and +therefore seem to be much more compact compared to observations +and some simulation studies (see Figure 9 in Buck et al. 2017). The +latter could be a result of the different treatments of star formation +and feedback in the simulations or the clump identification, as the +numerical resolution of TNG50 is largely comparable to those of +the previous studies. Additionally, in TNG50 clumps seem to form +continuously throughout cosmic time (see Figure E1), which has not +been investigated in other studies of clump formation. +These clumps can migrate to the center of their host galaxies due +to dynamical friction, as well as merge with each other while doing +so (Bournaud et al. 2007; Dekel & Krumholz 2013; Bournaud et al. +2014; Mandelker et al. 2014; Dekel et al. 2021). The migration time +to the center is found to be of the order of a few hundred Myr, which +is similar for clumps in TNG50, where most of the clumps arrive at +their respective galaxy’s center after 1 − 2 snapshots (∼ 200 Myr) +(see Figure E1). This mechanism contributes to the formation of +the bulge (e.g. Bournaud et al. 2007; Elmegreen et al. 2007; Dekel +et al. 2009). Consequently, the properties of the clumps need to be +specific, such that they survive their own internal stellar feedback and +the tidal forces on their way to the center with enough stellar mass +to significantly contribute to the formation of the bulge. In TNG50, +the fraction of clumpy migration stars to the total stellar mass in +the center is greater than 40% for about 12% of galaxies with any +clumpy migrated stars in their centers (all of those galaxies have +stellar masses above 1010.5 M⊙; see Figure E1). Thus, according +to TNG50, the stellar mass transported by the migration of clumps +is likely not very important for bulge formation for the majority of +high mass galaxies. Nevertheless, the clumps often retain a large +amount of gas until the center, or drag gas along (see clump closest +to the galaxy’s center in the top right of Figure E1), from which +stars might form. We have not checked explicitly if this increases the +contribution of stellar mass in the center significantly, or if such gas +is lost by directly funneling into the SMBH. +In contrast to that, some simulations report clump formation, but +then no migration due to almost immediate dissolution or disruption +(Hopkins et al. 2012, 2013a; Mayer et al. 2016; Oklopčić et al. +2017; Buck et al. 2017). This is likely due to the different simulation +set ups, as well as the exact implementation of stellar feedback, or +simply because not enough galaxy diversity is probed with isolated +or zoom-in galaxy simulations. For example, in Figure E1, we see +that galaxies above 1010 M⊙ in stellar mass start to exhibit more +than one clump on average, however significant amounts of clumps +that are able to migrate to the center reside in galaxies with stellar +masses around 1011 M⊙ and above. Hence, an investigation of clump +formation and migration in a fully cosmological box is necessary to +not only capture galaxies of different masses but also different galaxy +assembly histories. In TNG50 mergers and other galaxy interactions, +such as flybys, can trigger a significant amount of clump formation +(although not exclusively; see also Di Matteo et al. 2008; Hopkins +et al. 2013a; Calabrò et al. 2019, for similar reports). +Still, within TNG50 we want to exercise caution when it comes +to the trustworthiness of clump formation and their exact properties. +Only follow-up zoom-in simulations with higher resolution for dif- +ferent galaxies, as well as different treatment of star forming gas and +stellar feedback (see e.g. Hopkins et al. 2013b; Smith et al. 2018; +Smith 2021; Smith et al. 2021, for the influence of highly resolved +star formation and different stellar feedback schemes in galaxy sim- +ulations), will allow for a more robust quantification of clumps in +TNG50. Nevertheless, the clump formation in TNG50 is unlikely to +be a numerical artifact in its entirety, as the adaptive mesh refinement +naturally allows for smaller cell sizes in areas of high gas density. +5.4 The predictive power of TNG50 at small(er) scales +While the TNG modelling framework is extremely successful in re- +producing key observational results of galaxy populations, numerical +resolution and the implementation of sub-grid physics are insupera- +ble limitations of the physical model of galaxy formation. Regarding +the former we demonstrate in Figure D1 in Appendix D that the total +stellar mass within the central 500 pc of galaxies in TNG50 is con- +verging (see also Pillepich et al. 2019). When splitting the central +mass into the contribution of in-situ, migrated and ex-situ stars, the +start of convergence is more difficult to assess due to the fixed size of +rcut (see Appendix D for a more detailed discussion) as well as the +overall influence of resolution on the amount of accreted stellar mass +(which should overall increase with resolution, see also Grand et al. +2021). Thus higher resolution runs are needed to fully determine the +amount of convergence. +Higher resolution zoom-in simulations of some TNG50 galaxies +- additionally with models variation of stellar feedback and a better +resolved cold gas phase of the star forming gas - are certainly inter- +esting and needed to properly evaluate the convergence of the central +stellar mass, the formation of stellar clumps and observed nuclear +galaxy components. Nevertheless, our study of TNG50 shows that +MNRAS 000, 1–34 (2022) + +Centers of TNG50 Galaxies +21 +0.1 +0.25 +0.5 +1.0 +rcut [kpc] +104 +105 +106 +107 +108 +109 +1010 +1011 +Central Stellar Mass [M ] +2 +37 +72 +74 +17 +48 +69 +36 +49 +z = 0 +Insitu +Migrated 'Smooth+Clumpy' +Exsitu +TNG50-1 +TNG50-2 +TNG50-3 +0.1 +0.25 +0.5 +1.0 +rcut [kpc] +96 +54 +21 +16 +100 +73 +43 +22 +100 +77 +52 +42 +z = 0 +1010.5 M +M , tot +1011 M +TNG50-1 +TNG50-2 +TNG50-3 +0.1 +0.25 +0.5 +1.0 +rcut [kpc] +0 +1 +4 +0 +1 +0 +z = 0 +TNG50-1 +TNG50-2 +TNG50-3 +Figure 12. Effects of numerical resolution and aperture size on the central stellar mass for in-situ, migrated and ex-situ stars in TNG50 galaxies with +1010.5−11 M⊙ in stellar mass at z = 0. Lines show the median central stellar mass for in-situ (pink), migrated (orange) and ex-situ (blue) stars for four choices of +rcut = 0.1, 0.25, 0.5 and 1 kpc and three resolution realizations of TNG50 (thicker lines indicate better resolution). TNG50-1 is the highest resolution (flagship), +followed by TNG50-2 and -3, which have 2 and 4 times lower spatial resolution. The mass resolution is 8 and 64 times lower respectively and indicated by the +dotted horizontal lines. The numbers indicate the respective central stellar mass fraction in percent. Decreasing size of the center means decreasing stellar mass. +However the fraction of migrated mass increases, while the in-situ fraction consequentially decreases. At a hypothetical higher resolution (TNG50-0) the latter +effect would be lessened as more stellar mass is formed within a given aperture size. Similar trends are recovered for other galaxy masses. +the cosmological context plays a major role in the assembly of galaxy +centers, which is unlikely to become less significant with numerical +resolution and other modelling aspects. Already at the resolution of +TNG50 it is rare to find a galaxy with no ex-situ stars in its central +500 pc, which is only the case for around 9% of all galaxies in our +sample spanning a range between 5 × 108 − 5 × 1012 M⊙. We note +that this percentage remains the same, even if we do not impose a +constraint on the physical size of galaxies included in our analysis. +In fact, our selection of galaxies with half-mass radii > 2 kpc does +not impose any strong differential effects on our results. We have ex- +plicitly checked Figures 4 and 6, which show the same trends when +galaxies with R1/2 < 2 kpc are included. Thus, we do not expect that +any other results of our study are significantly effected by our choice. +Therefore, our findings highlight that the high density, nuclear +regions of galaxies can survive tidal forces and contribute to the +build-up of the centers of other galaxies, including low-mass galaxies. +Additionally, if the two merging galaxies are massive enough to both +host a black hole, the merger of the central regions of galaxies will +contribute to the growth of SMBHs (see e.g. Schweizer et al. 2018; +Voggel et al. 2021, for a recent observational confirmation of such a +system). +Currently there are no large-box, cosmological, hydrodynami- +cal simulations (including TNG50) that can resolve smaller central +galaxy components, such as nuclear star clusters (NSCs), until z = 0 +(see Brown et al. 2018, that investigate NSC formation in a cosmo- +logical set-up until z = 1.5). However, by extending the trends in +our study to even smaller scales, it is not impossible to think that +the formation and evolution of NSCs are also governed by galaxy +interactions. Even though the relative fraction of ex-situ stars on tens +of pc scales is likely very small for the majority of galaxies, it is clear +that the in-situ star formation and the migration of stars to the center +is closely connected to the formation pathway of the entire galaxy +(see Figure 11), because galaxy interactions are able to create the +conditions needed to funnel gas and stars to the center. Therefore, it +is important to treat NSC formation in the context of the hierarchi- +cal build-up of galaxies in a ΛCDM cosmology (Brown et al. 2018) +and consider the influence of galaxy interactions in (semi-)analytical +models (Leaman & van de Ven 2021). +We have explicitly checked within TNG50 if we can make predic- +tions at smaller scales than 500 pc, by repeating our entire analysis +for rcut of 250 pc (approximately the softening length of TNG50) +and 100 pc, as well as 1 kpc for a consistency check. In addition +to TNG50-1 (the highest resolution), we repeated this for the two +lower resolution realizations, which are TNG50-2 and TNG50-3 re- +spectively. The results are shown in Figure 12 for galaxies between +1010.5 and 1011 M⊙. +With decreasing size of the center (rcut) the absolute mass de- +creases for all three origins. However, the fraction of the in-situ +population decreases with decreasing central size, while the mi- +grated fraction increases; a consequence of the smaller volume that +is proved. At the same time, this behaviour is also affected by the res- +olution, which not only sets the absolute normalization of the mass +fraction, but also the spatial size at which the relative contribution of +the in-situ and migrated fraction swaps. We therefore conclude that a +hypothetical higher resolution (TNG50-0) would increase (decrease) +the in-situ (migrated) fraction below 250 pc due to the convergence +behaviour of the absolute stellar mass at fixed aperture size. Simi- +larly, the contribution of ex-situ stars will increase at a given aperture +and also likely reach scales smaller than 500 pc (see Appendix D for +more details). +This behaviour emphasizes that the contribution of all three origins +will likely remain relevant on scales of 100 pc. +MNRAS 000, 1–34 (2022) + +22 +Boecker et al. +5.5 Galaxy centers as tracers of overall galaxy assembly +Unveiling the merger history of galaxies proves difficult to tackle +outside our own Galaxy due to many reasons. Perhaps the most +severe one is the fact that accreted material is not necessarily visually +apparent in the forms of streams and shells (or any other form of +irregularity), especially when the merger coalesced many Gyr ago. +Since deep photometry of galaxies (initially stacked for many +galaxies) revealed the need for an additional Sérsic component to +accurately fit their surface brightness profiles beyond tens of kpc +(e.g. Zibetti et al. 2004; Tal & van Dokkum 2011; D’Souza et al. +2014), the focus of quantifying accreted material has primarily been +on the outskirts of galaxies, i.e. their stellar halos (e.g. Monachesi +et al. 2016; Merritt et al. 2016; Spavone et al. 2017; Huang et al. +2018; Spavone et al. 2020). It is understood that the excess of light +at large galactic radii should mark the transition from the in-situ to +ex-situ dominated areas of a galaxy, as (significant) stellar mass can +be build-up through minor merging there. +However, the new era of cosmological hydrodynamical simula- +tions suggest that such a transition does not necessarily exist for +every galaxy, as especially high mass galaxies can be dominated by +ex-situ stars at all radii (Tacchella et al. 2019; Pulsoni et al. 2021). +Furthermore, Remus & Forbes (2021) showed that the transition in +surface brightness profiles traced by two Sérsic fits does not corre- +spond to the true in-situ and ex-situ dominated regions. Similarly, +changes in kinematic profiles at large radii, which can, for example, +be obtained with globular clusters (e.g. Arnold et al. 2014) or plane- +tary nebulae (e.g. Pulsoni et al. 2018), do not, in general, correspond +to transitions between in-situ and ex-situ dominated regions (Schulze +et al. 2020; Pulsoni et al. 2021). +While detailed studies of stellar halos are certainly important and +necessary, our study suggests that there lies potential in using the +centers of galaxies to study their accretion history (see Figure 5). Not +only are the centers the brightest region of a galaxy and hence deliver +the highest quality data, but they are also increasingly covered in +numbers by IFU surveys, which provide detailed kinematic and stellar +population information (e.g. SAMI: Bryant et al. 2015, MaNGA: +Bundy et al. 2015). In particular, our results in Figures 9 and 10 +show that in-situ and ex-situ stars in the center are (on average) well +separated in age-metallicity-circularity space for galaxies with stellar +masses ≤ 1010.5 M⊙. Newly developed techniques are able to extract +such distributions in ages and metallicity as well as circularities +from IFU measurements, and already have been proven to be able +to estimate the true underlying accreted stellar material much more +realistically (Boecker et al. 2020a,b; Zhu et al. 2020; Davison et al. +2021; Zhu et al. 2021). +Even though in-situ and ex-situ stars separate better in their stellar +population and dynamical properties for these lower mass galaxies, +the ex-situ stellar mass fraction in the central 500 pc is on average +tenth of percent, thus picking up accreted signatures in the very +centers will still be challenging even with these new techniques. +However, low redshift IFU observations easily cover 1 − 2 half-light +radii, which extend beyond 500 pc and hence should encompass more +accreted material. More follow up work will be needed to quantify +the optimal extent needed from a galaxy’s center to reliably pick up +ex-situ fractions in observations. +On top of that, the large spread seen in the central ex-situ mass +at fixed total ex-situ mass (see Figure 5) points towards significant +spatial variation of ex-situ stars in the host galaxy, regardless of +whether the galaxy has accreted a lot of stellar material or not. It is +likely that measuring these spatial variations, will inform us about the +types of mergers that have happened. Typical characteristics could be +the merger ratio, for example major mergers will have the ability to +deposit more of their stars in the center of galaxies, but also the gas +content or orbital infall parameters. We plan to exploit this in future +work. +5.6 Hints for (SDSS-like) observations +TNG50 predicts a diverse mass build-up of galaxy centers. What are +the prospects to learn about a galaxy’s central in-situ, migrated and +ex-situ fraction from more “traditional” observations? For example, +from SDSS DR7 (Abazajian et al. 2009), that provides single 3′′ fiber +spectra for centers of hundred of thousand of galaxies, average ages, +metallicities and [𝛼/Fe] abundances can be determined (Gallazzi +et al. 2021). How much information do such measurements contain +about the contribution of stellar populations of different origins to a +galaxy’s center? +In Figure 13 we show the mass-weighted average central age and +metallicity for our sample of TNG50 galaxies color-coded by the +mass fraction attributed to each origin (LOESS smoothed; Cappellari +et al. 2013). +If the measured average central age and metallicity of a galaxy +lies on the respective mass-age and mass-metallicity relation, the +galaxy has likely a high fraction of in-situ stars in its center, except +if its larger than 1011 M⊙ in stellar mass. If the measured average +metallicity is below the 16th percentile at fixed stellar mass, it is +more likely that the galaxy’s center is dominated by ex-situ stars. +Similarly, if the galaxy has an average age below the 16th percentile, +it has likely a high amount of migrated stars in its center. High mass +galaxies above 1011 M⊙ with significant amounts of ex-situ stars +in their centers, also have a slightly younger age (between the 16th +percentile and the median) than the typical average galaxy in that +mass regime. +Naturally, a proper mocking of observed average stellar population +properties from TNG50 is needed to compare accurately to measure- +ments from Gallazzi et al. (2021). However, Figure 13 seems to +acknowledge that such measurements provide some leverage in de- +termining the fraction of stars with different origins in the centers of +galaxies. +With respect to comparisons to the whole SDSS galaxy sample, it +would be necessary to repeat the analysis of this paper for different +spatial apertures, The fixed 3′′ diameter of the SDSS fibers will +already encompass larger physical sizes than 1 kpc for galaxies with +z > 0.02. It would be interesting to understand how the relative +contribution of stars from the different origins change with greater +spatial extent, especially for the ex-situ stars. +6 SUMMARY AND CONCLUSIONS +Galaxies growth hierarchically in a ΛCDM universe. Their centers +are the regions where usually the highest quality observations are +available. What information about the hierarchical growth of galaxy +formation is encoded in this observationally favourable region? To +answer this, we investigated the central 500 pc mass assembly of +galaxies ranging from 5 × 108 M⊙ to 5 × 1012 M⊙ with half-mass +radii > 2 kpc in the TNG50 simulation. +Stars that are found at the center of TNG50 galaxies at z = 0 +originate from one of the three possibilities: in-situ (formed inside +the center), migrated (formed inside the host galaxy, but outside the +center), ex-situ (brought in by mergers). Stars can migrate to the +center either as as continuous distribution of individuals (smooth) or +in clumps. +MNRAS 000, 1–34 (2022) + +Centers of TNG50 Galaxies +23 +0.2 +0.0 +0.2 +0.4 +0.6 +Central Metallicity +[log10(Z/Z )] +TNG50 +z = 0 +Insitu +TNG50 +z = 0 +Migrated +TNG50 +z = 0 +Exsitu +109 +1010 +1011 +1012 +Total Stellar Mass [M ] +0 +2 +4 +6 +8 +10 +12 +14 +Central Age [Gyr] +TNG50 +z = 0 +Insitu +109 +1010 +1011 +1012 +Total Stellar Mass [M ] +TNG50 +z = 0 +Migrated +109 +1010 +1011 +1012 +Total Stellar Mass [M ] +TNG50 +z = 0 +Exsitu +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Central Stellar Mass Fraction +Figure 13. Information about the central (500 pc) fractional mass associated with in-situ, migrated and ex-situ stars contained in average age and +metallicity measurements from TNG50 galaxies at z = 0. Top row: Mass-weighted average metallicities for all central stars as a function of the galaxy’s total +stellar mass, but color-coded in each panel (from left to right) according to their fraction of in-situ, migrated and ex-situ stars. The colors are LOESS (Cappellari +et al. 2013) smoothed to show the average around neighbouring galaxies. The thick dashed line shows the median relation in each panel, and the thin dotted lines +show the 16th and 84th percentile respectively. Bottom row: The same but for the mass-weighted average age. Galaxies that are more metal-poor than the 16th +percentile for their corresponding stellar mass are more likely to have high ex-situ fractions in their centers, while galaxies with high central migrated fractions +are younger than the 16th percentile. +For each origin we characterized their radius with respect to their +host galaxy at birth to understand the travelling distances for the +migrated stars as well as the spatial environment of ex-situ stars at +the time of birth and deposit into the z = 0 host. +We then investigated the amount of the central stellar mass con- +tained in each of three origins and their relative contribution as well +as their correlation to each other across the entire TNG50 galaxy +mass range. Additionally, we studied differences in central in-situ, +migrated and ex-situ for different galaxy types at z = 0. +To address whether the different origins of central stars leave a +discernible imprint on their (observable) features, we characterized +and correlated their ages, metallicities, [Mg/Fe] abundances as well +as dynamical properties with their distributions in circularity as a +function of the galaxy’s total stellar mass. We summarize our most +important findings below: +• In-situ stars are on average the dominant component in stellar +mass in the central 500 pc of TNG50 galaxies across the entire mass +range of 5 × 108−12 M⊙. Migrated stars contribute on average 20% +to the total stellar mass in the center, where below (above) 5 × +1010 M⊙ in galaxy stellar mass smoothly (clumpy) migrated stars +encompass their majority. The central stellar mass fraction of ex- +situ stars becomes on average non negligible above galaxy masses +of 5 × 1010 M⊙ with a large scatter of up to 80%. However, it is +the exception to find a galaxy without any ex-situ stellar mass in its +central 500 pc, which is only the case for about 9% of galaxies in our +total sample. (Figure 4) +• The majority of smoothly migrated stars originate close to the +center (radii between 500 pc and 1 kpc), whereas ∼ 15% come from +larger distances up until 10 kpc. Compared to that, clumpy migrated +stars possess a distinctively different distribution of birth radii which +peaks around 20−30 kpc for galaxies with stellar masses greater than +5 × 1010 M⊙. Most of the ex-situ stars originate in the central 1 kpc +of their birth galaxies, where they remain until they are deposited +inside their z = 0 host galaxy. (Figure 3) +• At fixed galaxy stellar mass the amount of central ex-situ stellar +mass exhibits a significant scatter between 4 − 6 dex, reflecting the +stochasticity of the merger history of individual galaxies. In some +cases, close to the entire total amount of ex-situ stellar material ever +deposited inside the host galaxy resides within the central 500 pc. +(Figure 5) +• In TNG at z = 0, star forming galaxies with stellar masses +above 1010 M⊙ have on average larger ex-situ central stellar masses +than their quenched counterparts. Only quenched galaxies that are +additionally bulgey and have no bar signature show a rise of central +ex-situ stellar mass above 1010 M⊙ similar to the star forming galax- +ies. Galaxies between 5 × 109 − 5 × 1010 M⊙ with an overmassive +(undermassive) SMBH in the center are more compact (extended) +and show on average a higher (lower) in-situ and migrated central +stellar mass. There is no difference in neither in-situ, migrated nor +ex-situ central stellar masses for central or satellite galaxies. (Figure +6) +• Central ex-situ stars have on average the lowest metallicities, the +oldest ages and the highest [Mg/Fe] abundances. The slope of their +mass-metallicity relation is slightly steeper than that of the in-situ +and migrated stars, and their mass-age relation is flat compared to the +positive correlation between central age and galaxy stellar mass for +the in-situ and migrated stars. Overall, the average stellar populations +MNRAS 000, 1–34 (2022) + +24 +Boecker et al. +properties of in-situ and migrated stars are very similar, with in-situ +stars being slightly more metal-rich and older. (Figure 7) +• The majority of central stars for galaxies with stellar masses +below 109 M⊙ and above 1011 M⊙ regardless of their origin are +on random motion dominated orbits. For galaxies in between those +stellar masses, the peak of the circularity distribution shifts by 0.5 +(0.25) towards rotationally supported orbits for migrated (in-situ) +stars for both disk and bulge dominated galaxies, whereas ex-situ +stars remain random motion dominated at all galaxy masses. (Figure +8) +• For star forming galaxies around 1010 M⊙ in stellar mass, in- +situ, migrated and ex-situ stars clearly separate in age-metallicity +space, while the distinction becomes less clear for star forming galax- +ies outside that mass range and quenched galaxies in general. (Figure +9) +• For +both +disk +and +bulge +dominated +galaxies +between +109.5−10 M⊙ in stellar mass, in-situ, migrated and ex-situ stars clearly +separate in age-circularity space. The migrated stars are the youngest +with the highest amount of rotational support and the ex-situ stars are +the oldest and purely random motion supported, whereas the in-situ +stars are situated in between. (Figure 10) +Furthermore, we have demonstrated the diversity of the central +500 pc of galaxies as governed by the hierarchical mass build-up in a +ΛCDM universe. Galaxy interactions are an important driver in not +only contributing ex-situ stars to the center of galaxies, but also in +dictating the formation of in-situ stars and the migration of stars to +the center. This leads to an entanglement of different mechanisms +that influence the formation history of stars in the center of galax- +ies. In Figure 11 we have qualitatively identified these mechanisms +that are present in TNG50, which includes episodes of in-situ star +formation and stellar migration to the center during times of peri- +center passages and/or coalescence of mergers or flybys, infall into +galaxy groups/clusters as well as depletion of the central gas reservoir +through kinetic AGN feedback and environmental effects. +In the future, higher resolution simulations (not only spatially but +also concerning star formation and stellar feedback prescriptions) +will be needed to fully address the formation and migration of stellar +clumps and to study the formation of nuclear galaxy structures, such +as nuclear disks and star clusters, in a fully cosmological context. +Bright galaxy centers have the potential to be used in observations +as tracers of the overall galaxy assembly history. TNG50 predicts +distinct stellar populations and dynamical properties for the stars of +different origins in the center of galaxies, which can be observed +with today’s IFU capabilities. Figure 13 demonstrates that there is +even promise to deduce the fractional contribution of central in-situ, +migrated and ex-situ stars from SDSS-like observations in a galaxy +population averaged sense. +In summary, TNG50 is a tremendous advancement in predicting +the stellar build-up of sub-kpc scales in a fully cosmological context. +Its predictive power is valuable to consider new pathways in mod- +elling formation scenarios of central stellar components as well as to +push forward novel observational techniques to unveil the formation +history of galaxies. +DATA AVAILABILITY +The IllustrisTNG simulations, including TNG50, are publicly avail- +able and accessible at www.tng-project.org/data (Nelson et al. +2019a). Data directly related to this publication and its figures are +available upon reasonable request from the corresponding author. +ACKNOWLEDGEMENTS +AB is grateful to Ignacio Martín-Navarro and Jesús Falcón-Barroso +for their support and scientific discussion throughout this project. +AB also likes to thank Glenn van de Ven and Francisco Aros for use- +ful discussions. We thank the anonymous referee for helpful com- +ments on the manuscript. This work was funded by the Deutsche +Forschungsgemeinschaft (DFG, German Research Foundation) – +Project-ID 138713538 – SFB 881 (“The Milky Way System”, sub- +project B08). NF was supported by the Natural Sciences and Engi- +neering Research Council of Canada (NSERC), [funding reference +number CITA 490888-16] through the CITA postdoctoral fellowship +and acknowledges partial support from a Arts & Sciences Postdoc- +toral Fellowship at the University of Toronto. RR acknowledges fund- +ing from the Deutsche Forschungsgemeinschaft (DFG) through an +Emmy Noether Research Group (grant number NE 2441/1-1). The Il- +lustrisTNG simulations were undertaken with compute time awarded +by the Gauss Centre for Supercomputing (GCS) under GCS Large- +Scale Projects GCS-ILLU and GCS-DWAR on the GCS share of the +supercomputer Hazel Hen at the High Performance Computing Cen- +ter Stuttgart (HLRS), as well as on the machines of the Max Planck +Computing and Data Facility (MPCDF) in Garching, Germany. The +computations for this work were performed on the ISAAC cluster +of the Max Planck Institute for Astronomy at the Rechenzentrum in +Garching. +REFERENCES +Abadi M. G., Navarro J. F., Steinmetz M., Eke V. R., 2003, ApJ, 597, 21 +Abazajian K. N., et al., 2009, ApJS, 182, 543 +Agarwal M., Milosavljević M., 2011, ApJ, 729, 35 +Aharon D., Perets H. 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They are +either directly available from the corresponding halo/subhalo, stellar +particle or supplementary data catalogues on the TNG website10. +The classification of ‘bar-like’ signatures for galaxies is available +upon reasonable request from the corresponding author. +A1 Bulk Properties +We here describe how different galaxy bulk properties are defined +and measured in order to define different galaxy populations that are +analyzed in Section 4.1. All properties refer to z = 0. +• Mass: Generally all galaxy masses are reported to be the total +mass of particles of a given type (or all types in the case of dynam- +ical mass) bound to a specific subhalo as identified by the Subfind +algorithm. +• Environment: We crudely define the environment of a galaxy +by distinguishing between centrals and satellites. A central galaxy +is the most massive subhalo in its corresponding FoF halo, all other +galaxies within the same FoF halo are satellite galaxies. +• Star formation activity: Whether a galaxy is actively forming +stars or not is classified according to Pillepich et al. (2019) (see +also Donnari et al. 2019, 2021a,b), who determined the logarith- +mic distance from the star forming main sequence for each galaxy +(Δ log10 SFR). For this, the instantaneous star formation rate (SFR) +of the gas cells as well as galaxy stellar masses were calculated +within twice the stellar half mass radius. Star forming galaxies have +Δ log10 SFR ≥ −0.5 and quenched galaxies have Δ log10 SFR ≤ −1, +whereas galaxies in the green valley are in between those two values. +Unless otherwise stated we will omit the distinction of green valley +galaxies and also classify them as quenched. +• Morphology: We quantify disk or bulge dominated galaxies +based on the kinematic classification by Genel et al. (2018). For +each stellar particle the circularity parameter 𝜖 is calculated, which +gives the ratio of the particle’s specific angular momentum in z- +direction and its theoretical maximum angular momentum at that +specific binding energy (Abadi et al. 2003; Marinacci et al. 2014). +Then the mass fraction of all stellar particles with 𝜖 > 0.7 and within +ten stellar half mass radii is computed. If that fractional mass is above +0.4 we classify that galaxy as disky (see Joshi et al. 2020), otherwise +the galaxy is bulge dominated. +• Bar-like signatures: We also provide a quick estimate of whether +a galaxy has a bar-like structure in its center. For this, we calculate +𝐴2, the ratio between the second and zeroth term of the amplitude +of the Fourier expansion, from the face-on stellar surface density of +each galaxy as a function of the 2D radius in ∼ 0.04 dex steps, where +each bin is ensured to have at least 100 stellar particles. We then +10 https://www.tng-project.org/data/docs/specifications/ +MNRAS 000, 1–34 (2022) + +Centers of TNG50 Galaxies +27 +identify peaks in the 𝐴2-radius plane with a prominence of at least +0.05. After that, the value of 𝐴2 for the largest peak within a radius of +10 kpc is recorded. We impose this radius cut to mitigate the effect of +other 𝐴2 features that may be present at larger radii (see also Frankel +et al. 2022). This is done for all snapshots between z = 0 (SnapNum +99) and z = 4.2 (SnapNum 20). Similarly to Rosas-Guevara et al. +(2020)11, we then define a bar-like structure, when the maximum +𝐴2 value (at a given time step) is above 0.2. Bar-like structures at +z = 0 are defined solely based on their instantaneous 𝐴2 value at +that snapshot. While this method leads to accurate identification of +symmetrically elongated ‘bar-like’ features, we do not check if this +is actually a ‘proper’ bar in the astrophysical sense. Nevertheless, +our classification leads to a bar fraction of around 40% (50%) for +disk (all) galaxies above 1010 M⊙ in stellar mass, which is consistent +with observations (see e.g. Sheth et al. 2008; Simmons et al. 2014; +Díaz-García et al. 2016a). +• AGN feedback: We quantify the severity of feedback from su- +permassive black holes by determining whether a specific galaxy +lies below or above the median scaling relation of TNG50 galax- +ies (e.g. similarly to Martín-Navarro et al. 2018b, for observations). +Typical AGN feedback defining properties could be the black hole +(BH) mass and the cumulative energy injection in the thermal and/or +kinetic feedback modes (see e.g. Weinberger et al. 2017; Zinger et al. +2020, for background). Such scaling relations are always computed +with respect to the total stellar mass of galaxies as well as for the +total TNG50 galaxy sample above 5 × 108 M⊙ (see Section 2.4.2). +Properties of black hole particles per galaxy are computed as the sum +of all black holes particles associated to a given galaxy via Subfind. +• Physical size: Similarly, we define extended or compact galaxies +depending on whether they are below or above the median stellar +mass-size relation of TNG50 galaxies. The size is the 3D stellar half +mass radius. +A2 Stellar Particle Properties +Stellar particle properties at z = 0 that are analyzed in Section 4.2 +are briefly described here: +• Age: We define the age of a stellar particle as the look- +back time in Gyr. This quantity is calculated from the field +GFM_StellarFormationTime, which provides the exact time of +birth of a star in scale factors. +• Metallicity: We convert the mass fraction in metals 𝑍 as pro- +vided by the field GFM_Metallicity to log10 𝑍/𝑍⊙ with 𝑍⊙ = 0.02. +This follows conventions adopted in observations, e.g. Gallazzi et al. +(2021). +• [Mg/Fe]: We calculate the magnesium-to-iron abundance from +the mass fraction in magnesium 𝑍Mg and iron 𝑍Fe provided by the +simulation (GFM_Metals) via [Mg/Fe] = log10(𝑍Mg/𝑍Mg, ⊙) − +log10(𝑍Fe/𝑍Fe, ⊙). The adopted solar values are +𝑍Mg, ⊙ += +0.00064298 and 𝑍Fe, ⊙ = 0.001218 respectively from Asplund et al. +(2009). +• Circularity 𝜖: We calculate the instantaneous circularity of each +stellar particle following Genel et al. (2018). For this we first com- +pute the specific angular momentum of each particle in z-direction +by aligning the z-axis of the simulation box with the total angular +momentum of stellar particles within twice the stellar half mass ra- +dius of a given galaxy. The theoretical maximum angular momentum +11 We find that our computed 𝐴2 values are slightly lower than those mea- +sured by methods of Rosas-Guevara et al. (2020), hence we adopt their ‘weak +bar’ threshold of 0.2, whereas their ‘strong bars’ have values of 𝐴2 ≥ 0.3. +each stellar particle can have at its recorded specific binding energy +(i.e. 1 +2 |v|2 + Φ) is calculated by sliding a maximum filter across the +particle list of specific angular momenta sorted by their total spe- +cific binding energy with a window size of one hundred. Stars with +circularities around zero are on random motion dominated orbits, +whereas values close to one indicate more circular orbits. Negative +circularities depict counter rotating orbits. +APPENDIX B: VALIDATION FOR ANALYSIS OF TNG50 +GALAXY CENTERS +We briefly validate and justify here the analysis choices we made in +Sections 2.4 and 3.1 of the main text using two TNG50 galaxies as +examples where applicable. Both of these galaxies are also contained +within the subboxes, i.e. smaller regions of the full simulation box, +which offer 3600 snapshots resulting in a time sampling of 2−3 Myr. +At z = 0, SubfindID 537941 is a Milky Way like galaxy, found in +‘Subbox0’, and SubfindID 35, found in ‘Subbox2’, is a compact +∼ 1010 M⊙ in stellar mass galaxy, that quenched approximately 9 Gyr +ago and is now found in the most massive halo (∼ 1014 M⊙) in +TNG50. Both of these subhalos stay inside the subbox across their +life time making it possible to compare their full histories in the higher +cadence outputs to that of the hundred full box snapshot outputs. +B1 Selection of central stars +We show in Figure B1 the energy and angular momentum distribution +of stars in the galaxy’s centers at z = 0 by selecting particles based on +their current radius as well as on their energies according to Equation +1. The selection was made with 𝑟cut = 500 pc. We also show their +peri- and apocenters which we calculated by recording their radii +from all 16 subbox outputs between the last full box snapshot, i.e. +z = 0 (SnapNum 99), and the one prior to that, i.e. ≈ 136 Myr +(SnapNum 98) before. We then found all minima and maxima and +took the average respectively. +Some particles selected by their instantaneous radius at z = 0 have +large energies and are hence able to move away from the center to +much larger radii, i.e. they are not actually spending the majority of +their orbital time within our selected spherical aperture. This is also +reflected by their larger (up to 10 kpc) apocenters, whereas particles +selected by their energy have time-averaged peri- and apocenters not +larger than 1 kpc. Even though that is larger than our selected 𝑟cut +value, probably due to our simplifying assumptions in calculating +𝐸cut, we argue that this selection gives a much cleaner selection of +stars actually belonging to the center without interloping particles on +much more eccentric orbits. Both selection criteria yield a compara- +ble number of stars, with around 21000 stars for SubfindID 537941 +(1.6% of total amount of stars) and ∼ 40000 stars for SubfindID 35 +(∼ 15% of total amount of stars). +B2 Definition of birth radii of stellar particles +In Figure B2 we show the difference between the center position of +the two galaxies once taken from the high cadence subbox outputs +(the most bound stellar particle)12 and once from the SubhaloPos +12 As there is in fact no subhalo information from Subfind available for the +subboxes, we start off with the interpolated center values from the full box +snapshots and then recalculate the center in a 5 kpc box around that, recenter +and recalculate the center again in a 5 kpc box around that. We verified that this +MNRAS 000, 1–34 (2022) + +28 +Boecker et al. +Figure B1. Selection of stellar particles belonging to a galaxy’s center, left column for SubfindID 537941 and right column for SubfindID 35. Blue points +show stellar particles that have radii smaller than 𝑟cut = 500 pc and orange points where selected based on their energies being smaller than 𝐸cut according to +Equation 1, also with 𝑟cut = 500 pc. Top row: Angular momentum of selected particles as a function of energy. The dashed lines emphasizes 𝐸cut, whereas the +two dotted lines show Lz = ±𝑟cut𝑣circ(𝑟cut). Grey points show all other stellar particles belonging to the respective galaxy. Bottom row: Pericenter of stellar +particles versus their apocenter time averaged from subbox outputs between 0 (full box snapshot 99) and 136 Myr (full box snapshot 98). Stellar particles purely +selected on their radius have a wider distribution in their energies and are hence able to move much further outside our selected spherical volume of 500 pc, +which is also reflected by their larger apocenters. +argument from the full box snapshots, which we interpolated onto +the finer time sampling of the subbox outputs using a cubic spline. +It is evident that deviations of around 2 − 5 kpc appear, specifically +at times of pericenter passages of either satellites merging with the +host (SubfindID 537941) or of the galaxy itself around its host +(SubfindID 35). +These deviations of a galaxy’s center position are enough to +severely miss-classify migrated and in-situ stars, when their in- +stantaneous birth positions, as provided by the simulations output +(BirthPos), are used in conjunction with the interpolated subhalo +center. For example, in Figure B3 we show histograms of migrated +and in-situ stars selected according to their instantaneous birth po- +sition using the proper center from the subbox and the interpolated +one from the full box snapshots. For both galaxies the number of mi- +gives the correct center by inspecting the galaxies by eye. We note however +that this approach will not yield correct results in a general black box fashion. +grated stars doubles when the interpolated center is used compared +to the correct center. +Ideally, we would like to apply our analysis to as many galaxies as +possible, but as only a handful of galaxies reside inside the subboxes +during their whole life time, we need an alternative measure for +classifying them into migrated and in-situ stars that can be applied +to the full box of TNG50. We therefore take the particle’s position +at the full box snapshot it first appeared in as its “birth” position and +compare it to the instantaneous one with the proper centering, also +shown in Figure B3. We see that the shapes of their histograms as +well as their percentages match. Of course, their classification is not +identical, as particles move between their exact formation time and +the time of when the snapshot was taken, however this measure seems +to be more accurate than using the instantaneous birth positions with +the interpolated center. +Finally, we compare the instantaneous birth radii using the correct +center of the migrated and in-situ stars (classified with the instanta- +neous positions using the correct center) with birth radii calculated +MNRAS 000, 1–34 (2022) + +SubfindID 537941 +E[kpc]Centers of TNG50 Galaxies +29 +0 +2 +4 +6 +8 +10 +12 +14 +Lookback Time [Gyr] +4 +2 +0 +2 +4 +10 +100 +[kpc] +SubfindID 537941 +Distance to two +most massive mergers +Centersubbox +fullbox +x +y +z +0 +2 +4 +6 +8 +10 +12 +14 +Lookback Time [Gyr] +10 +5 +0 +5 +10 +100 +1000 +[kpc] +SubfindID 35 +Distance to host +Centersubbox +fullbox +x +y +z +Figure B2. Differences in galaxy centering between full box and sub- +box snapshots. Comparison between the interpolated (cubic spline) center +position from the 100 full box snapshots (using SubhaloPos from the sub- +halo catalogue) and the center position (most bound stellar particle) from +the higher cadence subbox outputs for two example galaxies (blue shaded +lines). The top panel shows SubfindID 537941, a Milky Way like galaxy +and the bottom panel shows SubfindID 35, a ∼ 1010 M⊙, quenched galaxy +that is now a satellite of the most massive halo in TNG50. We see that the +interpolated center of the galaxies starts to deviate significantly from the true +center position when there are pericenter passages from satellites that merge +with galaxy or when the galaxy itself is a satellite and approaches its host +(thick orange lines). +by the other two methods. The birth radii determined from the full +box snapshots scatter around the one-to-one relation, whereas the +ones with the interpolated center do not. Hence, we conclude that +applying the migrated and in-situ classification based on their birth +snapshot position to the whole TNG50 box seems to provide us with +similar knowledge about their origin as if we would have used their +instantaneous birth position. +APPENDIX C: THE TOTAL EX-SITU STELLAR MASS +FRACTION OF TNG50 +In Figure C1 we show the total ex-situ stellar mass fraction as a +function of total stellar mass for TNG50 galaxies. They are further +divided into star forming and quenched galaxies according to their +distance to the star forming main sequence at z = 0. +Most importantly, we see that the total ex-situ stellar mass fraction +does not change worryingly, when comparing the fiducial definition +of ex-situ stars according to Rodriguez-Gomez et al. (2016) as well as +our definition, where we exclude stars from accreted satellites classi- +fied as clumps (i.e. SubhaloFlag = 0). Even though the clumps do +not play an important role for the definition of the ex-situ fraction, +they are abundant in TNG50 (see Appendix E). Only due to our defi- +nition, we could confirm the migration of these clumps contributing +significantly to the build-up of the galaxy center for galaxies above +1011 M⊙ in stellar mass. +Additionally, Figure C1 shows that star forming galaxies have +a higher total ex-situ fraction on average than quenched galaxies +across all stellar masses. Until galaxy stellar masses of 1010 M⊙ +the ex-situ fraction stays roughly constant with values around 4-5% +and 3% for star forming and quenched galaxies respectively. There +is however a large scatter associated with the ex-situ fraction for +galaxies in this mass regime. Above 1010 M⊙ the ex-situ fraction +sharply increases with galaxy stellar mass reaching approximately +50% for galaxy between 1011 − 1012 M⊙ for TNG50. The scatter +decreases accordingly. We have checked this exact relation with the +lower resolution run TNG50-2 as well as TNG100 and obtained +similar results. +Figure C2 shows the mass-size relation for TNG50 galaxies col- +ored according their relative ex-situ fraction, i.e. if they have high +or low ex-situ fractions with respect to the average typical for their +respective stellar mass (Merritt et al. 2020, following). Galaxies with +stellar masses ≲ 5×1010 M⊙ and above average ex-situ fractions are +on average more extended. This median trend is not observed for the +high-mass end, however compact galaxies at 1011 M⊙ have almost +exclusively below average ex-situ fractions in agreement with other +studies (see Davison et al. 2020, for EAGLE; Merritt et al. 2020, for +TNG100; Zhu et al. 2021, for TNG50). +APPENDIX D: RESOLUTION CONVERGENCE TESTS +To test numerical convergence we perform the exact same analy- +sis from the main text for three different resolution realizations of +TNG50: TNG50-1 (flagship), TNG50-2 and TNG50-3. Throughout, +the galaxy sample selection is the same as in Section 2.4.2 except we +omit cutting galaxies with less then one hundred stellar particles in +their centers. +For reference, the mass resolution of TNG50-2 and TNG50-3 +is 8 and 64 times and the gravitational softening length is 2 and +4 times worse than TNG50-1 respectively. The latter translates to +physical sizes of 288 pc (TNG50-1), 576 pc (TNG50-2) and 1.152 kpc +(TNG50-3) for collisionless particles at z = 0. +In the top row of Figure D1 we show the results of the central stellar +mass of the in-situ, migrated and ex-situ stars as a function of total +dynamical mass for all three resolutions. We see that the total stellar +mass in the central 500 pc of TNG50 galaxies is converging at all halo +masses, meaning that the distance between TNG50-1 and TNG50-2 +(∼ 1 dex) is around 50% smaller than the distance between TNG50-2 +and TNG50-3 (∼ 0.5 dex). The same is true when looking at the +central mass for just the in-situ stars, even though the converging +MNRAS 000, 1–34 (2022) + +30 +Boecker et al. +Figure B3. Effect of erroneous galaxy centering on the classification of migrated stellar particles. Left panels: Histograms of particles being classified +as migrated (orange) and in-situ (pink) using three different approaches: the instantaneous position at birth with the proper centering from the subbox outputs +(thick solid line), the same but with the interpolated center (thin dashed line) as well as the position of particles taken from the full box snapshots they first +appeared in (thin solid line). Right panels: Comparing the values of the instantaneous birth radii with the centering from the subbox outputs with the ones with +the interpolated center (fainter, smaller points) as well as the radii from the full box snapshots in which the particles first appeared in (bolder, bigger points). +The one-to-one relation is also shown (black dashed lines). The top panel shows SubfindID 537941, a Milky Way like galaxy and the bottom panel shows +SubfindID 35, a ∼ 1010 M⊙, quenched galaxy that fell into the most massive halo of TNG50. Using the instantaneous birth position with the interpolated center +leads to a wrong classification of migrated and in-situ stars, whereas using the positions from the full box snapshots at birth gives comparable results to the +instantaneous birth positions with the proper centering in the actual number of particles in the two categories as well as the values of their birth radii. +MNRAS 000, 1–34 (2022) + +SubfindID 537941 +1000 +5 +Migrated +Instantenous (proper center): 21 % +les +From Snapshot: 20 % +Particle +800- +Instantenous (interpolated center): 43 % +600 +of +Number +400- +200- +1 +2500 +Insitu +Instantenous (proper center): 79 % +8 +From Snapshot: 80 % +2000 +Instantenous (interpolated center): 57 % +arti +Instantenous +Number +1000 +500 + From Snapshot + Instantenous (interpolated center) +0 +0.02 +14 +12 +10 +8 +2 +0.02 +6 +0 +0.1 +5 +Age [Gyr] +Birth Radius (see legend) 「kpciSubfindID 35 +2000 +5 +Migrated +Instantenous (proper center): 26 % +les +Birth Radius (proper center) [kpc] +From Snapshot: 26 % +1500 +Partic + Instantenous (interpolated center): 51 % + 1000 +Number +500- +0 +2500 +Insitu +Instantenous (proper center): 73 % +8 +From Snapshot: 74 % +2000 + Instantenous (interpolated center): 49 % +Instantenous +Number +1000- +500 +From Snapshot +Instantenous (interpolated center) +0 +0.02 +14 +12 +10 +2 +0.02 +8 +6 +0 +0.1 +5 +Age [Gyr] +Birth Radius (see legend) [kpc]Centers of TNG50 Galaxies +31 +Figure C1. Total ex-situ stellar mass fraction versus total stellar mass in +TNG50 at z = 0. The solid lines show the fiducial ex-situ fraction as classified +by methods in Rodriguez-Gomez et al. (2016), whereas the dashed lines +show total ex-situ fractions excluding spurious (SubhaloFlag = 0) galaxies +or “clumps” (see Section 3.1.2). The faint bands depict the 16th and 84th +percentiles. There is no significant difference between the two classifications +regarding the total ex-situ stellar mass fraction. We also divide by star-forming +(blue lines) and quenched galaxies (red lines) at z = 0. The star forming +galaxies have higher ex-situ fractions on average at fixed stellar mass than +quenched galaxies. +of the lines becomes much less obvious. For the migrated stars it +looks like the central mass is better converged for galaxies with total +dynamical masses below 1012 M⊙. However, when comparing the +central stellar mass fraction of in-situ and migrated stars (bottom row +of Figure D1), the convergence behaviour seems to be much more +complex. For galaxies outside masses of 1012±0.5 M⊙, the central +stellar mass fractions seem to be converging. Overall though, the +central migrated mass becomes on average larger than the in-situ +mass with decreasing resolution for all galaxy masses. +This is due to the fact that the numerical resolution also influ- +ences which stars are classified as in-situ and migrated, which is a +consequence of how (spatial and mass) resolution effects star for- +mation and feedback in the TNG model. Better resolution allows +for higher gas densities and better spatial sampling of the gas cells, +which produces galaxies with higher stellar mass and more compact +sizes. Thus, in an absolute sense more stellar mass resides overall +in the center of TNG50 galaxies for higher resolution runs, however +differentially more stellar mass comes from outside the fixed 500 pc +aperture for lower resolution runs (see also Section 5.4). Interest- +ingly, the complexity in the central in-situ and migrated fractions +seen around 1012 M⊙ coincides with the complexity in the conver- +gence behaviour of the mass-size relation in TNG50 (see Pillepich +et al. 2019, Figure B1). +For the absolute stellar mass of the central ex-situ stars the start of +convergence is not yet apparent, but the fractions is clearly converg- +ing. Again, the ex-situ mass in the center increases with increasing +resolution. Additionally, the minimum galaxy mass that exhibits any +ex-situ stellar mass in the center is extended towards lower mass +galaxies for increasing resolution. Again, this behaviour is a conse- +1.0 +0.5 +0.0 +0.5 +1.0 +fexsitu +109 +1010 +1011 +1012 +Total Stellar Mass [M ] +10 +1 +2 +50 +3D Stellar Half Mass Radius [kpc] +TNG50 +z = 0 +fexsitu +< fexsitu +Figure C2. Connection between the galaxy mass-size relation and total +ex-situ fractions in TNG50 at z = 0. The 3D half mass radius against total +stellar mass for 4344 galaxies in TNG50 (M★, tot > 5 × 108 M⊙). The points +are color-coded according to the relative ex-situ fraction Δfexsitu indicating +whether the total ex-situ fraction of a given galaxy above or below the average +at fixed galaxy stellar mass. Median mass-size relations are shown separately +for galaxies having above (solid black line) and below (dashed black line) +average ex-situ fractions. Below ∼ 5 × 1010 M⊙, galaxies with high relative +ex-situ fractions are on average more extended. +quence of the complex interplay between mass and spatial resolution. +Subhalos that were either not sufficiently resolved with enough stellar +particles or even remained dark at lower resolution are able to form +enough stars at higher resolution. As a consequence, galaxies that +become accreted are not only more massive in stellar mass, but also +more abundant, especially at the low mass end. Thus, the ex-situ stel- +lar mass is higher in general and also contributes at the lower galaxy +masses when resolution is increased. Numerical resolution has also +been shown to impact the disruption of merging satellite galaxies, +which could lead to overmerging, especially in the outskirts of galax- +ies (see van den Bosch & Ogiya 2018; Merritt et al. 2020). Hence, +a higher resolution will result in more accreted stellar material in +the center of galaxies. Furthermore, the resolution of TNG50 will +impact the ability to resolve the dynamics of its resolution elements +(stars, clumps, gas cells, etc.), including dynamical friction, at scales +much below the spatial resolution of the simulation, i.e. certainly at +the tens of pc scales. +Even though TNG50 is a tremendous improvement in resolution +for cosmological box simulations, we still need a higher resolution +to fully assess the convergence behaviour of the absolute central ex- +situ stellar mass (see Grand et al. 2021, for a study of the satellite +galaxy population of a highly resolved Milky Way like galaxy in a +cosmological context and comparison to lower resolution models). +MNRAS 000, 1–34 (2022) + +Iotal Exsitu Stellar Mass Fraction +1.00 +TNG50 +Z=0 +fiducial +without clumps +0.10 +Star Forming +Quenched +0.01 +109 +1010 +1011 +1012 +Total Stellar Mass [Mo]32 +Boecker et al. +1011 +1012 +1013 +Total Dynamical Mass [M ] +104 +105 +106 +107 +108 +109 +1010 +Central Stellar Mass [M ] +z = 0 +All +Insitu +Migrated 'Smooth+Clumpy' +Exsitu +TNG50-1 +TNG50-2 +TNG50-3 +1011 +1012 +1013 +Total Dynamical Mass [M ] +z = 0 +TNG50-1 +TNG50-2 +TNG50-3 +1011 +1012 +1013 +Total Dynamical Mass [M ] +z = 0 +TNG50-1 +TNG50-2 +TNG50-3 +1011 +1012 +1013 +Total Dynamical Mass [M ] +z = 0 +TNG50-1 +TNG50-2 +TNG50-3 +1011 +1012 +1013 +Total Dynamical Mass [M ] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Central Stellar Mass Fraction +z = 0 +Insitu +Migrated 'Smooth+Clumpy' +Exsitu +TNG50-1 +TNG50-2 +TNG50-3 +1011 +1012 +1013 +Total Dynamical Mass [M ] +z = 0 +TNG50-1 +TNG50-2 +TNG50-3 +1011 +1012 +1013 +Total Dynamical Mass [M ] +10 +4 +10 +3 +10 +2 +10 +1 +100 +z = 0 +TNG50-1 +TNG50-2 +TNG50-3 +Figure D1. Influence of the numerical resolution on the central (500 pc) in-situ, migrated and ex-situ stars of TNG50 galaxies at z = 0. Top panel: Median +trends of the central stellar mass as a function of a galaxy’s total dynamical mass for all (all), in-situ (pink), migrated (orange) and ex-situ (blue) stars and +different numerical resolution realizations of the same cosmological volume. The thicker the line the better the numerical resolution. Shaded areas show the +16th and 84th percentiles for the highest resolution run. Bottom panel: Instead of the absolute stellar mass we now show the central stellar mass fraction of the +in-situ, migrated and ex-situ stars. The central ex-situ mass faction is converging, whereas the behaviour is more complex for the in-situ and migrated fraction. +APPENDIX E: CLUMPS IN TNG50 +Here, we show some examples and properties of clumps, i.e. sub- +halos detected by the Subfind algorithm, but which were formed +not through the collapse of a dark matter halo, that are present in +TNG50. Instead these clumps form in the gaseous disk of galaxies +or fragments of it. Particularly, we could observe qualitatively that +significant clump formation predominantly occurs during galaxy in- +teractions at z > 1, such as mergers or fly-bys, but it can also take +place when a galaxy evolves in isolation (see also Di Matteo et al. +2008, for a similar finding). +We present a visual example of such clumps at the top of Figure +E1 for a galaxy that lives in an environment with many galaxy inter- +actions. Clumps embedded in the stellar disk of that galaxy clearly +exhibit a gaseous counterpart, whereas clumps further out and not +within the disk are only seen in the stellar surface mass density (at +the time of inspection). The clumps visible in both the stars and +gas are clearly distributed along the spiral (or tidal) arms of that +galaxy. All of them migrate to the center and deposit stars (and also +gas) there (marked by orange arrows). In that process some of them +(marked by lighter orange arrows) merge with other clumps, which +we reconstructed with the merger trees. +The statistics on clumps in TNG50 shown on the bottom left hand +side of Figure E1 reveals that around 36% of all TNG50 galax- +ies posses clumps at any point in their lifetime of which a fourth +have clumps that migrate to the center. Above 5 × 1010 M⊙ in stel- +lar mass all galaxies exhibit clumps at some point. The amount of +clumps per galaxy starts to rise above one for galaxy stellar masses +larger than 1010 M⊙. On average a Milky Way mass galaxy has five +clumps, whereas the most massive galaxies in TNG50 can have up +to a hundred clumps. The number of clumps that actually migrate +to a galaxy’s center starts to rise at higher galaxy masses at around +5 × 1010 M⊙ reaching an average of about 5 migrated clumps per +galaxy at the high mass end. We remind the reader that these and +following numbers do not account for clumps that merged with other +clumps, thus the true number of all clumps ever formed is actually +even higher. +We also show the distribution of four properties of the clumps in +TNG50 on the bottom right hand side of Figure E1. Each property +is shown for all clumps at the time of formation, for all clumps that +migrated at the time of formation and for all clumps that migrated at +the time they arrive at their galaxy’s center. +We see that clumps are formed with a broad distribution of stellar +masses with a peak at 108 M⊙13. Only clumps that are formed with +such stellar masses or higher are able to migrate to the center. The +13 We caution the reader to not trust the few clumps that have stellar masses +of around 1010 M⊙, which likely originate from switching between the host +galaxy and the actual clump during the halo/subhalo finding process. +MNRAS 000, 1–34 (2022) + +Centers of TNG50 Galaxies +33 +16 kpc +TNG50 +z = 0.46 +Stars +16 kpc +TNG50 +z = 0.46 +Gas +106 +107 +108 +109 + [M /kpc2] +106 +107 +108 +gas [M /kpc2] +SubfindID 220595 +Figure E1. Example and summary statistics of clumps in TNG50 galaxies. Top panel: Stellar (left, full projection) and gas (right, thin slice with |𝑧 | ≤ 5 kpc) +surface mass density of a TNG50 galaxy (SubfindID 220595) in face-on projection exhibiting clumps at z = 0.47. Clumps migrating to the center (pink dot) +are marked with orange arrows, whereas clumps not belonging to the galaxy are marked with white arrows. Clumps with lighter orange arrows will first merge +with other clumps (darker orange arrows), before arriving at the center. Lower panel, left: Distribution of host galaxy stellar masses (top) displaying clumps in +general (blue) and clumps that migrate to the center (orange) at any time, compared to the total galaxy sample (black). Numbers inside the brackets display the +total amount. All galaxies in our sample above ∼ 5 × 1010 M⊙ exhibit clumps at some point in their life time. The median number of clumps that ever existed per +galaxy as a function of host galaxy stellar mass are also shown (bottom). The shaded area shows the 16th and 84th percentile. The number of clumps arriving at +a galaxy’s center is about a decade lower than the total amount of clumps formed. Note however that the true number of all clumps ever formed is higher due +to clump-clump mergers. Lower panel, right: Distribution of certain clump properties (blue solid line: all clumps at the time of formation, orange solid line: +clumps that migrate to the center at the time of formation, light orange dashed-dotted line: clumps that migrated to the center at the time of arrival at the center): +from left to right: total stellar mass, gas fraction (ratio of gas mass to total baryonic mass), formation time (solid lines)/time taken to arrive at the center (dashed +line) and distance from the host galaxy of the clumps. +MNRAS 000, 1–34 (2022) + +All Galaxies (2531) +Galaxies with clumps (916) +Galaxies with clumps +All Clumps +Migrated Clumps +Migrated Clumps +migrating to center (235) +at Formation +at Formation +arriving at Center +103 +TNG50 +TNG50 +TNG50 +103 +Z=0 +z=0 +Z=0 +Clumps +Galaxies +102 +TT +# 10 +TT +10 +# +108 +1010 +104 +106 +0.0 0.2 0.4 0.6 0.8 1.0 +Total Stellar Mass +Gas Fraction +of Clumps [Mo] +of Clumps +TNG50 +z=O +TNG50 +TNG50 +All Clumps +103 +z=0 +z=0 +LL +per +(12885) +Clumps +Migrated +Clumps +Clumps +102 +(648) +# +10 += +# +LLLI +TTTT +T +TT +TTTTT +TTTTT +109 +1010 +1011 +24681012 +1012 +214 +10 +102 +0 +103 +Total Stellar Mass +Lookback Time [Gyr] +Distance from Host [kpc +of Host Galaxy [Mo]34 +Boecker et al. +stellar mass that they then actually deposit at the center is a flattened +out distribution all the way down to a few stellar particles. Thus +clumps can suffer significant stellar mass loss while travelling to the +galaxy center. +Even though there is a broader peak of clumps formed with high +gas fractions (0.6 − 1), the gas fraction distribution of clumps that +migrate to the center is significantly flatter. Thus, a high gas fraction +is not necessarily an indication of whether a clump is able to migrate +to the center or not. +Furthermore, the clumps are formed all throughout cosmic time in +TNG50, with a slight increase towards younger lookback times. The +number of clumps that migrated with formation times younger than +6 Gyr ago drops compared to clumps formed at older ages. This is +understandable as it takes a few Gyr for the clumps to migrate to the +center. Most clumps need around 1 Gyr to do so, but there is a long +tail towards higher migration times up until 6 Gyr. +The peak distance from the host galaxy, where clumps form, is +around 20 kpc and flattens out towards smaller distances. Whereas +the number of migrated clumps drop sharply after that distance, the +distance of clumps formed overall extends all the way until several +hundred kpc. Hence, clumps in TNG50 can form in the halo of +galaxies, possibly in gaseous tidal or ram-pressure-stripped tails and +gas fragments during galaxy interactions. We empathize here that the +clumps formed at such large distances are not part of any satellite +galaxy that then merged with the host (at least according to the +Subfind algorithm and the merger trees). This is because we identify +clumps as subhalos with SubhaloFlag = 0 that directly merged onto +the main progenitor branch of the host galaxy. Hence, clumps that +are part of satellite galaxies that are then brought in by merging with +the host galaxy, are also not accounted for in our statistics. +We provide further discussion and comparison to clumps in other +simulations in Section 5.3. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–34 (2022) + diff --git a/xNFJT4oBgHgl3EQfgiz6/content/tmp_files/2301.11562v1.pdf.txt b/xNFJT4oBgHgl3EQfgiz6/content/tmp_files/2301.11562v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bf4ee9a051beb058444b782089ed3760de66be59 --- /dev/null +++ b/xNFJT4oBgHgl3EQfgiz6/content/tmp_files/2301.11562v1.pdf.txt @@ -0,0 +1,11785 @@ +Variance, Self-Consistency, and Arbitrariness in Fair Classification +A. Feder Cooper 1 Solon Barocas 2 3 Christopher De Sa 1 Siddhartha Sen 2 +Abstract +In fair classification, it is common to train a model, +and to compare and correct subgroup-specific +error rates for disparities. However, even if a +model’s classification decisions satisfy a fairness +metric, it is not necessarily the case that these deci- +sions are equally confident. This becomes clear if +we measure variance: We can fix everything in the +learning process except the subset of training data, +train multiple models, measure (dis)agreement +in predictions for each test example, and interpret +disagreement to mean that the learning process +is more unstable with respect to its classification +decision. Empirically, some decisions can in fact +be so unstable that they are effectively arbitrary. +To reduce this arbitrariness, we formalize a notion +of self-consistency of a learning process, develop +an ensembling algorithm that provably increases +self-consistency, and empirically demonstrate its +utility to often improve both fairness and accu- +racy. Further, our evaluation reveals a startling +observation: Applying ensembling to common +fair classification benchmarks can significantly +reduce subgroup error rate disparities, without +employing common pre-, in-, or post-processing +fairness interventions. Taken together, our results +indicate that variance, particularly on small +datasets, can muddle the reliability of conclusions +about fairness. One solution is to develop larger +benchmark tasks. To this end, we release a toolkit +that makes the Home Mortgage Disclosure Act +datasets easily usable for future research. +1. Introduction +Consider the following experiment: We fit 10 logistic +regression models on different training sets from the +COMPAS benchmark (Larson et al., 2016), and compare +the resulting classifications for two individuals reserved +in a test set. As shown in Figure 1, while the 10 models +1Department of Computer Science, Cornell University, Ithaca, +NY, USA 2Microsoft Research, New York, NY, USA 3Department +of Information Science, Cornell University, Ithaca, NY, USA. Cor- +respondence to: A. Feder Cooper . +Preprint. Copyright 2023 by the author(s). +Ind. 1 +Ind. 2 +Two individuals from COMPAS +0 +2 +4 +6 +8 +10 +# of Model predictions +y = 1 +y = 0 +Figure 1. Comparing predictions ˆy for 2 individuals, according to +10 logistic regression models trained on bootstrap replicates. +indicate complete agreement for how to classify Individual +1, they disagree completely on Individual 2. If we were +to pick one model to use, there would be no effect on how +Individual 1 is classified; however, for Individual 2, the +prediction is effectively random. We can interpret this to +mean that the learning process that trained these models +is more unsure about how to classify Individual 2. The +classifications for Individual 2 exhibit high variance; they +are very sensitive to the training data on which the models +were trained. This presents a problem if we only analyze +one model with one deterministic classification decision. If, +by happenstance, we had sampled slightly different training +examples from the available data, the decision could very +well flip to the other class. In this respect, the classification +is effectively arbitrary, which is especially unsettling +for decisions that can have significant consequences on +individuals’ lives (Citron & Pasquale, 2014; Cooper et al., +2022a;b; Creel & Hellman, 2022; Black et al., 2022b). +Intuitively, it does not seem fair for individuals to be sub- +ject to decisions for which the learning process is arbitrary. +Moreover, this arbitrariness can also bring about discrimi- +nation, if a model’s decisions are systematically more arbi- +trary for certain demographic groups. However, such unfair- +ness is not captured in popular fairness definitions, which +are commonly applied to evaluate the fairness of a single +model (Hardt et al., 2016; Pleiss et al., 2017; Kleinberg et al., +2017; Chouldechova, 2017; Calders et al., 2009). Instead, it +is made visible by examining empirical approximations of +the distribution over possible models that a learning process +could produce by training on different samples of the dataset. +At this level, it becomes clear that even if a particular deter- +ministic classifier achieves fairness with respect to subgroup- +specific error rates, it is not necessarily the case that its +underlying classification decisions are equally confident. +In this paper, we formalize this intuition, conceiving of clas- +arXiv:2301.11562v1 [cs.LG] 27 Jan 2023 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +sifications like those for Individual 2 as reflective of a lack +of self-consistency in the learning process that produced +them. Our aim is to develop a procedure for improving +self-consistency, in order to improve both fairness and +reliability in conclusions drawn about model performance. +To these ends, we make the following contributions: +• We motivate a definition of self-consistency based on vari- +ance, and illustrate how it reveals novel insights concern- +ing arbitrariness and unfairness in ML (Section 3); +• We propose an ensembling algorithm that provably +increases self-consistency. Our algorithm predicts for +inputs that attain a user-specified level of confidence, and +abstains otherwise (Section 4); +• We validate our algorithm across datasets and models. +These results demonstrate the importance of self- +consistency in fair classification. Further, they dispute the +validity of COMPAS and South German Credit as +reliable fairness benchmarks (Section 5). +Taken together, our results indicate that variance, particu- +larly in small datasets, can muddle the reliability of conclu- +sions about fairness interventions. One possible solution is +to develop larger benchmark tasks. To this end, we build and +release a toolkit that makes the Home Mortgage Disclosure +Act datasets (HMDA) easily usable for future research. +2. Preliminaries +To begin, we need to establish some background notation +and definitions for supervised binary classification in algo- +rithmic fairness settings. Consider a distribution p(·), from +which we can sample examples (x, g, o), where x ∈ Rm +are instances with m features, g is a group of protected +attributes that we do not use for learning (e.g., race, gender), +and o ∈ O are the associated observed labels. O ⊆ Y, +where Y = {0, 1} is the label space. From p(·), we can +sample training datasets of size n, i.e., {(x, g, o)}n +i=1, with +D representing the set of all such n-sized datasets. A learn- +ing process runs a training procedure A on training dataset +Dk ∈ D to produce a classifier hDk ∈ H, where H is the +hypothesis class consisting of possible valid models. That +is, a trained classifier hDk provides a deterministic mapping +from the instance space to the label space, i.e. hDk : X �→ +Y. ˆy = hDk(x) is the predicted label (or, simply, predic- +tion) for x. A classifier hDk has an underlying regressor +rDk : X �→ [0, 1], which computes the probability of posi- +tive (i.e., 1) classification for each x. We produce hDk from +rDk by applying a threshold τ ∈ [0, 1] to the outputs of rDk; +the classification decision rule is hDk(x) = 1[rDk(x) ≥ τ], +which evaluates to 0 when the output regressor probability +for x is less than τ, and to 1 otherwise. +We refer to the distribution over possible trained models as +µ. Training procedure A produces hDk ∼ µ by minimizing +the loss of predictions ˆy with respect to their associated +observed labels o in Dk. +This loss is computed by a +chosen loss function ℓ : Y × Y �→ R. To reason about the +expected performance of a particular trained model hDk, +we compute predictions for a test set of fresh examples +and calculate their loss. Since these examples have not +influenced training, we can understand their predictive loss +to be an estimate of the error of hDk for the task at hand. +In practice this estimate is dependent on the specific dataset +Dk used for training. To reason more generally about the +error of possible models produced by a specific learning +process, we instead need to consider the expected error, +ED,O[ℓ(o, ˆy)]. This computes the loss with respect to the +distribution of all possible trained models µ. +Of course, in practice we typically only have access to one +training dataset, not a distribution p(·) from which we can +sample fresh training examples. As a result, to train multiple +hDk and estimate expected error, we bootstrap the available +data (Efron & Tibshirani, 1993) to generate replicates +D1, D2 . . . , DB, which simulates drawing different training +datasets from a distribution. In fair classification, to evaluate +expected error it is common to use 0-1 loss ≜ 1[ˆy ̸= o] or +cost-sensitive loss, which assigns asymmetric costs for false +positives, FP, and false negatives, FN (Agarwal et al. (2018); +Elkan (2001), Appendix B.2.1). Common fair classification +definitions, such as Equality of Opportunity (Hardt et al., +2016), further analyze error by computing disparities across +group-specific error rates FPR and FNR. +3. Variance and Self-Consistency +FPR and FNR represent only one way to decompose er- +ror. Additionally, we can analyze error’s different statistical +sources — its constituent noise, bias, and variance (Abu- +Mostafa et al., 2012; Geman et al., 1992). Noise and bias +depend on the Bayes optimal classifier, which we typically +do not have access to in practice (Appendix B.1). In con- +trast, since variance just depends on the underlying training +data, we can estimate it empirically. Following our intu- +ition from Figure 1, we can bootstrap the training dataset +to simulate drawing from a distribution (Efron, 1979), use +these bootstrap replicates to train multiple models, and then +compare the resulting models’ predictions for the same test +example to see how confident the learning process is with +respect to classifying the example. +Defining variance. We therefore begin to formalize our +analysis of arbitrariness by considering variance. Formally, +Definition 3.1. Given the distribution over possible trained +models µ and loss ℓ, the variance for fresh instance (x, g) is +Var +� +A, D, (x, g) +� +≜ EhDi∼µ,hDj ∼µ +� +ℓ +� +hDi(x), hDj(x) +�� += +1 +n(n − 1) +� +i̸=j +ℓ +� +hDi(x), hDj(x) +� +. +This definition uses the loss to compare all possible model +predictions to each other for the same test example, and + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Non-white (NW) +White (W) +(a) COMPAS split by g = race ∈ {NW, W} +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female (F) +Male (M) +(b) Old Adult split by g = sex ∈ {F, M} +Figure 2. Cumulative proportion of test instances that attain the given level of self-consistency. We train random forest classifiers to +estimate ˆ +SC with 101 bootstrap replicates, and repeat with 10 train/test splits to produce confidence intervals. +Table 1. Mean ± STD in error rates for the experiments in Figure 2. See Appendix E for more detailed results. +COMPAS +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.366 ± .005 +.173 ± .008 +.193 ± .007 +.73 ± .003 +g = NW +.369 ± .005 +.18 ± .007 +.19 ± .008 +.732 ± .004 +g = W +.359 ± .013 +.16 ± .012 +.199 ± .011 +.727 ± .008 +Old Adult +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.173 ± .003 +.077 ± .003 +.096 ± .001 +.878 ± .002 +g = F +.09 ± .003 +.037 ± .001 +.053 ± .003 +.938 ± .002 +g = M +.212 ± .003 +.097 ± .003 +.116 ± .001 +.85 ± .002 +computes the average over these pairwise comparisons. +For 0-1 or cost-sensitive loss, we can make the following +simplifying observation. +We denote ˆY the multiset of +predictions for models hD1, hD2, . . . , hDn on (x, g), with +| ˆY | = n = α + β > 1, and α and β representing the counts +of 0 and 1 predictions, respectively. It follows that +Var +� +A, D, (x, g) +� += (C01 + C10)αβ +n(n − 1) +, +(1) +with C01, C10 ∈ R+ denoting the FP- and FN-associated +costs, respectively, which we can relate directly to the +decision threshold τ used in A (Appendix B.2). Using the +bootstrap method (Efron, 1979; Efron & Tibshirani, 1993), +we can train a concrete number of models n and compute +an approximation of Definition 3.1, +ˆ +Var. +We can now formally support the claim that average error +rates like FPR and FNR (and their empirical counterparts, +ˆ +FPR and +ˆ +FNR), on which popular fairness definitions de- +pend, do not expose variance (Hardt et al., 2016; Jiang et al., +2020; Calders et al., 2009; Pleiss et al., 2017). This is clear +because, even when averaging over multiple models n, FPR +and FNR compare classifications ˆy = hDk to the observed +labels o; they do not compare the different ˆy to each other. +Defining and illustrating self-consistency. By consider- +ing costs C01 and C10, variance encodes a measure of mag- +nitude. However, magnitude is not especially meaningful +for our purposes; it is the relative, not absolute, cost of +C01 and C10 that define the classification decision thresh- +old τ (Section 2). In order to focus unambiguously on +the (dis)agreement part of variance, we define a notion of +self-consistency: +Definition 3.2. Given the distribution over possible trained +models µ, the self-consistency for fresh instance (x, g) is +SC +� +A, D, (x, g) +� +≜ PhDi∼µ,hDj ∼µ{hDi(x) = hDj(x)} += +1 +n(n − 1) +� +i̸=j +1[hDi(x) = hDj(x)]. +In words, SC models the probability that two models +produced by the same learning process, on different training +data subsets, agree on their predictions for the same test +example. In practice, we can compute +SC +� +A, D, (x, g) +� += 1 − +2αβ +n(n − 1), +(2) +where this equivalence follows from writing Definition 3.2 +in terms of Definition 3.1 for 0-1 loss SC is defined +on [0.5, 1], with 0.5 representing minimal and 1 repre- +senting maximal self-consistency. We choose to define +self-consistency on this range so that its measure coheres +with the intuition in Figure 1, with Individuals 1 and 2 +exhibiting 100% and 50% self-consistency, respectively +(Appendix C). As with variance, we can compute empirical +approximations of self-consistency, +ˆ +SC, with larger n +corresponding to higher quality approximations. +Beyond estimating self-consistency for an individual test +example, we can also do so across the entire test set and with +respect to subgroup membership. We provide illustrative +examples from two of the most common fair classification +benchmarks, COMPAS and Old Adult (Fabris et al., +2022). In Figure 2, we plot the distribution of ˆ +SC over the +test set: The y-axis shows the cumulative proportion of the +test set that has attained the x-specified level of ˆ +SC (defined +on [0.5, 1]). To obtain these results, we split the available +data into train and test sets, and bootstrap the train set 100 +times to train different models. We repeat this process on +10 train/test splits, and the resulting confidence intervals +(inset) indicate that our ˆ +SC estimates are stable. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Figure 2 further illuminates the importance of self- +consistency with respect to its relationship to arbitrariness +and discrimination. For one, both sub-figures show that +self-consistency varies drastically across test examples, un- +derscoring that classification decisions are far from equally +confident. In COMPAS, about one-half of test examples are +under 70% self-consistent; nearly one-quarter are effectively +50% self-consistent, meaning they resemble Individual 2 in +Figure 1 and thus their predictions are essentially arbitrary. +These differences in confidence persist despite the fact that +the 100 models plotted in Figure 2a exhibit relatively small +disparity between subgroup +ˆ +Err, +ˆ +FPR, and +ˆ +FNR (Table 1, +left). In short, it is possible to come close to satisfying some +fair classification metrics, even if the underlying classifica- +tions carry very different levels of confidence (Section 5). +The plot for Old Adult (Figure 2b) shows that self- +consistency can also differ according to subgroup-g mem- +bership. It is possible for the degree of arbitrariness to be +systematically worse for a particular demographic g. While +the lack of self-consistency is not as extreme as it is for +COMPAS — the majority of test examples for Old Adult +exhibit over 90% ˆ +SC — it is unequally distributed on the +Male subgroup. Given the relationship between variance +(Definition 3.1) and self-consistency (Definition 3.2), these +differences in subgroup-conditional self-consistency are +likely responsible for at least some of the observed discrep- +ancies in subgroup-conditional error rates (Table 1, right). +Of course, noise and bias likely also contribute to unfairness. +Due to these other sources of error, it is possible for models +to produce consistent predictions, but for those predictions +to be consistently wrong (with respect to observed label +alignment i.e., ˆy ̸= o). There are test examples for both +Algorithm 1 (Super-) Ensembling with Confidence +input training dataset (X, O)train, training procedure A, ensemble +size η, ˆ +SC threshold κ ∈ [0.5, 1], test instance xtest +output Prediction ˆy with ˆ +SC ≥ κ or Abstain +1: function BagConfidently +� +(X, O)train, A, η, κ, xtest +� +2: +ˆyA := list() // To store ensemble predictions +3: +for 1 . . . η do +4: +Dη ← Bootstrap +� +(X, O)train +� +5: +hDη ← A(Dη) //A may Bag using Dη +6: +ˆyA.add +� +hDη(xtest) +� +//ˆy1, . . . , ˆyη +7: +end for +8: +return Aggregate(ˆyA, κ) +9: end function +10: // An aggregate function that meets our framework’s semantics +11: function Aggregate +� +ˆy1, . . . , ˆyη, κ +� +12: +if SelfConsistency(ˆy1, . . . , ˆyη) ≥ κ then +13: +return arg maxy′∈Y +� �η +i=1 1[y′ = ˆyi] +� +14: +end if +15: +return Abstain +16: end function +plots in Figure 2 for which all 100 predictions ˆy ̸= o (Ap- +pendix E.3). Even in these cases, measuring ˆ +SC can still +highlight useful information. For example, if models tend to +be consistently wrong for examples in a particular subgroup, +it is worth considering that there may be label bias. That is, +the observed label available in the dataset may be tainted by +past unfairness in human decision processes, and thus not +reflective of desired learning outcomes (Cooper & Abrams, +2021; Wick et al., 2019; Jiang & Nachum, 2020). We defer +this line of investigation to future work. +4. Accounting for Self-Consistency +In the remainder of this work, we study the impact of im- +proving self-consistency in fair classification tasks. Our +goal is to reduce arbitrariness induced by a lack of self- +consistency in classifying individual examples, and to see +if doing so can also reduce the systematic differences in ar- +bitrariness that may exist across subgroups (Figure 2b). At +first glance, bootstrap aggregation (bagging) seems a promis- +ing candidate, as it has remained one of the most successful +variance reduction algorithms for decades (Breiman, 1996). +However, it does not naturally handle the arbitrariness prob- +lem we describe in Section 3. The bagging aggregation rule +picks the majority-vote classification, and therefore embeds +the notion that predicting slightly-better-than-random is a +sufficient tie-breaking strategy. +Instead, we suggest a simple framework that modifies bag- +ging to account for self-consistency (Algorithm 1). The key +difference is to add a user-selected, minimally-acceptable +level of self-consistency κ ∈ [0.5, 1] to threshold the test +examples being classified, with higher κ instilling a more +significant degree of confidence. For examples that fail to +achieve ˆ +SC that is at least κ, Algorithm 1 opts to Abstain +rather than produce a ˆy ∈ [0, 1]. The easiest way to imple- +ment the semantics of Algorithm 1 is to modify traditional +bagging with the example Aggregate function shown in +lines 11-16,1 Algorithm 1 can also be instantiated to use a +super-ensembling strategy. In this case, rather than training +an ensemble of η single models, we can train and ensemble +η bagged classifiers to reduce variance (line 5), and then +aggregate their outputs to account for ˆ +SC-level κ. +Measuring performance. To validate our approach, we +need to confirm that it improves self-consistency. And to +examine its impact on fairness, we need to measure changes +in the subgroup-conditional error rates. To measure self- +consistency, we adapt Definition 3.2 to account for absten- +tions. We define abstentions to agree with both 0 and 1 +predictions. This makes sense intuitively: Algorithm 1 ab- +stains to avoid making predictions that lack self-consistency, +so abstaining should not increase disagreement between +1We also experiment with averaging the outputs of regressors +r and applying threshold τ to produce predictions. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +Figure 3. Alg. 1, RFs, Old Adult, g = {Female, Male}. +Table 2. Mean ± STD across 10 train/test splits. +Ensembling random forests with confidence +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.155 ± .004 +.023 ± .001 +.06 ± .002 +.228 ± .004 +g = F +.048 ± .002 +.007 ± .001 +.035 ± .001 +.112 ± .003 +g = M +.219 ± .005 +.032 ± .002 +.076 ± .003 +.284 ± .007 +Super-ensembling random forests with confidence +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.204 ± .004 +.05 ± .002 +.088 ± .001 +.043 ± .001 +g = F +.077 ± .003 +.017 ± .002 +.049 ± .002 +.02 ± .003 +g = M +.267 ± .005 +.067 ± .003 +.107 ± .003 +.054 ± .002 +predictions. It follows that we can continue to use Defi- +nition 3.2, but with one small adjustment. Instead of the +total number of predictions n = α + β, with α and β cor- +responding to 0 and 1 predictions, respectively, we now +allow for n ≥ α + β, in order to account for possibly some +nonzero number of abstentions. It is easy to show that any +algorithm that meets the semantics of this framework will +have improved self-consistency (Appendix D.2). Of course, +this means that we could achieve perfect self-consistency +by always abstaining. It is therefore important to confirm +empirically that our super-ensembling algorithm does not +abstain too frequently, which necessarily depends on the +datasets examined in practice. +A remark on cost. It can be considerably more computa- +tionally intensive to train an ensemble of models to com- +pute ˆ +SC than to train a handful of models and perform +cross-validation, as is the standard practice in fair classi- +fication. However, as our empirical analysis in the next +section demonstrates, this cost comes with a huge bene- +fit: It enables us to improve self-consistency and to root +out the arbitrariness of producing predictions that are ef- +fectively close-to-random, which is especially important in +high-stakes fairness settings (Cooper et al., 2021a). More- +over, for common fair classification datasets, the increased +cost on modern hardware is relatively small; (super-) en- +sembling with confidence takes under an hour to execute +(Appendix E.5). +5. Experiments and Discussion +We provide extensive empirical results in two areas: We +validate the effectiveness of both ensembling and super- +ensembling variants of Algorithm 1, and we explain how our +results reveal key insights about the reliability of popular fair +classification benchmarks. For all experiments that follow, +we illustrate Algorithm 1 with κ = .75, but note that, in +practice, appropriate choice of κ is task-dependent. +Code. We build and release an extensible software suite +(https://github.com/pasta41/variance) +of different Aggregate methods, which we apply to +common fair classification datasets and models: COMPAS, +Old Adult, +German and Taiwan Credit, +and +3 +large-scale +New Adult - CA +tasks +on +logistic +regression (LR), decision trees (DTs), random forests +(RFs), multi-layer perceptrons (MLPs) and support vector +machines (SVMs). We consulted several well-cited prior +fair classification studies to inform our choice of models and +hyperparameter optimization search spaces (Appendix E). +We also examine the NY and TX 2017 subsets of the the +Home Mortgage Data Disclosure Act (HMDA) 2007-2017 +dataset (Federal Financial Institutions Examination Council, +2017), which have 244,107 and 576,978 examples, respec- +tively. These datasets are less commonly used in current +fairness research (Fabris et al., 2022), possibly because +the over-100-million data examples are only available in +bulk files. Following the example of Ding et al. (2021), one +of our contributions is to pre-process these datasets — all +locations and years — and release them with a standalone +software package that makes them easy to explore. Our +hope is that effortless access to HMDA will further reduce +the community’s dependency on small datasets.2 +Validating Algorithm 1. Algorithm 1 is naturally paralleliz- +able, so we designed our code for a batch processing cluster +environment. This enabled us to train and compare several +million different models over the course of our study. Unfor- +tunately, we necessarily defer most results to Appendix E.5, +and present a representative portion of results from different +datasets and models, beginning with Old Adult in Fig- +ure 3 and Table 2. We visualize Algorithm 1 by plotting +the self-consistency of the underlying models that we bag +with confidence κ = 0.75 (delineated as a dashed dark blue +line). We simultaneously plot the results of ensembling +individual random forests with confidence (the faded set of +curves), and super-ensembling bags of random forests with +confidence (the darker set of curves). We show our results +in terms of the ˆ +SC of the underlying bagged models because +2See https://github.com/pasta41/hmda for the lat- +est information on this PyPI package, which aligns with the terms +of service for HMDA. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +HL - before +NHL - before +HL - after +NHL - after +Figure 4. Alg. 1, DTs, HMDA-TX, g = {Hisp./Lat., Not Hisp./Lat.} +Table 3. Mean ± STD across 5 train/test splits. +Baseline: Decision tree classifiers +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.776 ± .001 +.083 ± .0 +.082 ± .001 +— +g = HL +.72 ± .002 +.101 ± .001 +.091 ± .001 +— +g = NHL +.794 ± .0 +.077 ± .0 +.079 ± .0 +— +Super-ensembling decision trees with confidence +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.832 ± .001 +.03 ± .001 +.039 ± .0 +.18 ± .001 +g = HL +.767 ± .001 +.033 ± .001 +.05 ± .0 +.21 ± .0 +g = NHL +.852 ± .001 +.029 ± .001 +.035 ± .0 +.17 ± .001 +doing so conveys how Algorithm 1 makes decisions to pre- +dict or abstain: 3 For both types of ensembling, Algorithm 1 +predicts for all examples captured by the area to the right of +the κ reference line, and abstains for all examples on the left. +By comparing the shaded regions between each set of curves, +it is clear that super-ensembling Old Adult improves +overall self-consistency and brings subgroup-conditional +self-consistency closer together. We can assess this more +formally by measuring the distance between each pair of +curves — before and after the internal round of bagging at +Algorithm 1, line 5 — and computing their difference. To +do so, we use the Wasserstein-1 distance (Appendix E.5.1), +which is the natural choice because it has a simple closed +form for CDFs. For the two subgroups, we can call their +respective ˆ +SC CDF curves A and B (with associated proba- +bility measures a, b), and compute +W1(a, b) = +� +R +|A(κ) − B(κ)| dκ, +(3) +which measures the summed differences at all values κ. +Subtracting (3) for the after curves from (3) for the before +curves yields a positive difference of .101. +Beyond ˆ +SC, we also show ˆ +PR, +ˆ +FPR, +ˆ +FNR, and the abstention +rate ˆ +AR in Table 2. We rely on Table 1 (right) in Section 3 +to serve as the baseline expected error of individual ran- +dom forests, and highlight the performance of ensembling +random forests (top) and super-ensembling bags of random +forests (bottom) with κ = .75. In addition to improving +ˆ +SC, both ensembling approaches improve error rates for +both subgroups, and also decrease the disparity in the rates +between subgroups: For Old Adult, our results are both +fairer and more accurate, according to common fairness def- +inition (Hardt et al., 2016). For example, in Table 1, g = F +has +ˆ +FPR = .037 and g = M has +ˆ +FPR = .097, for a disparity +of .06; both instantiations of Algorithm 1 improve upon this +disparity as a byproduct of improving self-consistency. +It is important to note that, for ensembling with confidence, +3Additionally, the ˆ +SC distribution of Algorithm 1 — computed +by doing a third round of ensembling — has nearly all of its mass +at ˆ +SC = 1, which makes it difficult to visualize (Appendix E.5). +we report corrected error rates and ˆ +PR that account for ab- +stentions; we compute these rates only in terms of cast 0 +and 1 votes, and separately report the abstention rate ˆ +AR in +relation to the total number of possible votes. Table 2 re- +veals some key points concerning ˆ +AR. ˆ +AR is unequal across +subgroups, with Algorithm 1 abstaining more frequently +for g = M. This makes sense in relation to our conceptual +contribution: The learning process is less confident about +how to predict examples with membership in g = M and, +rather than predicting arbitrarily, Algorithm 1 opts not to +predict at all. It is also useful to abstain from the perspective +of error and the impact on fairness metrics. We can see +this by analyzing the error for the classifications traditional +bagging would have made on the examples for which Algo- +rithm 1 abstains. For Old Adult, the average total +ˆ +Err +of examples with κ < .75 is close to 40%, as compared to +17% for random forests (Table 1), and 8% for ensembling +and 14% for super-ensembling with confidence, respectively +— with a significantly larger proportion of that 40% error +being ascribed to members of g = M (Appendix E.5). By +abstaining, Algorithm 1 also serves to identify examples that +could be investigated in more detail for other interventions. +To represent a large-scale task, we also highlight analo- +gous results for HMDA-2017-TX using decision trees in +Figure 4 and Table 3. In comparison to more traditional +smaller tasks, our results for larger datasets generally demon- +strate lower variance, and thus better ˆ +SC, (Appendix E). As +with Old Adult, Algorithm 1 significantly improves self- +consistency, and has a smaller (but appreciable) improve- +ment in subgroup differences in ˆ +SC, with the W1 distance +difference yielding .022. In Table 3, we show the baseline +results for decision trees (top) and super-ensembling with +confidence (bottom), and defer the associated results and +discussion for ensembling with confidence to the Appendix. +Both these results for HMDA and for Old Adult indicate +another important aspect of the abstention rate. For the +class of methods that meet the semantics of Algorithm 1, +there is a trade-off between ˆ +AR and error. Super-ensembling +involves an inner loop of traditional bagging for variance + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +NW - before +W - before +NW - after +W - after +Figure 5. Alg. 1, LR, COMPAS, g = {Non-white, White}. +Table 4. Mean ± STD across 10 train/test splits. +Baseline: Logistic regression models +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.403 ± .011 +.139 ± .011 +.191 ± .01 +— +g = NW +.453 ± .012 +.147 ± .013 +.183 ± .011 +— +g = W +.308 ± .015 +.126 ± .013 +.207 ± .011 +— +Super-ensembling with confidence +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.389 ± .006 +.123 ± .006 +.19 ± .015 +.041 ± .003 +g = NW +.442 ± .007 +.129 ± .008 +.18 ± .013 +.043 ± .005 +g = W +.286 ± .006 +.111 ± .006 +.208 ± .021 +.038 ± .005 +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +Figure 6. Alg. 1, DTs, German Credit, g = {Female, Male}. +Table 5. Mean ± STD across 50 train/test splits. +Baseline: Decision tree classifiers +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.711 ± .018 +.147 ± .024 +.17 ± .018 +— +g = F +.712 ± .042 +.145 ± .059 +.177 ± .047 +— +g = M +.711 ± .019 +.148 ± .023 +.169 ± .019 +— +Super-ensembling with confidence +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.847 ± .024 +.148 ± .03 +.054 ± .017 +.175 ± .03 +g = F +.879 ± .06 +.152 ± .07 +.052 ± .051 +.199 ± .059 +g = M +.842 ± .023 +.148 ± .031 +.054 ± .015 +.171 ± .033 +reduction that improves ˆ +SC before applying a confidence +threshold, as is clear when comparing the area between the +different sets of curves and the change in W1 distance. This +increase in ˆ +SC naturally leads to a decrease in the abstention +rate when we perform the outer loop to bag with confidence. +For Old Adult, ensembling with κ = .75 yields total ˆ +AR +of 22.8%, while super-ensembling yields 4.3% (Table 2); +HMDA-TX yields 35% and 18% ˆ +AR, respectively (Table 3, +Appendix E.5). However, this reduction in ˆ +AR comes at +the cost of predicting incorrectly more often; the super- +ensembled classifier has a larger number of examples that +are more self-consistent and wrong (i.e., ˆy ̸= o). This +can also have the effect of increasing the disparity between +subgroup error rates, if improving ˆ +SC benefits subgroups +asymmetrically. This occurs with +ˆ +FNR HMDA-TX (Table 3), +with the relative +ˆ +FNR disparity increasing from 1.2% to +1.5%. Even so, both subgroups exhibit more substantial +absolute +ˆ +FNR improvements after super-ensembling. +Reliability of common fairness benchmarks. +To close +out the results in the main paper, we emphasize some sur- +prising results that came about for two of the three most +common fair classification tasks – COMPAS and German +Credit (Fabris et al., 2022). Both exhibit a significant +lack of self-consistency, but it is more equally distributed +across subgroups than in the results above for Old Adult +and HMDA.4 This observation holds even for higher bias +4COMPAS W1 difference is .003; German Credit is .007. +models like logistic regression (Figure 5), which gener- +ally exhibit much higher ˆ +SC in comparison to other mod- +els, especially those trained on much larger datasets (Ap- +pendix E.4.6). This effect is particularly pronounced for +German Credit, which, for supervised fair classifica- +tion, has only 670 examples. The confidence intervals for +ˆ +SC measurements are more scattered, so we have to average +over substantially more train/test splits to produce them reli- +ably (Figure 6, Appendix E.4.2). For both datasets, the lack +of self-consistency leads to very high abstention rates — for +some model types, over 50% — when performing ensem- +bling with confidence on individual models (Appendix E.5). +Furthermore, in estimating the expected average error — an +artifact of computing ˆ +SC over 100 models — we find that av- +erage expected subgroup error rates for both of these tasks +are quite similar. COMPAS exhibits a couple of percent- +age points of disparity (Table 4), while South German +Credit subgroup rates are statistically equivalent (Ta- +ble 5). Notably, this is true even for the +ˆ +FPR baseline for +COMPAS, which lowers to 1.8% after super-ensembling (Ta- +ble 4). Put differently, by training and averaging over many +models, we produce a better estimate of the expected model +than can be achieved by training a small handful of models +and performing cross-validation; and, in doing so, our esti- +mates indicate both close-to-parity in common fairness met- +rics (Hardt et al., 2016) and, for COMPAS, lower overall ex- +pected error than the typically-reported 35% +ˆ +Err (Lin et al., + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +2020). We examine this result in detail in Appendix E.6. +The key takeaway has to do with variance. The underlying, +individual models that compose the expected error and con- +stitute our ensembles exhibit radically different subgroup- +specific error rates, with a skew toward slightly higher +ˆ +FPR +for g = NW. As a result, while any individual model drawn +from the distribution of possible models is more likely to +exhibit +ˆ +FPR unfairness for examples in g = NW, aggregat- +ing many models can have the effect of reducing the overall +magnitude of individual model disparities. +Nevertheless, the conclusion to draw from our results is +not that COMPAS and German Credit are close-to-fair. +Rather, the takeaway is that using standard fair classification +modeling approaches leads to a large amount of variance. +In turn, it is unlikely that reliable conclusions can be drawn +about satisfying fairness metrics from the typical practice of +training and cross-validating small sets of models. In this re- +spect, our work supports an observation that has been made +by much prior research. While it is possible to produce more +reliable estimates of error from bootstrapping and ensem- +bling, it is likely more prudent to test fair classification in- +terventions on larger datasets (Chen et al., 2018; Ding et al., +2021; Cooper & Abrams, 2021). Future scholarship should +transition to using datasets like New Adult, made easily +usable by Ding et al. (2021), and HMDA, which we package +for greater accessibility as a part of the present work. +6. Related Work on Variance and Fairness +While we have discussed related work throughout this paper, +we now provide additional discussion specifically on the +relatively small handful of papers on fair classification that +contend specifically with variance. One influential prior +work, Chen et al. (2018), adopted the definition of variance +from Domingos (2000a), which has since been taken up by +(Black & Fredrikson, 2021; Black et al., 2022a). This work +elects to rely on this variance definition because, for binary +classification and 0-1 loss, Domingos (2000a) permits a +formal decomposition of expected error into its constituent +noise, bias, and variance. However, this definition presents +some limitations. Notably, it depends on a notion of a +“main prediction,” which we show can be quite brittle in +high-variance tasks (like those we study here), and it also +does not cleanly extend to cost-sensitive loss in practice +(Appendix B.3). These drawbacks encouraged us to use +what we believe is a more natural definition of variance, +from which we are able to neatly derive the notion of self- +consistency that is the focus of our study (Section 3). +While our definition forgoes the decomposition that Domin- +gos (2000a;b) affords, our work does not require such a +decomposition for the results that we develop. Notably, the +prior work on fair classification that leverages Domingos +(2000a) does not directly employ the decomposition, either. +Further, Chen et al. (2018) does not measure variance or +self-consistency empirically, as we do in our work. Their +experiments on Old Adult alter training dataset size as +a proxy for understanding variance-induced error, rather +than measuring disagreement between predictions on the +same test example and its implications for fairness and +arbitrariness. +In contrast, Black et al. (2022a) studies +variance directly, and independently also develops an +ensembling-based strategy to contend with it. However, +their results are directly tied to estimating the “main +prediction” (Domingos, 2000a), and thus are fundamentally +different from our work, which is free from the issues +presented by this definition (Appendix B.3). +More distant related work studies variance in deep learning +on fair classification (Qian et al., 2021; Forde et al., 2021). +Studying variance in deep learning settings presents distinct +challenges that we do not address here. +In particular, +non-determinism introduced from using GPUs makes it +difficult to study a fixed learning process, for which the only +source of non-determinism originates from the stochasticity +of resampling the training dataset (Cooper et al., 2022a). +7. Conclusion and Future Work +Prior work on fair classification has focused overwhelm- +ingly on relative subgroup error rates, treating such +disparities as unfair because they amount to a form of dis- +crimination. We instead focus on a lack of self-consistency +in predictions, explaining why this, too, can be a important +type of unfairness, due to the arbitrariness that it introduces +into ML-based decision-making. We further show that +these phenomena can interact: Certain subgroups may be +subject to more arbitrary decisions than others. +To contend with arbitrariness, we develop an intuitive +ensembling-based approach that helps to improve self- +consistency. In contrast to traditional bagging, however, +our approach requires some minimum level of agreement in +model predictions, rather than relying on a simple-majority +vote, since anything slightly better than chance is rarely +enough to address concerns with arbitrariness. While we +find empirically that this approach reduces arbitrariness, and +can even narrow the gap in error rates across subgroups, our +method may abstain on a nontrivial fraction of predictions. +These results imply that there may be limits to how much +we can improve self-consistency—and that there may be +times when it is simply inappropriate to make predictions +given the remaining arbitrariness. +Altogether, our work suggests three directions for future +research. First, what other methods can we develop to im- +prove self-consistency? Second, what strategies should we +rely on when it is not possible to improve self-consistency? +Third, what can we learn by studying examples that models +predict incorrectly with a high degree of self-consistency? + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Acknowledgements +Much of this work was done as part of an internship at +Microsoft Research, FATE Lab, New York. A. 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The organization +is as follows: +• Appendix A: Extended Preliminaries +– A.1: Notes on notation on our choice of terminology +– A.2: Measuring error +* A.2.1: Estimating error in practice with the bootstrap +– A.3: Constraints on our setup +• Appendix B: Additional Details on Variance +– B.1: Types of statistical error +* B.1.1: Irreducible error +* B.1.2: Reducible error +– B.2: Our variance definition +* B.2.1: Measuring variance for different choices of costs +– B.3: Related work on alternative notions of variance +– B.4: Why we choose to avoid computing the main prediction +* B.4.1: The main prediction and cost-sensitive loss +– B.5: Defining variance: The main takeaways from above +• Appendix C: Additional Details on Self-Consistency +– C.1: Deriving the SC +� +A, D, (x, g) +� +metric +– C.2: Cost-independence of self-consistency +• Appendix D: Additional Details on Our Algorithmic Framework +– D.1: Super-ensembling for self-consistency +– D.2: Proof of improved self-consistency +• Appendix E: Additional Empirical Results and Details for Reproducibility +– E.1: Hypothesis classes, datasets, and code +* E.1.1: The standalone HMDA tookit +– E.2: Cluster environment details +– E.3: Details on motivating examples in the main paper +– E.4: Additional illustrative results +* E.4.1: COMPAS and Old Adult +* E.4.2: South German Credit +* E.4.3: Taiwan Credit +* E.4.4: New Adult - CA +* E.4.5: HMDA +* E.4.6: Extended discussion of illustrative examples of self-consistency +– E.5: Validating our algorithm in practice +* E.5.1: Measuring the distance between empirical self-consistency curves +* E.5.2: COMPAS +* E.5.3: Old Adult +* E.5.4: South German Credit +* E.5.5: Taiwan Credit +* E.5.6: New Adult - CA +* E.5.7: HMDA +* E.5.8: Discussion of extended results for Algorithm 1 +– E.6: Reliability and fairness metrics in COMPAS and South German Credit + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +A. Extended Preliminaries +A.1. Notes on notation on our choice of terminology +Sets, random variables, and instances. We use capital letters to denote random variables, curly capital letters to denote +sets, and lower case letters to denote specific elements. For example, for features, x is an arbitrary feature vector, X is the +set representing the space of all features x (x ∈ X), and X is the random variable that can take on specific values of x ∈ X. +We use this notation consistently, and thus do not always define all symbols explicitly (e.g., y and Y are defined explicitly, +but we do not define Y explicitly). +Observed labels. Traditionally, what we term “observed labels” (Section 2) is often referred to instead as the “ground +truth”, “true”, or “correct” labels (Domingos, 2000a;b; Abu-Mostafa et al., 2012; Murphy, 2022; Kong & Dietterich, 1995; +Hastie et al., 2009). In some domains, there is a valid notion of a “true” label, such as classifying if an image shows a cat +or a dog. In algorithmic fairness domains, the contestability of what a “true” label is becomes less straight-forward, and +can elude quantification. As a result, in the present work, when we refer to “true” labels, it is for cases in which there is a +generally accepted, reasonable ground truth (as in the cats/dogs example). In those cases, we will represent the label as t, +and t ̸= o would indicate that the observed label is an instance of data example mislabeling or noise, as ideally t = o. We +deliberately alter this terminology to avoid conflating this conception of truth (i.e., the label a data point has in a dataset) +with “objective” truth (i.e., what the label of the data point should be, regardless of its recorded value in a dataset), as the +two may not always be equivalent. That is, an observed label does not necessarily connote anything about ground truth, +particularly in learning problems involving social processes. +A.2. Measuring error +In the main paper, we elide a formal definition of the learning process and the training procedure A, which we provide here: +Definition A.1. A training procedure A is a randomized function that uses training data D to produce a deterministic +classifier hD. That is, A(D) �→ hD, where hD is the distribution over possible models. We denote this distribution µ. A +learning process runs a training procedure A, and can optionally contain other steps, such as hyperparameter optimization +over different instantiations of A. +We can consider specific runs of A on concrete D = Dk, which produce concrete hDk, for which we consider hDk ∼ µ. +We restrict the sources of randomness to that provided by the underlying training data. We therefore do not consider learning +processes that include steps like hyperparameter optimization. +During the execution of A on a concrete dataset Dk, we compute the loss in order to minimize the error on the training +dataset — the in-sample error. This minimization problem uses a well-behaved loss function, such as logistic loss, in order +to make optimization feasible. When estimating the expected performance of a particular trained model hDk in practice, we +generate predictions for a test set of previously un-seen, fresh example instances supplied to hDk. These instances should +be drawn from the same distribution p(·) as those drawn for training. This test loss, or out-of-sample error, is intended +to give a sense of how well a trained model will generalize to new examples that reflect the same distribution — to see +if the learning process has generated a model that over-fits to the training data. For classification, it is common to use 0-1 +loss or cost-sensitive loss to measure out-of-sample error. These loss functions do not have well-defined derivatives at +all values, which is why they are not used during training. Logistic loss, mentioned above, is a good stand-in. Moreover, in +practice, since we do not have access to the process for generating data examples, we typically estimate out-of-sample-error +on a set-aside portion of the dataset we have available to us (Abu-Mostafa et al., 2012; Murphy, 2022). +We formally define test error in terms of expectations; we reason about test error as if we have access to all possible datasets +in D (i.e., reasoning about the expected behavior of the random variable D, which can take on any concrete Dk of size n). +We want to reason about the behavior of hD — the aggregate, expected behavior as if we had generated and tested each +concrete hDk ∼ µ on an arbitrary (x, g). We choose to reason about error this way because, due to stochasticity in the +learning process, a training procedure A produces models that behave differently as a result of the particular dataset Dk on +which they are trained, and thus exhibit different losses for the same example (x, g, o). +Analytically, we can reason about the effect of different Dk on the resulting loss by averaging over the loss for all Dk ∈ D. +That is, basing our definitions on Kong & Dietterich (1995), we define the expected error (sometimes called just the error, +expected test error, expected total loss, or empirical risk) on an example instance x as + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Definition A.2. The expected error for fresh example instance (x, g) is +Error +� +A, D, O, (x, g) +� += ED,O[ℓ(o, ˆy)|X = x, G = g]. +This expectation is taken with respect to all n-sized training datasets, i.e. the set D, and all possible labels o, i.e. the set O, +and depends on the choice of training procedure A. +In words, the expected error is a weighted average of the loss; the weights correspond to the probability of sampling each con- +crete training dataset D = Dk and the probability of each possible label O = o, given (x, g). Informally, we can think of this +as the training procedure A being applied to each Dk ∈ D to produce corresponding model hDk; each respective prediction, +ˆy = hDk(x), can thus be produced to compute the loss in relation to the probability of each o, which is weighted by (x, g). +It is immediately clear from above that the expected error depends not just on the data distribution p(·) from which we draw +datasets to train; there is a formal dependence on the choice of training procedure A and the size of the training datasets n. +Even if it is not immediately quantifiable how much A and n impact error, it is nevertheless the case that the selection of +both will impact resulting error measurements in practice. +A.2.1. ESTIMATING ERROR IN PRACTICE WITH THE BOOTSTRAP +We of course do not have access to the data distribution p(·), from which we can sample fresh examples (x, g, o). We +typically have one dataset for a task that we can use for training. As a result, to estimate the error (Definition A.2), we can +reserve a random portion of the dataset to not be included in training (the test set), and evaluate the error of a trained model +using this set. The hope is that, since these examples should reflect the same distribution as the data on which we have +trained but have no affected the learning process, the test set should give us a less-biased estimate of the model’s error in +practice. +To simulate drawing different datasets from the sample of the distribution of data that we have available, we use the bootstrap +method (Efron & Tibshirani, 1993; Efron, 1979) to produce different, concrete {Dk}i +k=1, i ∈ N. More specifically, we +select a random seed to determine a train/test split. We reserve the test set; for the resulting training set x = {xi, gi, oi}m +i=1, +we generate B ∈ N bootstrap replicates (using B fresh random seeds) {x†i}B +i=1, where each x†i = {x† +j, g† +j, o† +j}m +j=1 by +sampling x with replacement. Random sampling with replacement enables guaranteeing independence, i.e. all m examples +are i.i.d. According to Efron & Tibshirani (1993, p. 13), it is useful to pick B ∈ [50, 200] for standard error estimation. +An overall estimate of the prediction error can be obtained by averaging over those B models’ predictions. For binary +classification using 0-1 loss, we can discretize this by taking the majority classification. This is traditionally what is done in +ensembling via bagging classifiers (Breiman, 1996). +In our formulation, which we implement empirically using bootstrapping, each dataset Dk is equally likely to be drawn +from the D. Moreover, we assume that we do not have access to sufficient data to approximate the conditional probability +distributions each o given different X, G, and thus treat the given label as fixed for the example (x, g, o) (akin to this +probability being set to 1). As a result, for a test example instance (x, g) with observed label o, we can approximate +Definition A.2 as follows: +˜ +Error +� +A, {Dk}B +k=1, (x, g, o) +� += 1 +B +B +� +k=1 +ℓ(o, hDk(x)) +(4) +Of course, this approximation will improve for larger i. We can also compute the overall error for all test examples by addition- +ally summing over each example instance (x, g, o), or for specific sub-groups by computing separate sums for the different g. +A.3. Constraints on our setup +Our setup is deliberately limited to studying the effects of variance due to changes in the underlying training dataset, with +such datasets drawn from the same distribution. Of course, variance can induce error from other sources in the training +pipeline. For example, stochastic optimization methods, such as SGD and Adam, can cause fluctuations in test error; as, +too, can choices in hyperparameter configurations (Cooper et al., 2021b). While each of these decision points is worthy of +investigation with respect to their impact on fair classification outcomes, we aim to fix as many sources of randomness as + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +possible in order to highlight the particular kind of arbitrariness that we describe in Sections 1 and 3. As such, we use the +Limited-memory BFGS solver and fix our hyperparameters based on the results of an initial search, for which we selected a +search space through consulting related work such as Chen et al. (2018). +B. Additional Details on Variance +In this Appendix, we provide additional details on variance, the type of statistical error with which we are concerned in this +work. To start, we extend our discussion of statistical error in Appendix A, clarifying how it traditionally gets decomposed +(Appendix B.1) into a combination of irreducible noise (Appendix B.1.2), and reducible bias and variance (Appendix B.1.1). +We then provide more details on our definition of variance in Section 3 (Appendix B.2) and an extended discussion of a +popular alternative definition used in prior work on fair classification, which depends on a notion of a main prediction B.3. +Presenting our definition and this prior work side-by-side enables us to explain in detail the mathematical reasons why we +choose not to use the main prediction (Appendix B.4), which have to do with computing variance for cost-sensitive loss +(which we detail in Appendix B.4.1) and issues of brittleness (Appendix B.4) for the main prediction. At the end of this +section, we briefly summarize the main takeaways (Appendix B.5). +B.1. Types of statistical error +Expected error provides an overall sense of the statistical error for an arbitrary example. This total picture does not disclose +underlying sources of error; it does not ascribe how much of the expected error is due to the irreducible noisiness of the data +or to our choices in setting up a learning process — the choice of training procedure A or the size n of the datasets D. +To do this analysis, it is useful to think about the minimal possible error that any machine-learned model using any learning +process could possibly produce. This is because the error of any model that we produce via a specific learning process is +lower-bounded by this minimal possible error. Typically, the model that provides this lower bound is called the Bayes optimal +predictor f. Informally, f can be understood as the function that provides the optimal prediction y∗ for each example +instance (x, g), i.e., y∗ = f(x).5. Irrespective of training data Dk or training procedure A, y∗ is the prediction that minimizes +the expected loss over all possible label values y ∈ Y (Domingos, 2000a;b; Abu-Mostafa et al., 2012).6 More formally, +Definition B.1. For (x, g), possible labels y ∈ Y, and loss function ℓ, the optimal prediction is defined as +y∗ = arg min +y′ +EY [ℓ(y, y′)|X = x]. +The optimal predictor f is the function that produces the optimal prediction y∗ for each (x, g), i.e., +OptimalPredictor(x, g) = y∗ = f(x) = arg min +y′ +� +i∈Y +P(Y = i|X = x) ℓ(i, y′) +In words, this says that we take a possible label y′ and compute the loss in comparison to each possible label i ∈ Y; we +weight the loss by the probability of the label i for the given features x. We sum these weighted losses for each possible label, +and the y′ that has the smallest weighted sum of losses is the optimal prediction for (x, g). Importantly, Definition B.1 makes +especially clear that the optimal predictor is not guaranteed to be error-free. It also makes clear (as alluded to informally +before Definition B.1) that the optimal predictor (and thereby its predictions) does not depend on our training decisions: f +indicates the best model that could possibly be produced in theory, and is therefore necessarily independent of the concrete +choices we make in practice when picking a specific training procedure to produce a model that we hope will reflect this +lowest possible error. +However, Definition B.1 also makes especially clear that the optimal prediction does depend on the the label space Y and the +conditional distribution over these labels given our examples (x, g). So, baked into this definition are the choices we have +made when modeling the data examples and labels, as these will of course impact this distribution. As noted in Domingos +5The Bayes optimal predictor in our setup deliberately does not consider g. Example instances with identical features but different +subgroup membership have the same Bayes optimal prediction. Of course, considering g could improve the Bayes error rate. +6This means that, for a given (x, g), y∗ may not exist in the observed labels O; it may be in Y \ O. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +(2000a), for classification contexts f is called the Bayes optimal classifier and its loss is the Bayes error rate. Lastly, except +in simple cases, since we typically do not have access to the true conditional distribution P(Y |X), we usually cannot +compute the optimal predictor in practice. +Decomposing error. A well-known result uses a notion of the optimal prediction to decompose error into its constituent +noise, bias, and variance for squared loss (Geman et al., 1992). This loss function is of course common in regression +settings, but not in classification. In the 1990s, there were several papers that debated how to come up with an analogous +result for 0-1 loss to fill this gap for classification contexts (e.g., Kohavi & Wolpert (1996); Domingos (2000a); Kong & +Dietterich (1995)). The key challenge, arguably, was to define an effective reference point for describing the expected +behavior of a model that uses a particular learning process. Similar to the optimal predictor (Definition B.1), which serves as +a reference point for the lowest-possible-error model, this reference point would ideally be useful for showing how different +models trained on different training sets in D deviate from the expected behavior of all such models. +The work by Domingos (2000a), in contrast to prior work, accounts for the role of noise in the decomposition. It relies +heavily on the notion of a “majority vote” predictor, adopted from Kong & Dietterich (1995) and renamed as the “main +predictor.” Given this importance of classification and the use of 0-1 loss in analyzing statistical definitions of fairness, +Domingos (2000a) has occasionally been taken up by work on algorithmic fairness (Chen et al., 2018; Black et al., 2022a). +In contrast to this prior work, we choose not to use this definition due to mathematical and conceptual issues that we take +with the definition of the main predictor. We discuss these issues briefly in the paper, and go into significantly greater detail +here. It is these issues that motivate us to deviate from prior work on fair classification that uses the main prediction, and to +provide our own definition of variance (Definition 3.1). +B.1.1. IRREDUCIBLE ERROR +Noise. The label o that we observe for x may not necessarily be the same as the optimal prediction y∗. Put differently, o, the +label we observe for (x, g) when drawing (x, g, o) ∼ p(·), may incur a larger error than if x had a different label. Drawing on +an example from Cooper & Abrams (2021) of pictures of cats and dogs, it is possible that a Pomeranian accidentally (though +perhaps understandably) gets mislabeled as a cat; the label of “dog” would incur a lower error than the observed label “cat.” +To further clarify this point, we can think of (x, o) being “noisy” in practice. An intuitive description of this noisiness is as +follows: It is not unreasonable to imagine drawing two identical feature vectors x from a distribution that have different +observed labels. Of course, identity between feature vectors is not required for there to be statistical noisiness in the observed +labels. Identical or similar x, which in algorithmic fairness often are compared for similarity across differences in protected +attributes — can have different observed labels. +In the case of algorithmic fairness domains, this difference can be due to g. Sometimes, differences due to g may be helpful, +as in the case of disease diagnosis, for which genetic etiology might be correlated with sex or gender. In other instances, this +is not the case. Differences can be due to inequity due to systemic discrimination that affected the past decisions in the +datasets that have been collected (e.g., past loan denials based on red-lining. Importantly (albeit somewhat confusingly in +relation to our discussion of noise), in algorithmic fairness this is commonly referred to as label bias, where “bias” is used +to connote discrimination as opposed to statistical bias. While label bias is not the same as statistical noise, it is possible the +it can have a similar effect on data distributions that ignore protected attributes g — giving different labels to examples x +that, separate from g are virtually indistinguishable when quantified for encoding as features. +It is important to note that noise has an effect on statistical error before we ever attempt to fit a specific model to data. It +does not depend on our choice of training procedure A (and, by extension, also not our choice of model class H) or specific +training datasets D. In other words, noise is irreducible because it is due to inherent randomness in the mapping of X to Y; +it constitutes the aleatoric or data uncertainty in the task at hand (Murphy, 2022). Moreover, because we do not typically +have access to the optimal predictor, it is difficult (if not intractable) to estimate noise accurately in practice, except in certain +simple cases. Nevertheless, heteroskedastic noise across subgroups is often hypothesized to be a source of unfairness in +machine learning (Cooper & Abrams, 2021; Chen et al., 2018; Fogliato et al., 2020; Chen et al., 2018). +As with expected error, we can extending thinking about noise on a per-example basis to being measured with respect to +being averaged over all possible example instances drawn from p(·) or according to group g. That is, we can reason about +noise being potentially different in expectation according to different g — i.e., noise being heteroskedastic when we consider +it for groups of examples conditioned on G = g. If this is the case, since noise is irreducible, it may not be possible to +achieve equal error rates among different subgroups g using machine learning. For a different framing of this observation in + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +relation to a “level of discrimination,” refer to Chen et al. (2018). +B.1.2. REDUCIBLE ERROR +While noise-induced error is irreducible, independent of the choices we make at training time, the remaining components +of statistical error are not. They are theoretically reducible and contingent on the learning process — the training procedure +A (and thus the model class H) and the training datasets D (and their chosen size n). Such reducible error is said to +contribute to epistemic uncertainty. This is distinguished from aleotoric uncertainty by the fact that it comes from our lack of +knowledge of the mapping from X to Y (Murphy, 2022), which in theory could be reduced if our learning process captured +greater knowledge about the mapping. The two types of error that reflect this lack of knowledge are bias and variance. +Bias. Informally, bias measures how much the models produced by a particular learning process deviate from the best- +possible machine-learned model for the task at hand. Slightly more formally, it is traditionally measured as the difference +between the behavior of the expected model (in the statistical expectation sense) produced by the learning process and the +optimal predictor (Definition B.1). The expected model is a function of all the possible models hDk produced by the learning +process, and is therefore dependent on A (and therefore choice of H). As a result, bias can be understood as the persistent +error of a specific learning process (Schapire & Freund, 2012).7 It is the same for all models hDk ∈ H produced by the +same such process, as all hDk produced by A share the same expected model and the optimal predictor f is independent of +the learning process. +There are multiple ways to mathematically formalize a notion of the expected model, which is why we have deliberately +left this discussion informal. For regression using squared loss, it is the average. For 0-1 loss, this definition has +involved considerable debate (which we discuss briefly in Appendix B.3). Nevertheless, it is worth noting that, like noise, +measurements of bias tend to depend on the optimal predictor, and thus we similarly cannot estimate it directly in practice. +In contrast to irreducible noise, this error can be thought of as reducible because we can theoretically employ or discover a +different learning process that better captures the patterns in the underlying data, reflected in the production of an expected +model that is more aligned with the optimal predictor. Such a learning process would reflect greater knowledge about the +task being modeled, reducing epistemic uncertainty. We similarly can think of bias of a single example instance, across all +example instances, or according to subgroup membership g. +Much literature in algorithmic fairness talks more generally about modeling biases, which can be related to but are not +necessarily the same as statistical bias. For example much prior work discusses how problem formulations and abstractions, +such as the particular choice of hypothesis class, can bias learning outcomes to favor particular groups (Selbst et al., 2019; +Passi & Barocas, 2019), or can bias away from more meaningful notions of fairness, justice, or equity (Cooper & Abrams, +2021; Jacobs & Wallach, 2021; Green, 2020). These ideas can be reasonably argued to date back to more-distant work +on the normative dimensions of bias in science and statistics. For example, Popper’s work significantly predates work on +inductive and modeling biases in machine learning. In summarizing work by Kant that critiques Hume, Popper (1962) says: +...[W]e may consider the idea of building an induction machine. Placed in a simplified ‘world’ (for example, one of sequences +of coloured counters) such a machine may through repetition ‘learn’, or even ‘formulate’, laws of succession which hold in its +‘world’. If such a machine can be constructed (and I have no doubt that it can) then, it might be argued, my theory must +be wrong; for if a machine is capable of performing inductions on the basis of repetition, there can be no logical reasons +preventing us from doing the same. +The argument sounds convincing, but it is mistaken. In constructing an induction machine we, the architects of the machine, +must decide a priori what constitutes its ‘world’; what things are to be taken as similar or equal; and what kind of ‘laws’ +we wish the machine to be able to ‘discover’ in its ‘world’. In other words we must build into the machine a framework +determining what is relevant or interesting in its world: the machine will have its ‘inborn’ selection principles. The +problems of similarity will have been solved for it by its makers who thus have interpreted the ‘world’ for the machine. +(Internal footnotes and citations omitted, pp. 47-48, bold emphasis added Popper (1962)) +About 20 years later, we believe that Mitchell (1980) puts machine learning in conversation with Popper (1962). He +acknowledges the inherent difficulty of biases: they bring about error, but they are also necessary for learning useful +generalizations. This reality requires that the uses of specific modeling biases be justified. In a similar vein, but in relation to +computing more broadly, the now-classic work of Friedman & Nissenbaum (1996) distinguishes between different types of +bias and how they can function in concert to cause injustice. Of particular note in this respect is how “technical bias” in +7See p. 117 in Schapire & Freund (2012). + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +modeling decisions can cause other types of bias to emerge when it fails to capture social phenomena (where modeling can +be statistical or non-statistical): +In contrast to preexisting bias, technical bias arises from the resolution of issues in the technical design. Sources of technical +bias can be found in several aspects of the design process, including limitations of computer tools such as hardware, software, +and peripherals; the process of ascribing social meaning to algorithms developed out of context; imperfections in pseudorandom +number generation; and the attempt to make human constructs amenable to computers, when we quantify the qualitative, +discretize the continuous, or formalize the nonformal. (p. 335, Friedman & Nissenbaum (1996)) +Bringing together all of this prior work — connecting older work on inductive bias to more recent observations about +fairness and to statistical bias — is an interesting area for future work. Unfortunately, a comprehensive analysis of the +normative valences of different types of bias in relation to the biases in machine learning is out of the scope of our present +work. We believe it would be a rich contribution to put these earlier works in conversation with ideas in algorithmic fairness, +e.g., work like Friedler et al. (2016).8 We instead use the above excerpts and citations just to point out that it is clearly not a +new idea that people necessarily encode inductive biases into models. It is arguably an idea that finds renewed interest and +vigor during different moments in technological development. +Variance. In contrast to bias, variance captures the error that comes from fluctuations in different models trained on +different examples. This is akin to comparing the outputs of different hDk to each other and measuring how much those +comparisons deviate from each other. Importantly, we can empirically estimate variance directly. This is because, unlike +statistical noise and bias, it does not depend on the Bayes optimal label y∗, which we do not tend to have available in +practice. The comparisons between models can be measured in various ways; one such way is to compare each concrete +hDk to the expected model produced by the learning process (Appendix B.3). In our work, discussed in the main paper +and below (Appendix B.2), we choose to directly compare predictions ˆy to each other without an embedded notion of an +expected model. +One of our main contributions is to relate variance to a notion of self-consistency of the learning process (Appendix C), +which we argue captures arbitrariness in the learning process (Section 3). In our work, we focus on pinning down a clear +notion of self-consistency and analyzing it in the context of fair classification tasks. In the future, we hope to complement +our computer science work with more normative precision concerning the relationship between variance, self-consistency, +and arbitrariness. Some prior work has examined variance in the context of fair ML, but does not address or connect these +ideas. We discuss this work in Section 6. +B.2. Our variance definition +As discussed in Section 2 and briefly above, we measure variance by directly comparing concrete predictions to each +other. For a given task, we train multiple models hDk on different training datasets of the same task (via bootstrapping, see +Appendix A.2), and compare the different ˆy produced for a fresh (x, g). This results in Definition 3.1, restated below: +Definition 3.1. Given the distribution over possible trained models µ and loss ℓ, the variance for fresh instance (x, g) is +Var +� +A, D, (x, g) +� +≜ EhDi∼µ,hDj ∼µ +� +ℓ +� +hDi(x), hDj(x) +�� += +1 +n(n − 1) +� +i̸=j +ℓ +� +hDi(x), hDj(x) +� +. +The above equivalence follows from the fact that, in our setup, drawing any two datasets Di, Dj ∈ D is equally likely. It +follows that drawing any hDi, hDj ∼ µ is equally likely. As a result, we can just compute the expectation as the average +over the sum of the losses. +As we note in the main text, for binary classification using 0-1 loss or cost-sensitive loss, we can equivalently re-write +Definition 1 as +8There has been some work that has started to tease apart related ideas, e.g., Danks & London (2017); Mehrabi et al. (2021). + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +(C01 + C10)αβ +n(n − 1) +, +(1) +where C01, C10 ∈ R+ and, for 0-1 loss, C01 = C10 = 1. We clarify this equivalence as follows: +Proof. For the models {hDk}n +k=1 that we produce, we denote ˆY to be the multiset of their predictions on (x, g). | ˆY | = n = +α + β, where α and β represent the counts of 0 and 1-predictions, respectively. We also set the cost of false positives to be +ℓ(0, 1) = C01 and the cost of false negatives to be ℓ(1, 0) = C10. +Looking at the sum in Definition 3.1 (i.e., � +i̸=j), each of the α 0-predictions will get compared to the other α − 1 +0-predictions and to the β 1-predictions. By the definition of ℓ, each of the α − 1 computations of ℓ(0, 0) evaluate to 0 and +each of the β computations of ℓ(0, 1) evaluate to C01. Therefore, the 0-predictions contribute +α × +�� +0 × (α − 1) +� ++ C01 × β +� += C01αβ +to the sum in Definition 1. By similar reasoning, each of the β 1-predictions will get compared to the other β−1 1-predictions +and to the α 0-predictions. By the definition of ℓ, each of the β − 1 computations of ℓ(1, 1) evaluate to 0 and each of the α +computations of ℓ(1, 0) evaluate to C10. Therefore, the 1-predictions contribute +β × +�� +0 × (β − 1) +� ++ C10 × α +� += C10αβ. +It follows that the total sum is +� +i̸=j +ℓ +� +hDi(x), hDj(x) +� += (C01 + C10)αβ. +Therefore, +Definition 1 +� +�� +� +1 +n(n − 1) +� +i̸=j +ℓ +� +hDi(x), hDj(x) +� += +(1) +� +�� +� +(C01 + C10)αβ +n(n − 1) +B.2.1. MEASURING VARIANCE FOR DIFFERENT CHOICES OF COSTS +Recall that, in our setup (Section 2), we learn models hDk(x) = 1[rDk(x) ≥ τ], where rDk is an underlying regressor +that approximates the probabilities P[Y = 1|X], to which we apply decision threshold τ to output discrete classification +decisions ˆY ∈ {0, 1}. We can directly relate the costs C01 and C10 in (1) to τ. We can therefore set up our learning +problem to produce rDk, pick a τ such that it reflects specific chosen costs C01 and C10, which we can then use to estimate +ˆ +Var +� +A, D, (x, g) +� +empirically for 0-1 loss and cost-sensitive loss. We explain these relationships in this Appendix. This +discussion also proves useful for explaining our choices for defining self-consistency SC +� +A, D, (x, g) +� +(Definition 3.2) to +be cost-independent, upon which we elaborate in Appendix C below. +The relationship between cost-sensitive loss and decision threshold τ. 0-1 loss assigns all misclassifications the same +cost (i.e., 1). We can visualize this with the following confusion matrix: +This matrix shows how misclassifications are composed of what are traditionally called “false positives” (FP) and “false +negatives” (FP); similarly, accuracy often measures the sum of “true positives” and “true negatives.”9 This view makes +9We use this terminology because it is standard in classification literature. However, we refer the reader to our note in Appendix A.1 +concerning how o is an observed label, versus reflective of ground truth, as these traditional terms would suggest. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Table 6. Confusion matrix for 0-1 loss, where o is the observed label and ˆy is the predicted label. +ˆy = 0 ˆy = 1 +o = 0 TN: 0 FP: 1 +o = 1 FN: 1 TP: 0 +especially clear that 0-1 loss treats the cost of different types of errors equally; false positives and false negatives are +quantified as equivalently bad – they are symmetric. +Of course, depending on the learning application context, this may be a bad modeling assumption. It may instead make +sense to model these costs as (potentially substantially) different. For example, when diagnosing a serious illness, for +which we will treat 1 as indicating the disease and 0 its absence, a false negative can certainly be more costly than a false +positive. In such cases, it may be better to use a loss that models asymmetric costs. The following confusion matrix, adapted +from Elkan (2001), illustrates how the differing costs for different types of error in binary classification can be framed +more generally. Rather than the confusion matrix for 0-1 loss in Table 6, we can account for different costs for different +outputs by assigning each classification decision combination a function-conditional value, i.e., ℓ : Y × Y �→ R, which +explicitly assigns costs to different input pairs. +Table 7. Confusion matrix for loss ℓ. +ˆy = 0 +ˆy = 1 +o = 0 TN: 0 +FP: C01 +o = 1 FN: C10 TP: 0 +This enables us to model C01 ̸= C10. Specifically, for the prior example concerning diagnosing an illness, C10 > C01. Based +on the above, we can therefore think of 0-1 loss as a special case of cost-sensitive loss. +Changing the decision threshold. Altering the asymmetric of costs shifts the classification decision threshold τ applied to +the underlying regressor r. We can see this by examining the behavior of r that we learn. r estimates the probability of a +each label Y = y given x (since we do not learn using g), i.e., that we devleop a good approximation of the distribution +P(Y |X). Ideally, r will be similar to the Bayes optimal model (Definition B.1). Recall that the Bayes optimal model’s rule +is to produce predictions y∗ that yield the smallest weighted sum of the loss, where the weights are the probabilities of a +particular label Y = i for a given (x, g). For binary classification, the loss in the weighted sum in Definition B.1 will always +either be 0 or 1, and the possible labels i and predictions y′ will also be 0 or 1. Thus, when computing the weighted sum +above for a particular y′, there will be possible cases to consider for the different terms in the term we are summing over, +P(Y = i|X = x) ℓ(i, y′). +(5) +There are two different cases to consider: +• i = y: By definition, ℓ(i, y′) = 0; therefore, (5) = 0. +• i ̸= y: By definition, ℓ(i, y′) = C01 or ℓ(i, y′) = C10, depending on the values of i and y′. So, (5) will weight the cost +by the probability P(Y = i|X = x) for the example instance (x, g). +For our fair binary classification setup, the total probability of Y given (x, g) is +P(Y |X = x) = P(Y = 1|X = x) + P(Y = 0|X = x) = 1. +(6) +We can therefore break down the Bayes optimal predictor from Definition B.1 into the following decision rule, which we +hope to approximate through learning. For an arbitrary (x, g) and Y = {0, 1}, we compute + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +min +� +Weighted cost of predicting positive (1) class +� +�� +� +Probability of FP +� +�� +� +P(Y = 0|X = x) ×C01 + +Probability of TP +� +�� +� +P(Y = 1|X = x) ×0, +Weighted cost of predicting negative (0) class +� +�� +� +Probability of TN +� +�� +� +P(Y = 0|X = x) ×0 + +Probability of FN +� +�� +� +P(Y = 1|X = x) ×C10 +� += min +� +Probability of FP +� +�� +� +P(Y = 0|X = x) ×C10, +Probability of FN +� +�� +� +P(Y = 1|X = x) ×C10 +� +That is, to predict label 1, the cost of mis-predicting 1 (i.e., the cost of a false positive) must be be smaller than the cost of mis- +predicting 0 (i.e, the cost of a false negative). By (6), we can assign P(Y = 1|X = x) = τ and P(Y = 0|X = x) = 1 − τ. +and rewrite the above as +min +� +(1 − τ)C01, τC10 +� +. +(7) +We can observe that the decision boundary is the case for which both of the arguments to min in (7) are equivalent (i.e., the +costs of predicting a false positive and a false negative are equal), such that we cannot distinguish between which is minimal. +We can therefore rewrite (7) for this case and solve for τ: +(1 − τ)C01 = τC10 +τ = +C01 +C01 + C10 +, +and write the binary classification decision rule as +f(x) = +� +1, +if P(Y = 1|X = x) ≥ τ = +C01 +C01+C10 +0, +otherwise. +(8) +For 0-1 loss, in which C01 = C10 = 1, τ evaluates to 1 +2. A threshold of 1 +2 intuitively makes sense for 0-1 loss; since +false positives and false negatives are equally costly, cost effectively does not weight the decision threshold in a particular +direction. f just evaluates which label is more likely and, since there are only two labels, that just means picking the label +that appears more than half the time in the distribution of Y given x. If we want to model asymmetric costs, then we need +to change this decision threshold. For example, let us say that false negatives are more costly than false positives, with +C01 = 1 and C10 = 3. This leads to a threshold of 1 +4, which biases f toward choosing the (generally cheaper to predict/ +more conservative) positive class. This also makes sense intuitively. In our example of the diagnosing a disease, missing +disease diagnoses (false negatives) is more expensive than diagnosing a disease that is not actually present (false positive), +so we bias toward predicting the presence of the disease. +The effect on variance. As discussed above, changing the costs C01 and C10 change the decision threshold τ applied to +rDk to produce classifier hDk. Costs greater than 1 can clearly change the magnitude of the computed variance in (1); the +minimum variance is still 0, (when either α or β is 0), but it is possible for the maximum to change, depending on the value +of C01 + C10. In more detail, let us examine the behavior of maximal and minimal variance as n → ∞, i.e., +lim +n→∞ +(C01 + C10)αβ +n(n − 1) +. +(9) +Recall that α + β = n. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +• Minimal variance. In this case, either α or β will be 0, with the other being n, making (9) +lim +n→∞ +(C01 + C10) × 0 +n(n − 1) += 0, +regardless of the value of C01 + C10. +• Maximal variance. In this case, α will represent half of n, with β representing the other half. More particularly, +α = n +2 and β = n +2 ; or, without loss of generality, α = n−1 +2 +and β = n+1 +2 . This means that +(C01 + C10)αβ +n(n − 1) += (C01 + C10)( n +2 )2 +n(n − 1) +� +Or = (C01 + C10)( n−1 +2 )( n+1 +2 ) +n(n − 1) +� += (C01 + C10)( n2 +4 ) +n2 − n +� +Or = (C01 + C10)( (n2−1 +4 +) +n(n − 1) +; it will not matter when we take the limit +� += (C01 + C10)n2 +4n2 − 4n +. +And, therefore, +lim +n→∞ +(C01 + C10)n2 +4n2 − 4n += C01 + C10 +4 +. +(10) +It follows that, analytically, Var +� +A, D, (x, g) +� +�→ [0, C01+C10 +4 +]; any intermediate amount of variance between the +minimal and maximal variance will fall on this interval. Of course, empirically for concrete n, +ˆ +Var +� +A, D, (x, g) +� +→ +[0, C01+C10 +4 ++ ϵ], for small positive ϵ as the number of models n increases; the maximal variance will better approximate +C01+C10 +4 +as n gets larger, but will be > C01+C10 +4 +. For example, for 0-1 loss C01+C10 +4 += 2 +4 = 0.5. For n = 100, the +maximal +ˆ +Var +� +A, D, (x, g) +� += 2×50×50 +100×99 = 50 +99 ≈ .505. +B.3. Related work on alternative notions of variance +As noted in Section 2, prior work that discusses variance and fair classification often relies on the definition of variance from +Domingos (2000a). We deviate from prior work and provide our own definition. The reasons for this choice are twofold +— mathematical and conceptual. Here, we discuss the mathematical reasons in details, which we elide in the main text +for brevity. There are two overarching mathematical reasons for this choice: 1) variance in Domingos (2000a;b) does not +cleanly extend to cost-sensitive loss, discussed in Appendix B.4.1, and 2) the reference point for measuring variance in +Domingos (2000a;b) — the main prediction — can be unstable/ brittle in practice. To make these points clear, we first +provide necessary background from Domingos (2000a;b), which we then use to highlight the benefits of our definition of +variance used in Section 2 and detailed in Appendix B. We also use this discussion to further clarify the conceptual reasons, +which we treat briefly in Section 3. +To begin, we restate the definitions from Domingos (2000a;b) concerning the expected model (called the main predictor), +informally discussed in Appendix B.1.2. We change the notation and provide additional intuition for these definitions to +provide what we believe to be greater clarity concerning their meaning, significance, and consequent takeaways. Nevertheless, +these definitions for quantifying error are equivalent to those in Domingos (2000b), and they fundamentally depend on +human decisions for setting up the learning process. +Defining variance in relation to the main prediction. Domingos (2000a;b) define predictive variance in relation to +a point of reference. At a high level, this strategy is similar to how this work defines both noise and bias in relation to +the optimal predictor — the theoretically best (in terms of loss minimization) model. The optimal predictor provides a +mechanism for reasoning about the lower bound on the possible error — the best one can do using machine learning — for +a specified learning task. In contrast, to account for error introduced by the chosen learning process (which is independent +from the error of the theoretical-best-case model trained by any such process) Domingos (2000a;b) introduce a point of +reference that captures the general, expected behavior of models that could be produced by the chosen learning process. +This reference, notably, has nothing to do with how optimal this behavior is; it is something that can be used to compare +produced models to each other, irrespective of whether those models are solving the learning problem well. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +For clarity, we elaborate on the discussion provided in Domingos (2000a;b), and begin by supplying an informal intuition of +how to capture this notion of expected behavior. We can imagine training all possible hypotheses using datasets of the same +size n drawn from the distribution p(·), i.e., all hD. Then, for a given example instance (x, g), we compute and collect all +the predictions made by those classifiers, i.e., hD1(x), hD2(x) . . . hDk(x) into a multiset ˆY . A natural question is to ask +how these predictions in ˆY compare to each other in terms of the loss ℓ. For example, we can ask if there is a (potentially +hypothetical) prediction in Y (the whole label space) that, in comparison to all the concrete predictions in ˆY , has the smallest +average loss. As Domingos (2000b) notes, we can think of such a prediction as being the one that “differs the least” from +the predictions we have collected into ˆY , in terms of the loss. We can think of this prediction as the “central tendency” of +predictions made by all hD on x. It is this central tendency prediction that Domingos (2000a) calls the main prediction. +More formally, +Definition B.2. The main prediction y is the prediction value y′ ∈ Y that generates the minimum average loss with respect +to all of the predictions ˆy ∈ ˆY generated by the different hDk. It is defined as the expectation over training sets D for a loss +function ℓ, given an example instance (x, g). That is, +y = arg min +y′ +ED[ℓ(ˆy, y′)|X = x, G = g]. +(11) +The main predictor h : X �→ Y is the function that produces the main prediction y for each (x, g). +Note that it is possible for the same h to result for the same x with different g, since we do not use group-membership +directly in h. We do, however, condition on g in our definition of the main prediction to clarify how we distinguish groups +for fairness analysis at test time. This is necessary for us to perform group-specific variance analysis. This of course requires +a slight notation change from Domingos (2000a;b) in our presentation of the main prediction, as the original paper is not +geared toward fairness. That is, we split features into x and g; we can equivalently consider both together as x to align with +the definition in the original paper. +What the expected value in Equation (11) evaluates to in practice of course depends on the loss function ℓ. For squared loss, +the main prediction is defined as the mean of all the hDk(x) (Domingos, 2000a; Kong & Dietterich, 1995). This makes sense +intuitively: The mean necessarily “differs least” from the values that compose it. Of course, as a mean, this may reflect a value +in Y that is not in the produced in the prediction multiset ˆY . In various domains, such as classification — and particularly +binary classification in algorithmic fairness — 0-1 loss is a common loss function to evaluate at test. For this setting, we +represent these two classes symbolically by the set {0, 1}. In algorithmic fairness decision contexts, 0 represents the negative +class and 1 represents the positive class. Following Kong & Dietterich (1995), for 0-1 loss Domingos (2000a) defines the +main prediction as the mode/ majority vote — the most frequent prediction for an example instance (x, g). This makes sense +intuitively. For 0-1 loss, there are only two possible labels in Y that we need to consider. As a result, ˆY will contain some +number of 0s and some number of 1s, and nothing else. We can think of the most frequent prediction (either 0 or 1) being in +more comparisons to itself when comparing the ˆy ∈ ˆY to y′ ∈ Y; these comparisons (computed via 0-1 loss) will evaluate +to 0, weighting the average more toward the 0, minimizing the expectation in Equation (11) for the majority prediction. +We provide a more formal discussion below when we discuss the main prediction for cost-sensitive loss (Appendix B.4). +Following from the above, Domingos (2000a;b) defines variance in relation to specific models hDk and the main predictor +h. That is, for a specific (x, g), it is possible to compare the individual predictions ˆy = hDk(x) to the central tendency of +all such models — the main prediction y = hDk(x). Using the main prediction as a reference point, one can compute the +extent of disagreement of individual predictions with the main prediction as a source of error. Formally, we reframe this +definition of variance from Domingos (2000a;b) as +Definition B.3. The variance-induced error for fresh example instance (x, g) is +Var +� +A, D, (x, g) +� += ED[ℓ(y, ˆy)|X = x, G = g], +where y = h(x) is the main prediction (Definition B.2) and the ˆy are the predictions for the different hDk, i.e., ˆy = hDk(x). +The expectation for variance-induced error is taken with respect to all n-sized training datasets, i.e. the set D and depends +on the choice of training procedure A. +In practice, we can think of this as the training procedure A being applied to each Dk ∈ D to produce corresponding models +hDk; each respective prediction, ˆy = hDk(x), can thus be produced to compute the loss in relation to the main prediction +y. The average of that loss, weighted by the probability with respect to drawing each training dataset, lets us compute the +expected variance-induced error concerning predictions for (x, g) over the different possible hDk. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Note on prior work. It is this definition (Definition B.3) that prior work on fair classification tends to reference when +discussing variance (Chen et al., 2018; Black et al., 2022a). However, as we discuss in more detail below (Appendix B.4), +many of the theoretical results in Chen et al. (2018) follow directly from the definitions in Domingos (2000a), and the +empirical work does not actually use those results in practice. Black et al. (2022a), in contrast, presents results that rely +heavily on the main prediction in Domingos (2000a). +Note on bias. In short, as discussed above, Domingos (2000a;b) relies on the main predictor as a way to capture the +behavior of the expected model produced by a learning process (discussed informally in Appendix B.1.2). As a result, this is +the definition used to measure bias in that work; it compares the optimal predictor f to the main predictor h. +B.4. Why we choose to avoid computing the main prediction +There are some noteworthy similarities between to our definition of variance (Definition 3.1, Appendix B.2) and that defined +in Domingos (2000a;b) (Definition B.3, Appendix B.3). First, in practice we can approximate the Domingos (2000a) +definition using a concrete number of models. Second, we can also similarly compute variance not just on a per-example +basis, but across examples by averaging across different example instances (x, g). Lastly, this definition does not explicitly +depend on its observed label o. As a result, the definition of variance does not convey information in relation to either the +Bayes optimal prediction or to the label we have in the dataset for the example. +There are, however, notable differences, which on the whole we believe demonstrate the benefits of our definition: +• No need to compute a “central tendency”: In Domingos (2000a;b), variance is defined in terms of both the loss +function ℓ and the main prediction y. This assumes that the main prediction is well-defined for the loss function, and +that it is well-behaved. While there is a simple interpretation of the main prediction for squared loss (the mean) and for +0-1 loss (the mode/ majority vote), it is significantly messier for cost-sensitive loss, which is a more general formulation +that includes 0-1 loss. Domingos (2000a;b) does not discuss this explicitly, so we derive the main prediction for +cost-sensitive loss ourselves (We defer these details to Appendix B.4). Showing the behavior of the main prediction for +cost-sensitive loss reveals that the decomposition result provided in the extended technical report (Theorem 4, Domingos +(2000b)) of the shorter conference paper (Domingos, 2000a) is in fact very carefully constructed to accommodate the +behavior of the main prediction. We believe that this construction is so specific that it is not practically useful (it is, in +our opinion, hardly “unified” in a more general sense, as it is so carefully adapted to specific loss functions and their +behavioral special cases). By decoupling from the need to compute a main prediction as a reference point, our variance +definition is ultimately much simpler and more general, with respect to how it accommodates different loss functions.10 +• No decomposition result: Following from above, it is worth noting that by not relying on the main prediction, we do +lose the applicability of the decomposition result that Domingos (2000a;b) develop. However, we believe that this is +fine for our purposes, as we are interested in the impact of variance specifically on fair classification outcomes; we +do not need to reason about bias or noise analytically in our results. It is also worth noting that prior work on fair +classification that leverages Domingos (2000a) also does not leverage the decomposition, either. Chen et al. (2018) +extends the decomposition to subgroups in the context of algorithmic fairness,11 and then informally translates the +takeaways of the Domingos (2000a) result to a notion of a “level of discrimination.” And, moreover, that work does not +actually measure variance directly in its experiments. +• Brittleness of the main prediction: In contrast, Black et al. (2022a) relies heavily on the main prediction. This work +develops a process for determining whether it is possible to reliably compute the main prediction and, for cases in +which this is not possible, abstains from predicting. As a result, this work does get at the idea that the main prediction +can be brittle, especially for examples that are high variance. We address problems with this brittleness differently. In +brief, our reasoning is as follows: For high variance instances, the main prediction can flip-flop from ˆy = 1 to ˆy = 0 +10This observation actually reveals a subtle ambiguity in the definition of the loss function ℓ in Domingos (2000a;b). Neither paper +explicitly defines the signature of ℓ; looking at the definitions of the main prediction (Definition B.2) and variance (definition B.3), there is +a lack of clarity in what constitutes a valid domain for ℓ. Computing the main prediction y suggests ℓ : ˆY × Y �→ R≥0, where y ∈ Y , +but, since ˆY ⊆ Y , it is possible that y ̸∈ ˆY . However, the definition of variance suggests that ℓ : Y × ˆY �→ R≥0. Since ˆY ⊆ Y , it is not +guaranteed that ˆY = Y . This may be fine in practice, especially for squared loss and 0-1 loss (the losses with which Domingos (2000a) +explicitly contends, but it does arguably present a problem formally with respect to generalizing. +11This just involves splitting the conditioning on an example instance of features x into conditioning on an example instance whose +features are split into (x, g). + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +and back. While the strategy in Black et al. (2022a) is to abstain on the prediction in these cases, we believe that a +better alternative is to understand that the main prediction is not super meaningful more generally for high-variance +examples. That is, for these examples, the ability (and reliability) of breaking close ties to determine the main (simply +majority) prediction is not the right approach. Instead, we should ideally be able to embed more confidence into +our process than simple-majority-vote determination.12 Put different, in cases for which we can reliably estimate +the main prediction, but the vote margin is slim, we believe that the main prediction is still uncertain, based on our +understanding of variance, intuited in Figure 1. The main prediction can be reliable, but it can still, in this view, be +arbitrary. With a simple-majority voting scheme, there can be huge differences between predictions that are mostly (or +completely) in agreement, and those that are just over the majority reference point. Freeing ourselves of this reference +point and encoding uncertainty in our self-consistency metric (Section 3, Appendix C), we can define thresholds of +self-consistency as our criterion for abstention (where simple-majority voting is one instantiation of that criterion).13 +B.4.1. THE MAIN PREDICTION AND COST-SENSITIVE LOSS +The main prediction, and thus variance, is more complicated for cost-sensitive (i.e., asymmetric) loss. In short, sometimes +the main prediction is the majority vote, but sometimes it is the minority vote, depending on how asymmetric the costs are. +Domingos (2000a) does not discuss this, but work done in an extended technical report mentions the case of asymmetric +loss (Domingos, 2000b). However, this extended work, which provides an error decomposition in Theorem 4, does not +explain the effects on the main prediction. We do so below, and also call attention to 0-1 loss as a special case of cost-sensitive +loss, for which the costs are symmetric (and equal to 1). +Proof. Let us consider cost-sensitive loss for binary classification, for which ℓ(0, 0) = ℓ(1, 1) = 0 and we have potentially- +asymmetric loss for misclassifications, i.e. ℓ(1, 0) = a and ℓ(0, 1) = b, with a, b ∈ R+. 0-1 loss is a special case for this +type of loss (Elkan, 2001; Agarwal et al., 2018), for which a = b = 1. +Let us say that the total number of models trained is k, which we evaluate on an example instance x. Let us set | ˆY | = k = +2w + 2n + 1, with w ≥ 0 and n ≥ 0. We can think of w as the common number of votes that each class has, and 2n + 1 as +the margin between the two classes. Given this setup, this means that k ≥ 1, i.e., we always have the predictions of at least 1 +model to consider, and k is always odd. This means that there is always a strict majority classification. +Without loss of generality, on input x let us say that, of these k model predictions ˆy ∈ ˆY , there are w class-0 predictions +and w + 2n + 1 class-1 predictions. That is, without loss of generality, we will do our analysis with class 1 as the majority +prediction. +To compute the main prediction y, each prediction ˆy ∈ ˆY will get compared to the values of possible predictions +y′ ∈ Y = {0, 1}. That is, there are two cases to consider: +• Case y′ = 0: +y′ = 0 will get compared w times to the w ˆy = 0s in ˆY , for which ℓ(0, 0) = 0; y′ = 0 will similarly get compared +w + 2n + 1 times to the 1s in in ˆY , for which (by Definition B.2) the comparison is ℓ(1, 0) (with ℓ(1, 0) = a, given +above). By definition of expectation, the the expected loss for these comparisons is +w × 0 + (w + 2n + 1) × a +2w + 2n + 1 += a(w + 2n + 1) +2w + 2n + 1 . +(12) +• Case y′ = 1: +Similarly, the label 1 will also get compared w times to the 0s in ˆY , for which (by Definition B.2) the comparison +is ℓ(0, 1) (with ℓ(0, 1) = b, given above); y′ = 1 will also be compared w + 2n + 1 times to the 1s in ˆY , for which +12This is also another aspect of the simplicity of not needing to define and compute a “central tendency” prediction. We do not need to +encode a notion of a tie-breaking vote to determine a “central tendency.” The main prediction can be unclear in cases for which there is no +“main outcome” (e.g., Individual 2 in Figure 1), as the vote is split exactly down the middle. By avoiding the need to vote on a main +reference point, we also avoid having to ever choose that reference point arbitrarily. +13Moreover, this problem is worse for cost-sensitive lost, for which the main prediction cannot be simply defined as the majority vote +(Appendix B.4); it depends on the majority class being predicted, the asymmetry of the costs, and occasional tie-breaking, such that the +main prediction can either be the majority vote or the minority vote. Again, freeing ourselves from this definition also, then, allows us to +avoid considering the magnitude of variance, and just the level of (dis)agreement in our self-consistency measure. Doing so with the main +prediction is more challenging, as it encodes the cost to determine the reference point to compare against. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +ℓ(1, 1) = 0. The expected loss for these comparisons is +w × b + (w + 2n + 1) × 0 +2w + 2n + 1 += +bw +2w + 2n + 1. +(13) +We need to compare these two cases for different possible values of a and b to understand which expected loss is minimal, +which will determine the main prediction y that satisfies Equation (11). The three different possible relationships for values +of a and b are a = b (symmetric loss), and a > b and a < b (asymmetric loss). +For reference, recall from above that ℓ(1, 0) = a and ℓ(0, 1) = b. Since the results of the two cases above share the same +denominator, we just need to compare their numerators, a(w + 2n + 1) (12) and bw (13). +Symmetric Loss (0-1 Loss) +When a = b = 1, the numerators in Equations (12) and (13) yield expected losses w + 2n + 1 and w, respectively. It +follows that +w + +≥1 +� �� � +2n + 1 ≥ w + 1. +(Given n ≥ 0) +Therefore, the comparison of the numerators can be written as +w < w + 1. +(i.e., Equation (13) ¡ Equation (12)) +This means that the case of y′ = 1 (13) is the minimal one. In other words, the expected loss for class 1, the most frequent +class, is the minimum, and thus the most frequent/ majority vote class is the main prediction. An analogous result holds +if we instead set the most frequent class to be 0. More generally, this holds for all symmetric losses, for which a = b. +▶ In brief, symmetric losses result in the main prediction y being the majority vote of the predictions in ˆY . +Asymmetric Loss +For asymmetric/ cost-sensitive loss, we need to examine two sub-cases: a > b and a < b. +• Case a > b: +bw < a(w + +≥1 +� �� � +2n + 1) +(Given n ≥ 0) +Therefore, since bw is minimal and associated with class 1 (the most frequent class in our setup), the most majority +vote is the main prediction. We can achieve an analogous result if we instead set 0 as the majority class. +▶ In brief, asymmetric losses result in the main prediction y being the majority vote of the predictions in ˆY , +if the majority class has a cheaper cost associated with misclassification (i.e., if the majority class is 1 and +ℓ(1, 0) < ℓ(0, 1), or if the majority class is 0 and ℓ(0, 1) < ℓ(1, 0)). +• Case a < b: +If a < b, it depends on how asymmetric the costs are and how large the margin (i.e., 2n + 1) between class votes is. +There are 3 sub-cases to evaluate: +– Case bw = a(w + 2n + 1): + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Let us first examine if it is possible for the costs to be equal. We can rearrange terms as a ratio of costs to votes, i.e., +bw = a(w + +≥1 +� �� � +2n + 1) +(The terms in this equality are > 0) +b +a = w + 2n + 1 +w +(Given the above, bw > 0 so w > 0) += 1 + 2n + 1 +w +b +a − 1 = 2n + 1 +w +b − a +a += 2n + 1 +w +b − a +a += (w + 2n + 1) − w +w +≥ 1 +w +(14) +In words, we can look at the relative asymmetric cost difference of the minority class cost (above b, without +loss of generality) and the majority class cost (above a, without loss of generality), (above b−a +a , without loss of +generality). If that relative cost difference is equal to the relative difference of the votes between the majority +and minority classes (i.e., (w+2n+1)−w +w +), then the costs of predicting either 1 or 0 are equal. +▶ In brief, for asymmetric loss when the majority-class-associated cost is less than the minority-class associated +cost, if the expected losses are equal, then we can treat the main prediction y to be either 1 or 0, as long as we +make this choice consistently. +– Case bw > a(w + 2n + 1): +Following (14) above, this is the same as saying +b − a +a +> (w + 2n + 1) − w +w +In words, we can look at the relative asymmetric cost difference of the minority class cost (above b, without +loss of generality) and the majority class cost (above a, without loss of generality), (above b−a +a , without loss +of generality). If that relative cost difference is greater than the relative difference of the votes between the +majority and minority classes (i.e., (w+2n+1)−w +w +), then the minority vote yields the minimum cost and is the +main prediction y (above y = 0, without loss of generality; an analogous result holds if we had set the majority +vote to be 0 and the minority vote to be 1). +▶ In brief, for asymmetric loss when the majority-class-associated cost is less than the minority-class associated +cost, it is possible for the minority class to have a greater associated loss. In this case, the minority vote is the +main prediction y. +– Case bw < a(w + 2n + 1): +Following (14) above, this is the same as saying +b − a +a +< (w + 2n + 1) − w +w +In words, we can look at the relative asymmetric cost difference of the minority class cost (above b, without +loss of generality) and the majority class cost (above a, without loss of generality), (above b−a +a , without loss of +generality). If that relative cost difference is less than the relative difference of the votes between the majority +and minority classes (i.e., +(w+2n+1)−w +w +), then the majority vote yields to minimum cost and is the main +prediction y (above y = 1, without loss of generality; an analogous result holds if we had set the majority vote +to be 0 and the minority vote to be 1). +▶ In brief, for asymmetric loss when the majority-class-associated cost is less than the minority-class associated +cost, it is possible for the majority class to have a greater associated loss. In this case, the majority vote is the main +prediction y. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Summary of the above analysis of the main prediction. We extract the key takeaways concerning the computation of +the main prediction below. +• Symmetric loss: The main prediction is the majority vote. +• Asymmetric loss: Compute +– The relative cost difference (i.e., b−a +a ) +– The majority class (and, as a result, the minority class) for the ˆy ∈ ˆY +– The relative difference in the number of votes in the majority and minority classes ( (w+2n+1)−w +w +) +Then, +– If the majority class in ˆY has the lower cost of misclassification, then the main prediction is the majority vote. +– If the majority class in ˆY has the higher cost of misclassification, then the main prediction depends on the +asymmetry of the costs and the vote margin, i.e., +* If b−a +a += (w+2n+1)−w +w +, we can choose the main prediction to be either class (but must make this choice +consistently). +* If b−a +a +> (w+2n+1)−w +w +, the minority vote is the main prediction. +* If b−a +a +< (w+2n+1)−w +w +, the majority vote is the main prediction. +B.5. Defining variance: The main takeaways from above +The work in Domingos (2000a;b) is just one of many decompositions of error into noise, bias, and variance for 0-1 loss. +Like those before it, it has some desirable qualities, and some drawbacks. Given that the math, in some respects, works +out better than prior work, Domingos (2000a) has been particularly influential. Given the importance of 0-1 loss in the +algorithmic fairness literature, Domingos (2000a) has been adopted in some fairness literature (Black et al., 2022a; Chen +et al., 2018). We discuss the limitations of these prior approaches above. +Altogether, the discussion and results in this Appendix show our attempt to make the Domingos (2000a;b) definitions work +for the present study. In short, we can summarize the above to say that we identified mathematical and conceptual problems +with the main prediction, which led us to go in a different direction with our definition. We cover the mathematical and +conceptual difficulties in this Appendix. In deviating from the definitions in Domingos (2000a;b), we lose the theoretical +analysis that he proves for explicitly decomposing 0-1 loss into a concrete relationship between noise, bias, and variance. +Given the issues we discuss with the main prediction (Definition B.2), upon which this decomposition relies, we believe that +this decomposition has limited practical utility.14 Our results emphasize empirical utility, and we believe that our definitions +(freed from a dependence on the main prediction) provide a better formulation of the problem. Furthermore, we do not +need a decomposition result to analyze variance and self-consistency, as we do in this paper. +C. Additional Details on Self-Consistency +In this appendix, we describe the relationship between variance (Definition 3.1) and self-consistency (Definition 3.2) in +more detail. This will complete our technical background on our choice of metrics, which informs our normative analysis +(Section 3) and our algorithm results (Section 4, Appendix D). +We return to Figure 1 from Section 1, which we use to provide a more detailed intuition for self-consistency. +This figure conveys different degrees of SC +� +A, D, (x, g) +� +for different (x, g) in our test set. Individual 1 is completely +self-consistent; for this empirical estimate of models produced by the learning process A, there is no uncertainty as to +how Individual 1 is classified. In contrast, Individual 2 is maximally self-inconsistent. Based on our empirical estimate, +14This is even more relevant for the decomposition result provided in the extended report for cost-sensitive loss (Domingos (2000b) +Theorem 4). Our proof (Appendix B.4.1) for the main prediction for cost-sensitive loss is one of the missing elements. The fact that the +main prediction changes dependent on the relative costs, and that the decomposition also changes conditioned on the main prediction, +makes this result (in our opinion) so carefully constructed as to not be practically useful. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Ind. 1 +Ind. 2 +Two individuals from COMPAS +0 +2 +4 +6 +8 +10 +# of Model predictions +y = 1 +y = 0 +Figure 1. Comparing predictions ˆy for 2 individuals, using 10 logistic regression models trained on bootstrap replicates. +the classification for Individual 2 is effectively random, with no tendency toward either class label. There are two further +observations we want to make, now that we are examining this figure after having defined variance and self-consistency +in the main paper. First, this figure confers no information about the magnitude of variance. This figure just shows the +extent of (dis)agreement between predictions for these two Individuals in the test set, without a notion of costs C01 and C10. +Second, we can understand Individual 1 to be 100% self-consistent, as is clear from the fact that all of the predictions are +the same. Similarly, we can understand Individual 2 to be 50% self-consistent, reflecting that there is an equal probability +of either class label being predicted. That is, we can 50% self-consistency to be the minimum amount of self-consistency +possible in theory, i.e. SC +� +A, D, (x, g) +� +is defined on [0.5, 1]. We clarify these details more formally in this Appendix, +relate SC +� +A, D, (x, g) +� +directly to Var +� +A, D, (x, g) +� +, and discuss the empirical approximation ˆ +SC. +C.1. Deriving the SC +� +A, D, (x, g) +� +metric +We derive our equation for SC +� +A, D, (x, g) +� +(2) from Definitions 3.1 and 3.2. We will then use (2) to more formally justify +why SC +� +A, D, (x, g) +� +�→ [0.5, 1] and that ˆ +SC �→ [0.5 − ϵ, 1], for small positive ϵ as the number of models n increases +(Appendix C.2). +Proof. Recall from Definition 3.2 that +SC +� +A, D, (x, g) +� +≜ PhDi∼µ,hDj ∼µ{hDi(x) = hDj(x)} = +1 +n(n − 1) +� +i̸=j +1[hDi(x) = hDj(x)]. +Note that, by the definition of 0-1 loss, Var +� +A, D, (x, g) +� +for 0-1 loss is +Var +� +A, D, (x, g) +� +0-1 = +1 +n(n − 1) +� +i̸=j +1[hDi(x) ̸= hDj(x)] += +2αβ +n(n − 1). +(15) +Observe that, by the definition of 1, +1 = +1 +n(n − 1) +� +i̸=j +� +From Var +� +A,D,(x,g) +� +0-1 +� +�� +� +1[hDi(x) ̸= hDj(x)] + +From SC +� +A,D,(x,g) +� +� +�� +� +1[hDi(x) = hDj(x)] +� += +(15) +� +�� +� +2αβ +n(n − 1) + +1 +n(n − 1) +� +i̸=j +1[hDi(x) = hDj(x)]. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Therefore, rearranging the above, +SC +� +A, D, (x, g) +� += 1 − +2αβ +n(n − 1). +We note that (2) is independent of specific costs C01 and C10. Nevertheless, the choice of decision threshold τ will of course +impact the values of α and β in practice (Appendix B.2). In turn, this will impact the degree of self-consistency that a +learning process exhibits empirically; the measured degree of self-consistency will depend on the choice of ℓ. +Following an analysis similar to what we present for Var +� +A, D, (x, g) +� +in Appendix B.2.1, (9), we can show that +SC +� +A, D, (x, g) +� +�→ [0.5, 1] by analyzing +1 − lim +n→∞ +2αβ +n(n − 1). +(16) +for minimal and maximal consistency, with minimal consistency approaching 1 +2 and maximal consistency equalling 1, with +any intermediate amount of self-consistency between these two values, as n → ∞. +In practice, we approximate SC +� +A, D, (x, g) +� +by ˆ +SC, which uses a concrete n to train hD1, hD2, . . . , hDn. As n increases, +the minimally self-consistent examples will increasingly measure closer and closer to 1 +2, but will never actually be exactly 1 +2. +As a result, ˆ +SC will be a value in [0.5 + ϵ, 1], for small positive ϵ. This reality is reflected in the results that we report for our +experiments in Sections 3 and 5, as well as our extended results in Appendix E. +C.2. Cost-independence of self-consistency +Intuitively, self-consistency of a learning process is a relative metric; it is a quantity that is measured relative to A. We +therefore conceive of it as a metric that is normalized with respect to the learning process; such a process can be maximally +100% self-consistent, but it does not make sense for it to be more than that (reflected by the maximum value of 1). +In contrast, as discussed in Appendix B, variance can measure greater than 1, depending on the magnitude of the sum of +the costs C01 and C10; in particular, for C01 + C10 > 4 (10). However, it is not necessarily meaningful to compare the +magnitude of variance across classifiers. Recall that the effect of changing costs C01 and C10 corresponds to a change in the +binary classification decision threshold, with τ = +C01 +C01+C10 (8). It is the relative costs that change the decision threshold; not +the costs themselves. For example, the classifier with costs C01 = 1 and C10 = 3 is equivalent to the classifier with costs +C01 = 1 +2 and C10 = 3 +2 (for both, τ = 1 +4), but the former would measure a larger magnitude for variance (See Definition 3.1 +and (1)). +It is this observation the grounds our cost-independent definition of self-consistency in Section 3 and Appendix C. Given the +fact that the magnitude of variance measurements can complicate our comparisons of classifiers, as discussed above, we +focus on the part of variance that encodes information about arbitrariness in a learning process: its measure of (dis)agreement +between classification decisions that result from changing the training dataset. +Put differently, we could alternatively conceive of self-consistency as the inverse of normalized variance. The idea here +would be to mitigate the effects of variance magnitude complicating our analysis by normalizing variance to be defined on +[0, 1] This way, classifiers with the same τ but different C01 + C10 can be compared to each other more easily. We could +then compute the the maximum possible variance (which is dependent on the choice of C01 and C10), normalize the variance +by that maximum possible amount. For example, for even n (and similarly for odd n), Var +� +A, D, (x, g) +� +max = (C01+C10)n2 +4n(n−1) +and a specific variance measure is Var +� +A, D, (x, g) +� += (C01+C10)αβ +n(n−1) +, from which we would compute + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Var +� +A, D, (x, g) +� +normalized = +Var +� +A, D, (x, g) +� +Var +� +A, D, (x, g) +� +max += +(C01+C10)αβ +n(n−1) +(C01+C10)n2 +4n(n−1) += 4αβ +n2 . +As expected, the sum of costs cancels as a part of normalization, so our normalized variance measure is independent of +the costs. If we were then to compute self-consistency as the additive inverse of variance, we could then compute it as +1−Var +� +A, D, (x, g) +� +normalized, which would produce a self-consistency metric that has its range be [0, 1] rather than [0.5, 1], +which is the range for (2). +We could of course compute self-consistency in this way for our analysis, and we would produce analogous results to +those presented in this paper. Since it is still a relative metric, this kind of measurement will behave similarly to the one +we propose. For practical reasons, we choose to measure self-consistency in this way for two reasons. First, it is more +cumbersome to compute the cost-sensitive maximal variance, which (2) does not require while still capturing a measure of +(dis)agreement between predictions. Second, we believe that, following the intuition in Figure 1 discussed above, we believe +that a minimum value of 1 +2 for SC +� +A, D, (x, g) +� +is easier to interpret than a minimum value of 0.15 +D. Additional Details on Our Algorithmic Framework +As we discuss in Section 3, even though our estimate of the average classifier is tight, there remains a lot of underlying +variance in the predictors that compose the average. We can produce stable estimates of group-specific error rates, but these +error rates do not convey how much instability there is in the underlying predictions that compose them. We made this clear +in Figure 2. We plot cumulative self-consistency to show both how much (dis)agreement there is between different models +trained on different replicates, and to give a sense of how that (dis)agreement is distributed. +A natural question is to see if we can improve self-consistency, with the hope that doing so would reduce arbitrariness in +the learning process, improve accuracy, and, for the cases in which there is different self-consistency across subgroups, +also improve fairness. To do so, we consider ways of reducing variance, as, based on our definitions (Definition 3.1 +and 3.2), doing so should improve self-consistency. We first consider the classic bootstrap aggregation — or, bagging — +algorithm (Breiman, 1996), which we use to explain our algorithm in extended detail (Appendix D.1), which builds upon +bagging to instill a user-specified tolerance of minimal self-consistency when making predictions. +D.1. Super-ensembling for self-consistency +It has been well-known since Breiman (1996) that bagging can improve the performance of unstable predictors. That is, +for models produced by a learning process that is sensitive to the underlying training data, it is (theoretically-grounded) +good practice to train an ensemble of models hD1, hD2, . . . hDB using bootstrapping (with B bootstrap replicates) (See +Appendix A.2.1; Efron (1979); Efron & Tibshirani (1993)). When classifying an example x, we leverage the whole ensemble +by aggregating the predictions produced by its members hDb. This aggregation process identifies the most common +prediction in the ensemble, and returns that label as the classification. Put differently, we have combined the information of +a lot of unstable classifiers, and averaged over their behavior in order to generate more stable classifications. +Given the the relationship between variance (Definition 3.1) and self-consistency (Definition 3.2), it is natural to see +if we can improve self-consistency by reducing variance. However, rather than relying on a simple-majority-vote to +decide the aggregated prediction, we will instill a notion of confidence in our predictions by requiring a minimum level of +self-consistency. +We present a framework that alters the semantics of classification outputs to 0, 1, and Abstain, and employ ensembling to +determine the ˆ +SC-level that guides the output process. We define abstentions to be consistent with both 0 and 1 predictions. +15Of course, since ˆ +SC approximates SC +� +A, D, (x, g) +� +, for n = 2 it is possible to empirically see ˆ +SC = 0. However, it does not make +sense to set n = 2 when measuring ˆ +SC; it should be set larger to better approximate SC +� +A, D, (x, g) +� +. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Intuitively, this makes sense; the point of abstaining is to avoid disagreement — to not provide a classification when it is +not possible to do so confidently. Algorithm 1 suggests many possible ways that this can be achieved. For example, we +can change the aggregation rule in regular bagging to use a self-consistency level κ rather than majority vote. Instead of +relying on votes, we can bag the underlying prediction probabilities and then apply κ a filter. We could take the top-n most +consistent predictions and let a super-ensemble of underlying bagged classifiers decide whether to abstain or predict. +In the experiments in the paper, we provide two examples: Changing the underlying bagging vote aggregation rule (simple +ensembling), and applying a round of regular bagging to do variance reduction and then bagging the bagged outputs +(super-ensembling) to apply a self-consistency threshold. Put slightly different, we build an ensemble that consists of +bagged classifiers — a super ensemble of ensembles. Each member of our super ensemble will perform the typical +bagging algorithm (Breiman, 1996). The super ensemble will then examine its members’ predictions and compute their +self-consistency. If their self-consistency exceeds a user-specified threshold (which, by Definition 3.2, has to be on [0.5, 1]), +then the super ensemble outputs the majority prediction. Otherwise, the super ensemble abstains from predicting. This will +force our algorithm to only return predictions that achieve a user-specified desired level of confidence, and will therefore +achieve our goal of removing the kind of arbitrariness that our paper is concerned about. (Of course, this says nothing about +removing or mitigating other sources of arbitrariness). Our model will not produce predictions for examples for which the +lack of confidence is too high. We describe our procedure more formally in Algorithm 1, which we reproduce below for +completeness. +Algorithm 1 Super-ensembling for self-consistency +input training dataset (X, O)train, training procedure A, ensemble size η, +ˆ +SC threshold κ ∈ [0.5, 1], test instance xtest +output Prediction ˆy with ˆ +SC ≥ κ or Abstain +1: function BagConfidently +� +(X, O)train, A, η, κ, xtest +� +2: +ˆyA := list() // To store ensemble predictions +3: +for 1 . . . η do +4: +Dη ← Bootstrap +� +(X, O)train +� +5: +hDη ← A(Dη) //A may Bag using Dη +6: +ˆyA.add +� +hDη(xtest) +� +//ˆy1, . . . , ˆyη +7: +end for +8: +return Aggregate(ˆyA, κ) +9: end function +10: // An aggregate function that meets our framework’s semantics +11: function Aggregate +� +ˆy1, . . . , ˆyη, κ +� +12: +if SelfConsistency(ˆy1, . . . , ˆyη) ≥ κ then +13: +return arg maxy′∈Y +� �η +i=1 1[y′ = ˆyi] +� +14: +end if +15: +return Abstain +16: end function +D.2. Proof of improved self-consistency +We briefly show the simple proof that any method that meets the semantics of Algorithm 1 will be more self-consistent than +its counterpart that cannot Abstain. Recall that we define abstentions to be in agreement with both 0 and 1 predictions. +This makes sense intuitively: Algorithm 1 abstains to avoid making predictions that lack self-consistency, so abstaining +should not increase disagreement between predictions. +It follows that we can continue to use Definition 3.2, but with one small adjustment. Instead of the total number of predictions +n = α + β, with α and β corresponding to 0 and 1 predictions, respectively, we now allow for n ≥ α + β, in order to +account for possibly some non-zero number of abstentions. +In more detail, let us + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +denote ˆY to be the multiset of predictions for models hD1, hD2, . . . , hDn on (x, g), with | ˆY | = n = α + β + γ. This is +where we depart from our typical definition of self-consistency, for which n = α + β (Section 3, Appendix C). We continue +to let α and β represent the counts of 0 and 1 predictions, respectively, and now include γ to denote the (possibly nonzero) +number of abstentions. This leads to the following adjustment of Definition 3.2: +SC +� +A, D, (x, g) +� += 1 − 2(αβ + αγ + βγ) +n(n − 1) +. +(17) +Equation (17) follows from a similar analysis of comparing 0s, 1s, and abstentions for Definition 3.2, which lead us to +derive (2) in Appendix C.1. However, since the costs of 0-to-Abstain comparisons and 1-to-Abstain comparisons are +both 0, the αγ and αβ terms in (17) reduce to 0. As a result, we yield our original definition for self-consistency, with the +possibility that n = α + β + γ > α + β, if there is a nonzero number of abstentions γ. +Of course, since n > 1 and α, β, γ ≥ 0, it is always the case that option to Abstain is at least as self-consistent as not +having the option to do so. This follows from the fact that α + β + γ = n ≥ α + β, which would make the denominator in +(17) greater than or equal to the corresponding method that cannot Abstain; when subtracted from 1, this would produce +a SC that is no smaller than the value for the corresponding method without that cannot Abstain. +Now, it follows that, given the choice between Abstain and predicting a label that is in disagreement with an existing +prediction label, choosing to Abstain will always lead to higher self-consistency. This is because the cost to Abstain is +less than disagreeing, so it will always be the minimal choice that maximizes SC. +E. Additional Empirical Results and Details for Reproducibility +The code for the examples in Sections 1, 3 and 5 can be found in https://github.com/pasta41/variance. +This repository also contains necessary and sufficient information concerning reproducibility. At the time of writing, we +use Conda to produce environments with associated package-versioning information, so that our results can be exactly +replicated and independently verified. We also use the Scikit-Learn (Pedregosa et al., 2011) toolkit for modeling and +optimization. More details on our choice of models and hyperparameter optimization can be found in our code repository, +cited above. In brief, we consulted prior related work (e.g., Chen et al. (2018)) and performed our own validation for +reasonable hyperparameters per model type. We keep these settings fixed to reduce impact on our results, in order to observe +in isolation how different training data subsets impact our results. During these early runs, we collected information on train +accuracy, not just test accuracy; while models ultimately have similar test accuracy in most cases for the same task, they can +vary significantly in terms of train accuracy (e.g., for logistic regression, COMPAS is in the low .70s; for random forests, it is +in the mid .90s). We do not include these results for the sake of space. +This section is organized as follows. We first present information on our datasets, models and code, including our HDMA +toolkit (Appendix E.1). We then provide details on our setup for running experiments on our cluster (Appendix E.2). +Appendix E.3 contains more detailed information concerning the experiments performed to produce Figures 1 and 2 in the +main paper. We then provide extensive analogous results in Appendix E.4 for additional models and datasets: COMPAS +and Old Adult (Appendix E.4.1), South German Credit (Appendix E.4.2), Taiwan Credit (Appendix E.4.3), +3 tasks from New Adult (Appendix E.4.4), and 2 states from HMDA 2017 (Appendix E.4.5). We then provide a +synthesized discussion of these results in Appendix E.4.6. In Appendix E.5, we provide more details on the results presented +in Section 5, as well as additional experiments. Lastly, in Appendix E.6, we discuss implications of these results for common +fairness benchmarks like South German Credit. We conclude that in many cases, without adequate attention to error +estimation, it is likely that training and post-processing a single model for fairness on these models likely is a brittle approach +to achieve generalizable fairness (and accuracy) performance. Based on our empirical results, it seems like high variance +can be a significant confounding factor when using a small set of models to draw conclusions about performance — whether +fairness or accuracy. There is an urgent need for future work concerning reproducibility. More specifically, our results +indicate that it would be useful to revisit key algorithmic strategies in fair classification to see how they perform in context +with more reliable expected error estimation and variance reduction. +Note on CDF figures.. It is worth noting (though hopefully obvious) that our CDF plots of ˆ +SC are not continuous, yet we +choose to plot them as interpolated curves. This are discrete because we train a concrete number of models (individual +models or bags) — typically 101 of them — that we treat as our approximation for n when computing ˆ +SC. This means that +there are a finite number of x-values for ˆ +SC, for which we plot a corresponding concrete number of heights y corresponding + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +to the cumulative proportion of the test set. In this respect, it would perhaps be more precise to plot our curves using a step +function, exemplified below: +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +HL - before +NHL - before +HL - after +NHL - after +Figure 7. Plotting Figure 4 to emphasize discrete steps. +We opted not to do this for two reasons. First, plotting steps for some of our figures, in our opinion, can make the figures +more difficult to understand. Second, in experiments for which we increase the number of models used to estimate ˆ +SC +(e.g., Appendix E.6), we found that the curves for 101 models were a reasonable approximation of the overall CDF shape. +We therefore concluded that plotting the figures without steps was worth the clarity of presentation, with a sacrifice in +correctness for the overall takeaways that we intend with these figures. +E.1. Hypothesis classes, datasets, and code +Models. According to a comprehensive recent survey study (Fabris et al., 2022), as well as related work like Chen et al. +(2018), we conclude that some of the most common models used in fair classification are logistic regression, decision tree +classifiers, random forest classifiers, SVMs, and MLPs. We opted to include comprehensive results for the first three, since +they capture different complexities, and therefore encode different degrees of statistical bias, that we expected to have an +impact on the underlying sources of error. We provide some results for SVMs and MLPs, which we include in this Appendix. +Since we choose not to use stochastic optimizers to reduce the sources of randomness, for our results, training MLPs is slower +than it could be. We consistently use a decision threshold of 0.5 (i.e., 0-1 loss) for our experiments, though our results can +easily be extended to other thresholds, as discussed in Section 3. Depending on the dataset, we reserve between 20% and 30% +of the available data for the test set. This is consistent with standard fair classification training settings, which we validated +during our initial experiments to explore the space (for which we also did preliminary hyperparameter optimization, before +fixing the hyperparameters for our presented results). Please refer to https://github.com/pasta41/variance +for more details. +Datasets. Also according to Fabris et al. (2022), the most common tasks in fair classification are Old Adult (Kohavi, +1996), COMPAS (Larson et al., 2016), and South German Credit (Gr¨omping, 2019).16 These three datasets arguably +serve as a de facto benchmark in the community, so we felt the need to include them in the present work. In recognition of +the fact that these three datasets, however standard, have problems, we also run experiments on 3 tasks in the New Adult +dataset, introduced by Ding et al. (2021) to replace Old Adult. We subset to the CA (California) subset of the dataset, and +run on Income, Employment, and Public Coverage, and consider sex and race as protected attributes, which we +binarize into {Male, Female} and {White, Non-white}. These are all large-scale tasks, at least in the domain of algorithmic +fairness — on the order of hundreds of thousands of example instances. However, the 3 tasks do share example instances +and some features. In summary, concerning common tasks in fair classification: +• COMPAS (Larson et al., 2016). We run on the commonly-used version of this dataset from Friedler et al. (2019), which +has 6167 example instances with 404 features. The target is to predict recidivism within 2 years (1 corresponding to +Yes, and 0 to No). The protected attribute is race, binarized into “Non-white” (0) and “White” (1) subgroups. +• Old Adult (Kohavi, 1996). We run on the commonly-used version of this dataset from Friedler et al. (2019), which +has 30,162 examples with 97 features. This version of the dataset removes instances with missing values from the +16Technically, Gr¨omping (2019) is an updated and corrected version of the dataset from 2019. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +original dataset, and changes the encoding of some of the features (Kohavi (1996) has 48842 example instnaces with +88 features). The target is to predict < $50, 000 income (0) >= $50, 000 income (1). The protected attribute is sex, +binarized into “Female” (0) and “Male” (1) subgroups. +• South German Credit (Gr¨omping, 2019). We download the dataset from UCI and process the data ourselves. +We use the provided codetable.txt to “translate” the features from German to English. We say “translate” because +the authors took some liberties, e.g., the column converted to “credit history” is labeled “moral” in the German, which +is not a translation. There are four categories in the protected attribute “personal status sex” column, one of which (2) +is used for both “Male (single)” and “Female (non-single).” We therefore remove rows with this value, and binarize the +remaining three categories into “Female” (0) and “Male” (1). What results is a dataset with 690 example instances (of +the original 1000) with 19 features. The target is “good” credit (1) and “bad” credit (0). +• Taiwan Credit (Yeh & Lien, 2009). This task is to predict default on credit card payments (1) or not (0). There are +30,000 example instances and 24 features. The protected attribute is binary sex. We download this dataset from UCI. +• New Adult (Ding et al., 2021). This dataset contains millions of example instances from US Census data, which can +be used for several different targets/tasks. We select three of them (listed below). These tasks share some features, and +therefore are not completely independent. Further, given the size of the whole dataset, we subset to CA (California), +the most populous state in the US. There are two protected attribute columns that we use: sex, which is binarized +“Female” (0) and “Male” (1) subgroups, and race, which we binarize into “Non-white” (0) and “White” (1). In future +work, we would like to explore extending our results beyond binary subgroups. +– Income. This task is designed to be analogous to Old Adult (Kohavi, 1996). As a result, the target is to +predict < $50, 000 income (0) >= $50, 000 income (1). In the CA subset, there are 195,665 example instances +with 8 features. +– Employment. This task is to predict whether an individual is employed (1) or not (0). In the CA subset, there +are 378,817 example instances with 14 features. +– Public Coverage. This task is to predict whether an individual is on public health insurance (1) or not (0). +In the CA subset, there are 138,554 example instances with 17 features. +E.1.1. THE STANDALONE HMDA TOOKIT +In addition to the above standard tasks, we include experiments that use the NY and TX 2017 subsets of the the Home +Mortgage Data Disclosure Act (HMDA) 2007-2017 dataset (Federal Financial Institutions Examination Council, 2017). +These two datasets have 244,107 and 576,978 examples, respectively, with 18 features. +The HMDA datasets together contain over 100 million examples of US home mortgage loans from 2007-2017 (newer data +exists, but in a different format). We developed a toolkit, described below, to make this dataset easy to use for classification +experiments. Similar to New Adult, we enable subsetting by US state. For the experiments in this paper, we run on the +NY (New York) and TX (Texas) 2017 subset, in order to add some geographic diversity to complement our New Adult +experiments. We additionally chose New York and Texas because they are two of the most populous states in the US, +alongside California17 +The target variable, action taken, concerning loan origination has 8 values, 2 of which we cannot meaningful conclude +approval or denial decisions. They are: Action Taken: 1 – Loan originated, 2 – Application approved but not accepted, 3 – +Application denied by financial institution, 4 – Application withdrawn by applicant, 5 – File closed for incompleteness, 6 +– Loan purchased by the institution, 7 – Preapproval request denied by financial institution, and 8 – Preapproval request +approved but not accepted (optional reporting). We filter out 4 and 6, and binarize into grant={1, 2, 8} = 1 and +reject={3, 5, 7} = 0. There are three protected attributes that we consider: sex, race, and ethnicity: +• sex has 5 possible values, 2 of which correspond to categories/non-missing values: Male – 1 and Female – 2. We +binarize sex into F = 0 and M = 1. +• race has 8 possible values, 5 of which correspond to categories/ non-missing information: 1 – American Indian +or Alaska Native, 2 – Asian, 3 – Black or African American, 4 – Native Hawaiian or Other Pacific Islander, and 5 – +17As of the 2020 Census, the top four most-populous states are California, Texas, Florida, and New York (Mackun et al., 2021). We +chose New York for northeastern representation. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +White. There are 5 fields for applicant race, which model an applicant belonging to more than one racial group. For our +experiments, we only look at the first field. When we binarize race, NW = 0 and W = 1. +• ethnicity has 5 possible values, 2 of which correspond to categories/ non-missing information: 1 – Hispanic or +Latino and 2 – Not Hispanic or Latino. We binarize ethnicity to be HL = 0 and NHL = 1. +After subsetting to only include examples that have values that do not correspond to missing information, HMDA has 18 +features. The NY dataset has 244,107 examples; the TX dataset has 576,978 examples, making it the largest dataset in our +experiments. As with our experiments using New Adult, we would like to extend our results beyond binary subgroups +and binary classification in future work. +Releasing a standalone toolkit. These datasets are less-commonly used in current algorithmic fairness literature (Fabris +et al., 2022). We believe this is likely due to the fact that the over-100-million data examples are only available in bulk +files, which are on the order of 10s of gigabytes and therefore not easily downloadable or explorable on most personal +computers. Following the example of Ding et al. (2021), one of our contributions is to pre-process all of these datasets +— all locations and years — and release them with a software toolkit. The software engineering effort to produce this toolkit +was substantial. Our hope is that wider access to this dataset will further reduce the community’s dependency on small (and +dated) datasets. Please refer to https://github.com/pasta41/hmda for the latest information on this standalone +software package. Our release aligns with the terms of service for this dataset. +E.2. Cluster environment details +While most of the experiments run in this paper can be easily reproduced on a modern laptop, for efficiency, we ran all of our +experiments (except the one to produce Figure 1) in a cluster environment. This enabled us to easily execute train/test splits +n in parallel on different CPUs, serialize our results, and then reconstitute and combine them to produce plots locally. Our +cluster environment runs Ubuntu 20.04 and uses Slurm v20.11.8 to manage jobs. We ran all experiments using Anaconda3, +which is why we used Conda to reproduce environments for easy replicability. +The experiments using New Adult and HMDA rely on datasets that are (in some cases) orders of magnitude larger than the +traditional algorithmic fairness tasks. This is one of the reasons why we recommend running on a cluster, and therefore do +not include Jupyter notebooks in our repository for these tasks. We also limit our modeling choices to logistic regression, +decision tree classifiers, and random forest classifiers for these results due to the expense of training on the order of 1000 +models for each experiment. +E.3. Details on motivating examples in the main paper +This appendix provides extended results for the experiments associated in Sections 1 and 3, which give an intuition for +individual- and subgroup-level consistency. The experimental results in the main paper are for logistic regression. We +expand the set of models we examine, and associated discussion of how to interpret comparisons between these results. +Reproducing Figure 1. The experiment to produce this figure in Section 1 (also shown in Appendix C) trains S = 10 +logistic regression models on the COMPAS dataset (Appendix E.1) using 0-1 loss. We use the bootstrap method to produce +each model, which we evaluate on the same test set. We then search for a maximally consistent and minimally consistent +individual in the test set, i.e., an individual with 10 predictions that agree and an individual with 5 predictions in each +class, which we plot in the bar graph. Please refer to the README in https://github.com/pasta41/variance +regarding which Jupyter notebook to run to produce the underlying results and figure. The experiments to reproduce +this figure can be easily replicated on a laptop. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Reproducing Figure 2. These figures were produced by executing S = 10 runs of B = 100 bootstrap training replicates to +train random forest classifiers for Old Adult and COMPAS. We reproduce these figures below, so that they can be examined +and treated in relation to our additional results for decision tree classifiers and logistic regression. For each s run, we take +train/test split, bootstrap the train split B = 100 times, and evaluate the resulting model classification decisions on the test +set. ˆ +SC can be estimated from the results across those 100 models. We Run this process S = 10 times to produce confidence +intervals, shown in the figures below. The intervals are not always clearly visible; there is not a lot of variance at the level of +comparing whole runs to each other. Please refer to the README in https://github.com/pasta41/variance +regarding which Jupyter notebook to run to produce the underlying results and figure. There are also scripted version of +these experiments, which enable them to be run in parallel in a cluster environment. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Non-white (NW) +White (W) +(a) COMPAS split by g = race +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female (F) +Male (M) +(b) Old Adult split by g = sex +Figure 2. Cumulative proportion of test instances that attain the given level of self-consistency. We train random forest classifiers to +estimate ˆ +SC with B = 100 bootstrap replicates, and repeat with S = 10 train/test splits to produce confidence intervals. +Table +1. +Mean +± +STD +across +S += +10 +train/test +splits +× +B += +100 +bootstrap-replicate +training +runs +for +random +forest +classifiers +on +COMPAS +and +Old Adult. +These +results +correspond +to +Figure +2. +COMPAS +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.366 ± .005 .173 ± .008 .193 ± .007 +.73 ± .003 +g = NW .369 ± .005 +.18 ± .007 +.19 ± .008 +.732 ± .004 +g = W +.359 ± .013 +.16 ± .012 +.199 ± .011 .727 ± .008 +Old Adult +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.173 ± .003 .077 ± .003 .096 ± .001 .878 ± .002 +g = F +.09 ± .003 +.037 ± .001 .053 ± .003 .938 ± .002 +g = M +.212 ± .003 .097 ± .003 .116 ± .001 +.85 ± .002 +Table 1 suggests the surprising result that, for COMPAS, the aggregated, average model demonstrates close-to-parity for +subgroup-specific error, +ˆ +FPR, and +ˆ +FNR rate. We return to this observation in Appendix E.6 +Self-consistency of incorrectly-classified instances. Last, we include figures that underscore how self-consistency is +independent from correctness that is measured in terms of observed label alignment. That is, it is possible for an instance +(x, g) to be self-consistent and classified incorrectly, with respect to its observed label o. We show this using stacked bar +plots. For the above experiments, we find the test examples that have the majority of their classifications incorrect (ˆy ̸= o, for +B = 100, we find the instances with ≥ 50 incorrect classifications) and the majority of their classification correct (ˆy = o, for +B = 100, we find the instances with > 50 correct classifications), and we examine how self-consistent they are. We bucket +self-consistency into different levels, and then plot the relative proportion of majority-incorrectly and majority-correctly +classified examples according to subgroup. Subgroups in COMPAS exhibit a similar trend, while subgroups in Adult Old +exhibit differences, with the heights of the bars corresponding to the trends we plot in our CDF plots. As we note briefly in + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Section 3, it may be interesting to examine patterns in examples about which learning processes are confident (i.e., highly +self-consistent) but wrong in terms of label alignment. If such issues correlate with subgroup, it may be worth testing the +counterfactual that such labels are indicative of label bias. We leave such thoughts to future work. +SC < . 6 +. 6 +SC < . 7 +. 7 +SC < . 8 +. 8 +SC < . 9 +. 9 +SC +1 +Level of self-consistency SC +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +Proportion of g-subgroup test examples +Visualizing test set SC and label alignment by g +Majority of y = o +Majority y +o +g = Non-white (NW) +g = White (W) +(a) COMPAS +SC < . 6 +. 6 +SC < . 7 +. 7 +SC < . 8 +. 8 +SC < . 9 +. 9 +SC +1 +Level of self-consistency SC +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Proportion of g-subgroup test examples +Visualizing test set SC and label alignment by g +Majority of y = o +Majority y +o +g = Female (F) +g = Male (M) +(b) Adult Old +Figure 9. Self-consistency broken down by g and label alignment with the observed label o. For each train/test split in S = 10, and for +each ˆ +SC range (x-axis), we find the examples that are incorrectly classified the majority of time (≥ 5 splits, we find that y ̸= o), and the +examples that are correctly classified the majority of the time (> 5, we find that y = o). We compute the average the proportion over +S = 10 in each ˆ +SC range (y-axis). We plot these proportions with respect to subgroup g (where the sums of the heights of bars for by +each g is equal to 1). + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +E.4. Additional illustrative results +Measuring self-consistency is useful across tasks and models, not just for the ones we had space to illustrate in Section 3. To +demonstrate that the observations we make about variance, self-consistency, and arbitrariness hold up in relation to other +models, we provide additional illustrative results here as follows: +• Old Adult and COMPAS (Appendix E.4.1): We provide analogous results to those for random forests that we include +in Section 3, in which we compare performance for logistic regression, decision trees, multi-layer perceptrons (MLPs), +and support vector machine (SVMs). Overall, the story is the same; for Old Adult and COMPAS, the logistic +regression, decision tree, MLP, and SVM models demonstrate similar trends as those for random forests. MLPs do not +always converge for the S = 10 × B = 100 models. +• South German Credit (Appendix E.4.2): We present results for logistic regression, decision trees, random forest +classifiers, MLPs, and SVMs. We demonstrate that on this dataset it is difficult to consistently quantify self-consistency, +likely due to small dataset size. +• Taiwan Credit (Appendix E.4.3): We present results for logistic regression, decision trees, random forest classifiers, +MLPs, and SVMs. +• New Adult - CA (Appendix E.4.4): We run experiments on New Adult-Income, -Employment, and +-Public Coverage for each protected attribute. We limit these runs to logistic regression, decision trees, and +random forests, due to the increased dataset size and the associated cost of running MLPs and SVMs. Decision tree +and random forest classifiers yield results that are useful for our purposes, at significantly reduced cost. +• HMDA 2017 - NY and TX (Appendix E.4.5): We run experiments on HMDA for NY and TX for each protected +attribute (sex, race, ethnicity). Similar to New Adult, we limit these runs to logistic regression, decision trees, +and random forests, due to the increased dataset size and the associated cost of running MLPs and SVMs. Decision +tree and random forest classifiers yield results that are useful for our purposes, at significantly reduced cost. +All figures show confidence intervals over the specified S runs, where S is the number of train/test splits produced with +unique seeds. Note that, for some experiments, these intervals are too small to be easily visible. We defer unified discussion +of these results to Appendix E.4.6, where we also address in more detail our decision to not run MLPs and SVMs on the +large-scale datasets. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +E.4.1. COMPAS AND OLD ADULT +To parallel our presentation in Section 3, we present the results for these datasets side-by-side for each model type. +Logistic regression. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +Cumulative Proportion of Test Set +Non-white +White +(a) COMPAS split by g = race +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.01 +0.03 +0.05 +0.07 +0.09 +0.11 +0.13 +Cumulative Proportion of Test Set +Female +Male +(b) Old Adult split by g = sex +Figure 10. Cumulative proportion of test instances that attain the given level of self-consistency. We perform logistic regression, estimating +ˆ +SC with B = 100 bootstrap replicates. We repeat S = 10 runs of this process to produce confidence intervals. We exclude perfectly +self-consistent instances to highlight differences (i.e., we plot (x, g) for which ˆ +SC < 1), as most examples in both datasets are completely +self-consistent for logistic regression. +Table 8. Mean ± STD across S = 10 train/test splits ×B = 100 bootstrap-replicate training runs for logistic regression on COMPAS +(g = 0 = NW; g = 1 = W) and Old Adult (g = 0 = F; g = 1 = M). These results correspond to Figure 10. +COMPAS +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.331 ± .009 .139 ± .011 +.191 ± .01 +.883 ± .005 +g = NW +.33 ± .013 +.147 ± .013 .183 ± .011 .882 ± .004 +g = W +.332 ± .011 .126 ± .013 .207 ± .011 .887 ± .008 +Old Adult +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.155 ± .003 .055 ± .002 +.01 ± .002 +.981 ± .001 +g = F +.078 ± .004 .023 ± .003 .055 ± .003 .991 ± .001 +g = M .191 ± .003 +.07 ± .003 +.122 ± .002 .976 ± .001 +For this set of experiments, we also show a sample of the underlying data that we collect for each run s (Tables 9 and 10) +used to produce the summary statistics we report in the main paper (akin to Table 8). These tables show the breakdown +of error and self-consistency measured for each s train/test split. Each statistic is the mean ± standard deviation, which +are computed across the B = 100 bootstrap replicates run for the split. Analogous results at this level of detail for all of +our illustrative examples can be found in .csv files at https://github.com/pasta41/variance. These tables +are themselves summaries; each column representing a train/test split s summarizes the performance of the B bootstrap +replicate models produced. An excerpt of these low-level, per-bootstrap-replicate model results is also reproduced below for +Old Adult logistic regression in Table 11. We defer discussion of these results to Appendix E.4.6. However, we do note +(and address later) that the scale of the y-axis for Figure 10 is much smaller than it is for decision trees and random forests. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Table 9. Mean ± STD of each train/test split run using seed s for Old Adult (10b), computed across the B bootstrap replicates run +for each train/test split. We report total and subgroup g ∈ {Female, Male} = {F, M} Error Rate, (Err) False Positive Rate (FP), and +False Negative Rate (FN). Mean Self Consistency (SC) is reported without STD, as it is computed once across each B bootstrap runs. +s +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Err +.151 ± .001 +.154 ± .001 +.151 ± .001 +.159 ± .001 +.155 ± .001 +.156 ± .001 +.156 ± .001 +.157 ± .001 +.152 ± .001 +.155 ± .001 +FP +.053 ± .002 +.052 ± .002 +.055 ± .002 +.057 ± .002 +.054 ± .001 +.06 ± .002 +.057 ± .002 +.056 ± .002 +.055 ± .002 +.055 ± .002 +FN +.098 ± .002 +.102 ± .002 +.096 ± .002 +.101 ± .002 +.1 ± .002 +.096 ± .002 +.099 ± .002 +.102 ± .001 +.096 ± .002 +.101 ± .002 +SC +.98 +.98 +.979 +.979 +.98 +.978 +.978 +.981 +.979 +.979 +ErrF +.073 ± .001 +.078 ± .001 +.074 ± .001 +.078 ± .001 +.078 ± .001 +.077 ± .001 +.08 ± .001 +.08 ± .001 +.073 ± .001 +.08 ± .001 +FPF +.019 ± .002 +.022 ± .002 +.02 ± .002 +.026 ± .002 +.019 ± .002 +.028 ± .002 +.022 ± .002 +.024 ± .002 +.02 ± .002 +.024 ± .002 +FNF +.054 ± .002 +.057 ± .002 +.054 ± .002 +.052 ± .002 +.058 ± .002 +.049 ± .002 +.058 ± .002 +.057 ± .002 +.053 ± .002 +.056 ± .001 +SCF +.99 +.989 +.99 +.989 +.99 +.989 +.989 +.991 +.991 +.991 +ErrM +.187 ± .001 +.19 ± .001 +.187 ± .001 +.198 ± .001 +.191 ± .001 +.193 ± .001 +.192 ± .002 +.194 ± .001 +.189 ± .001 +.191 ± .001 +FPM +.069 ± .002 +.066 ± .002 +.071 ± .002 +.073 ± .002 +.071 ± .002 +.076 ± .002 +.073 ± .003 +.071 ± .002 +.072 ± .002 +.069 ± .002 +FNM +.119 ± .003 +.124 ± .002 +.116 ± .002 +.125 ± .003 +.12 ± .002 +.118 ± .003 +.119 ± .002 +.123 ± .002 +.117 ± .002 +.122 ± .003 +SCM +.975 +.975 +.974 +.974 +.975 +.973 +.973 +.977 +.974 +.973 +Table 10. Mean ± STD of each train/test split run using seed s for COMPAS (10a), computed across the B bootstrap replicates run for +each train/test split. We report total and subgroup g ∈ {NW, W} Error Rate, (Err) False Positive Rate (FP), and False Negative Rate +(FN). Mean Self Consistency (SC) is reported without STD, as it is computed once across each B bootstrap runs. +s +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Err +.328 ± .005 +.343 ± .005 +.339 ± .005 +.338 ± .005 +.344 ± .006 +.336 ± .004 +.339 ± .006 +.326 ± .005 +.325 ± .006 +.328 ± .005 +FP +.138 ± .008 +.149 ± .008 +.144 ± .008 +.131 ± .007 +.135 ± .007 +.135 ± .008 +.157 ± .008 +.137 ± .009 +.152 ± .009 +.14 ± .01 +FN +.189 ± .008 +.194 ± .007 +.195 ± .007 +.207 ± .008 +.209 ± .007 +.201 ± .008 +.182 ± .007 +.189 ± .009 +.173 ± .007 +.188 ± .009 +SC +.88 +.873 +.88 +.88 +.882 +.882 +.873 +.877 +.873 +.879 +ErrNW +.325 ± .006 +.353 ± .006 +.346 ± .006 +.339 ± .006 +.341 ± .007 +.333 ± .006 +.336 ± .007 +.327 ± .007 +.335 ± .008 +.322 ± .006 +FPNW +.149 ± .01 +.167 ± .008 +.147 ± .007 +.139 ± .008 +.139 ± .007 +.142 ± .009 +.164 ± .011 +.151 ± .011 +.163 ± .011 +.145 ± .01 +FNNW +.177 ± .01 +.186 ± .008 +.199 ± .008 +.2 ± .009 +.201 ± .008 +.191 ± .01 +.172 ± .009 +.176 ± .011 +.172 ± .009 +.177 ± .009 +SCNW +.875 +.872 +.882 +.877 +.881 +.881 +.873 +.871 +.873 +.877 +ErrW +.332 ± .008 +.323 ± .011 +.325 ± .01 +.336 ± .008 +.349 ± .008 +.342 ± .009 +.344 ± .009 +.325 ± .008 +.306 ± .009 +.341 ± .01 +FPW +.119 ± .011 +.114 ± .012 +.137 ± .015 +.115 ± .01 +.125 ± .013 +.123 ± .013 +.143 ± .014 +.109 ± .011 +.131 ± .013 +.132 ± .015 +FNW +.212 ± .011 +.209 ± .012 +.188 ± .01 +.221 ± .01 +.224 ± .01 +.219 ± .01 +.201 ± .012 +.217 ± .01 +.176 ± .011 +.209 ± .012 +SCW +.89 +.876 +.875 +.886 +.883 +.884 +.873 +.889 +.874 +.882 +Table 11. Excerpt of results for all logistic regression models produced to generate Figure 10b. We use a seeds {n}N to generate a +train/test split, and then use seeds {b}B to generate bootstrap replicates of the training dataset, which are used to produce models hDn,b, +with S = 10 and B = 100. That is, for 10 train/test splits, we sample 100 bootstrap replicates for training, which we evaluate on the +test set. We report total and subgroup g ∈ {Female, Male} = {F, M} Error Rate (Err), False Positive Rate (FP), and False Negative +Rate (FN). s represents the concrete seed used for the train/test split; b represents the concrete seed used for a specific bootstrap replicate +run (we use the default scheme of selecting n ∈ {1, 2, . . . , 10}; for each s, we use B = 100 seeds b ∈ {n × 1, n × 2, . . . , n × 100}). +s +b +Errtotal +FPtotal +FNtotal +ErrF +FPF +FNF +ErrM +FPM +FNM +1 +1 +0.151 +0.051 +0.1 +0.073 +0.017 +0.055 +0.188 +0.066 +0.121 +1 +2 +0.151 +0.055 +0.096 +0.074 +0.023 +0.052 +0.187 +0.07 +0.117 +1 +3 +0.151 +0.054 +0.097 +0.073 +0.018 +0.056 +0.188 +0.072 +0.116 +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +10 +980 +0.154 +0.053 +0.101 +0.08 +0.024 +0.057 +0.19 +0.067 +0.122 +10 +990 +0.154 +0.055 +0.099 +0.079 +0.023 +0.056 +0.19 +0.071 +0.119 +10 +1000 +0.154 +0.054 +0.1 +0.08 +0.023 +0.057 +0.19 +0.069 +0.121 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Decision trees. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Non-white +White +(a) COMPAS split by g = race +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female +Male +(b) Old Adult split by g = sex +Figure 11. Cumulative proportion of test instances that attain the given level of self-consistency. We perform classification with decision +tree classifiers, estimating ˆ +SC with B = 100 bootstrap replicates. We repeat for S = 10 train/test splits. +Table 12. Mean ± STD across S = 10 train/test splits ×B = 100 bootstrap-replicate training runs for decision tree classifiers on +COMPAS (g = 0 = NW; g = 1 = W) and Old Adult (g = 0 = F; g = 1 = M). These results correspond to Figure 11. +COMPAS +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.386 ± .005 .187 ± .008 .198 ± .007 .694 ± .005 +g = NW .388 ± .005 .188 ± .008 .199 ± .007 .698 ± .006 +g = W +.382 ± .014 .186 ± .012 +.196 ± .01 +.688 ± .008 +Old Adult +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.191 ± .003 .099 ± .002 .092 ± .002 .839 ± .003 +g = F +.108 ± .003 .057 ± .002 .051 ± .003 .903 ± .004 +g = M .231 ± .003 .119 ± .003 .112 ± .002 .809 ± .003 +MLPs. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Non-white +White +(a) COMPAS split by g = race +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female +Male +(b) Old Adult split by g = sex +Figure 12. Cumulative proportion of test instances that attain the given level of self-consistency for MLPs, estimating ˆ +SC with B = 100 +and repeating for for S = 10. Unfortunately, not all runs converge, despite substantially increasing the max iters. +Table 13. Mean ± STD across S = 10 train/test splits ×B = 100 bootstrap-replicate training runs for MLPs on Old Adult +(g = 0 = F; g = 1 = M) and COMPAS (g = 0 = NW; g = 1 = W). These results correspond to Figure 12. +COMPAS +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.398 ± .005 .195 ± .008 .203 ± .008 .718 ± .004 +g = NW .399 ± .006 .194 ± .007 .205 ± .009 .722 ± .005 +g = W +.397 ± .008 .197 ± .011 .201 ± .012 .712 ± .009 +Old Adult +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.196 ± .003 .094 ± .003 .102 ± .001 .852 ± .001 +g = F +.11 ± .003 +.056 ± .002 .054 ± .002 .911 ± .004 +g = M .237 ± .003 .113 ± .003 .124 ± .001 .824 ± .002 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +SVMs. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +Cumulative Proportion of Test Set +Non-white +White +(a) COMPAS split by g = race +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +0.18 +0.20 +Cumulative Proportion of Test Set +Female +Male +(b) Old Adult split by g = sex +Figure 13. Cumulative proportion of test instances that attain the given level of self-consistency with SVMs, estimating ˆ +SC with B = 100 +bootstrap replicates. We repeat with S = 10 test/train splits, and exclude perfectly self-consistent instances to highlight differences (i.e., +we plot (x, g) for which ˆ +SC < 1). +Table 14. Mean ± STD across S = 10 train/test splits ×B = 100 bootstrap-replicate training runs for MLPs on Old Adult +(g = 0 = F; g = 1 = M) and COMPAS (g = 0 = NW; g = 1 = W). These results correspond to Figure 12. +COMPAS +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.353 ± .007 .151 ± .012 .202 ± .009 .886 ± .015 +g = NW .356 ± .009 .159 ± .013 +.197 ± .01 +.887 ± .016 +g = W +.347 ± .007 .135 ± .015 .212 ± .013 .885 ± .015 +Old Adult +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.159 ± .002 +.05 ± .003 +.11 ± .002 +.962 ± .001 +g = F +.08 ± .004 +.021 ± .002 +.06 ± .003 +.981 ± .001 +g = M .196 ± .002 .063 ± .003 .133 ± .002 .954 ± .001 +Note that for ˆ +SC < 1, SVMs have difficulty consistently estimating self-consistency for COMPAS when S = 10. We will +see this be a common issue for South German Credit in the next subsection. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +E.4.2. SO U T H GERMAN CREDIT +We first present the results for logistic regression and decision trees together, and then results using MLPs and SVMs. We +close with results for random forests using different S. As above, we defer detailed discussion to Appendix E.4.6. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +Cumulative Proportion of Test Set +Female +Male +(a) Logistic regression +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female +Male +(b) MLPs +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female +Male +(c) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female +Male +(d) SVMs +Figure 14. Cumulative proportion of test instances that attain the given level of self-consistency for South German Credit (g = sex) +for (a) logistic regression, (b) MLPs, (c) decision tree classifiers, and (d) SVMs. For all, we estimate ˆ +SC with B = 100 bootstrap +replicates and repeat for S = 10 test/train splits. For logistic regression, we exclude perfectly self-consistent instances to highlight +differences (i.e., we plot (x, g) for which ˆ +SC < 1). +Table 15. Mean ± STD across S = 10 × B = 100 runs for South German Credit (Figure 14). +Model +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.242 ± .03 +.157 ± .019 .085 ± .024 +.898 ± .013 +Logistic regression +g = F +.223 ± .069 +.162 ± .062 .071 ± .037 +.901 ± .036 +g = M +.242 ± .029 +.155 ± .023 .087 ± .026 +.897 ± .011 +Total +.319 ± .018 +.14 ± .012 +.18 ± .012 +.681 ± .017 +Decision tree classifiers g = F +.307 ± .044 +.115 ± .04 +.192 ± .036 +.681 ± .032 +g = M +.321 ± .018 +.144 ± .017 .177 ± .015 +.681 ± .015 +Total +.297 ± .021 +.134 ± .016 .162 ± .014 +.759 ± .013 +MLPs +g = F +.276 ± .041 +.126 ± .051 .151 ± .031 0.762 ± .036 +g = M +.3 ± .023 +.136 ± .022 .164 ± .016 +.759 ± .013 +Total +.248 ± .024 +.148 ± .014 +.1 ± .021 +.873 ± .01 +SVMs +g = F +.229 ± 0.061 .161 ± .061 .069 ± .032 +.886 ± .024 +g = M +.251 ± .027 +.146 ± .019 .105 ± .022 +.87 ± .01 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Computing ˆ +SC does not have tight confidence intervals for S = 10. Our estimates for self-consistency are not consistent for +the Female subgroup; there are significant differences in ˆ +SC across test/train splits s. We attempt to tighten our estimates +by increasing S, and show the results for random forest classifiers. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female +Male +(a) S = 10 +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female +Male +(b) S = 50 +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female +Male +(c) S = 25 +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female +Male +(d) S = 100 +Figure 15. Cumulative proportion of test instances that attain the given level of self-consistency for South German Credit (g = sex) +for random forest classifiers. For all, B = 100 to estimate ˆ +SC, and we vary train/test split S ∈ {10, 25, 50, 100}. +Table 16. Mean ± STD across S ∈ {10, 25, 50, 100}×B = 100 runs for South German Credit, random forests (Figure 15). STD +is computed across each S runs +S +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.278 ± .021 +.16 ± .015 +.118 ± .012 +.76 ± .017 +10 +g = F +.258 ± .051 +.138 ± .048 +.119 ± .031 +.773 ± .032 +g = M +.281 ± .02 +.163 ± .018 +.117 ± .014 +.758 ± .015 +Total +.28 ± .023 +.166 ± .022 +.113 ± .013 +.769 ± .015 +25 +g = F +.279 ± .061 +.166 ± .063 +.113 ± .042 +.767 ± .033 +g = M +.279 ± .023 +.166 ± .023 +.113 ± .014 +0.769 ± .016 +Total +.278 ± .022 +.168 ± .027 +.11 ± .015 +.769 ± .015 +50 +g = F +.284 ± .058 +.171 ± .68 +.113 ± .043 +.764 ± .035 +g = M +.277 ± .022 +.167 ± .026 +.11 ± .016 +.77 ± .016 +Total +.281 ± .021 +.17 ± .026 +.111 ± .017 +.767 ± .017 +100 +g = F +.293 ± .06 +.182 ± .068 +.11 ± .039 +.763 ± .038 +g = M +.279 ± .022 +.169 ± .027 +.111 ± .019 +.768 ± .018 +These results (as well as SVMs on COMPAS, Figure 13a) show how metrics like average self-consistency do not always pro- +vide a clear picture. We can see in this table that estimates of the cumulative self-consistency level can vary significantly, while +overall estimates of the average self-consistency can remain fairly constant, as a function of different S. Similar to COMPAS, +fairness improves between subgroups as we train more models, with differences between subgroups effectively vanishing. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +E.4.3. TA I W A N CREDIT +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female (F) +Male (M) +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female (F) +Male (M) +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +Cumulative Proportion of Test Set +Female (F) +Male (M) +(c) Logistic regression +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female (F) +Male (M) +(d) MLPs +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +Cumulative Proportion of Test Set +Female (F) +Male (M) +(e) SVMs +Figure 16. Cumulative proportion of test instances that attain the given level of self-consistency for Taiwan Credit (g = sex). For +all, we estimate ˆ +SC with B = 100 bootstrap replicates and repeat on S = 10 test/train splits. For logistic regression and SVMs, we +exclude perfectly self-consistent instances to highlight differences (i.e., we plot (x, g) for which ˆ +SC < 1). +Table 17. Mean ± STD across S = 10 train/test splits × B = 100 bootstrap-replicate training runs on Taiwan Credit, g = sex ∈ +{F, M}. These results correspond to Figure 16. +Decision trees +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.277 ± .002 .146 ± .002 .131 ± .003 .726 ± .002 +g = F +.268 ± .002 .144 ± .002 .124 ± .004 .731 ± .003 +g = M .291 ± .003 +.15 ± .003 +.141 ± .002 .718 ± .002 +Random forests +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.216 ± .002 .076 ± .002 +.14 ± .003 +.857 ± .002 +g = F +.205 ± .003 .072 ± .002 .133 ± .004 .863 ± .002 +g = M .233 ± .004 .082 ± .003 +.15 ± .003 +.847 ± .003 +Logistic regression +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.19 ± .004 +.021 ± .002 +.17 ± .004 +0.989 ± 0 +g = F +.178 ± .005 .018 ± .001 +.16 ± .006 +.99 ± .001 +g = M +.209 ± .004 .025 ± .003 .184 ± .004 .988 ± .001 +MLPs +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.234 ± .002 .096 ± .002 .139 ± .003 +.83 ± .003 +g = F +.223 ± .003 .092 ± .003 .131 ± .004 .836 ± .004 +g = M .252 ± .003 .101 ± .003 +.15 ± .003 +.821 ± .003 +SVMs +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.182 ± .003 .032 ± .001 .151 ± .003 .986 ± .001 +g = F +.171 ± .004 .029 ± .001 .142 ± .004 .987 ± .001 +g = M +.199 ± .004 .036 ± .002 .163 ± .003 .984 ± .001 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +E.4.4. NE W AD U LT - CA +We present the results for binarized sex and race separately, in two panels each with the 3 model types we examine +for this dataset – decision tree classifiers, random forest classifiers, and logistic regression. As above, we defer detailed +discussion to Appendix E.4.6. +Income. +By sex. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female +Male +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female +Male +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.0025 +0.0075 +0.0125 +0.0175 +0.0225 +Cumulative Proportion of Test Set +Female +Male +(c) Logistic regression +Figure 17. Cumulative proportion of test instances that attain the given level of self-consistency for New Adult - CA - Income +(g = sex). For all, we estimate ˆ +SC with B = 100 bootstrap replicates and repeat on S = 10 test/train splits. For logistic regression, we +exclude perfectly self-consistent instances to highlight differences (i.e., we plot (x, g) for which ˆ +SC < 1). +Table 18. Mean ± STD across S = 10 train/test splits × B = 100 bootstrap-replicate training runs on New Adult - CA - Income, +g = sex ∈ {F, M}. These results correspond to Figure 17. +Decision trees +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.252 ± .001 .127 ± .001 .124 ± .001 .764 ± .001 +g = F +.244 ± .001 .135 ± .001 .109 ± .001 .769 ± .001 +g = M .258 ± .001 .121 ± .001 .138 ± .001 +.76 ± .001 +Random forests +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.218 ± .001 .107 ± .001 +.111 ± 0 +.839 ± .001 +g = F +.21 ± .001 +.114 ± .001 .096 ± .001 .842 ± .001 +g = M .224 ± .001 +.1 ± .001 +.124 ± .001 .836 ± .001 +Logistic regression +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.22 ± .001 +.104 ± .001 .116 ± .001 .995 ± 0 +g = F +.215 ± .002 .125 ± .002 +.09 ± .001 +.995 ± 0 +g = M +.224 ± .002 .085 ± .001 .139 ± .001 .996 ± 0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +By race. +For these results we binarize the protected attribute into two groups: “Non-white” (0) and “White” (1). +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Non-white +White +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Non-white +White +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.0025 +0.0050 +0.0075 +0.0100 +0.0125 +0.0150 +0.0175 +0.0200 +0.0225 +0.0250 +Cumulative Proportion of Test Set +Non-white +White +(c) Logistic regression +Figure 18. Cumulative proportion of test instances that attain the given level of self-consistency for New Adult - CA - Income +(g = race). For all, we estimate ˆ +SC with B = 100 bootstrap replicates and repeat on S = 10 test/train splits. For logistic regression, we +exclude perfectly self-consistent instances to highlight differences (i.e., we plot (x, g) for which ˆ +SC < 1). +Table 19. Mean ± STD across S = 10 train/test splits × B = 100 bootstrap-replicate training runs on New Adult - CA - Income, +g = race ∈ {NW, W}. These results correspond to Figure 18. +Decision trees +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.252 ± .001 .127 ± .001 .124 ± .001 .764 ± .001 +g = NW .245 ± .001 .129 ± .001 .116 ± .001 .771 ± .001 +g = W +.256 ± .001 .127 ± .001 +.13 ± .001 +.76 ± .001 +Random forests +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.218 ± .001 .107 ± .001 +.111 ± 0 +.839 ± .001 +g = NW +.21 ± .001 +.103 ± .001 .107 ± .001 .846 ± .001 +g = W +.222 ± .001 .109 ± .001 .114 ± .001 .835 ± .001 +Logistic regression +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.22 ± .001 +.104 ± .001 .116 ± .001 .995 ± 0 +g = NW +.216 ± .002 +.1 ± .002 +.116 ± .002 .995 ± 0 +g = W +.222 ± .001 .106 ± .001 .116 ± .001 .995 ± 0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Employment. +By sex. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.05 +0.15 +0.25 +0.35 +0.45 +0.55 +Cumulative Proportion of Test Set +Female +Male +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.05 +0.15 +0.25 +0.35 +0.45 +0.55 +Cumulative Proportion of Test Set +Female +Male +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.0025 +0.0050 +0.0075 +0.0100 +0.0125 +0.0150 +0.0175 +0.0200 +0.0225 +Cumulative Proportion of Test Set +Female +Male +(c) Logistic regression +Figure 19. Cumulative proportion of test instances that attain the given level of self-consistency for New Adult - CA - +Employment (g = sex). For all, we estimate ˆ +SC with B = 100 bootstrap replicates and repeat on S = 10 test/train splits. +We exclude perfectly self-consistent instances to highlight differences (i.e., we plot (x, g) for which ˆ +SC < 1). +Table 20. Mean ± STD across S = 10 train/test splits × B = 100 bootstrap-replicate training runs on New Adult - CA - +Employment, g = sex ∈ {F, M}. These results correspond to Figure 19. +Decision trees +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.225 ± .001 .135 ± .001 +.09 ± .001 +.87 ± .001 +g = F +.248 ± .001 .165 ± .001 .083 ± .001 +.87 ± .001 +g = M +.2 ± .001 +.104 ± .001 .096 ± .001 .871 ± .001 +Random forests +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.216 ± .001 .125 ± .001 .091 ± .001 .884 ± .001 +g = F +.24 ± .001 +.155 ± .001 .085 ± .001 .882 ± .001 +g = M .192 ± .001 .095 ± .001 .097 ± .001 .885 ± .001 +Logistic regression +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.233 ± .001 .161 ± .001 .072 ± .001 .996 ± 0 +g = F +.258 ± .001 .201 ± .001 .057 ± .001 .995 ± 0 +g = M +.208 ± .001 +.12 ± .001 +.088 ± .001 .996 ± 0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +By race. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +Cumulative Proportion of Test Set +Non-white +White +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +Cumulative Proportion of Test Set +Non-white +White +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.0025 +0.0050 +0.0075 +0.0100 +0.0125 +0.0150 +0.0175 +0.0200 +0.0225 +Cumulative Proportion of Test Set +Female +Male +(c) Logistic regression +Figure 20. Cumulative proportion of test instances that attain the given level of self-consistency for New Adult - CA - +Employment (g = race). For all, we estimate ˆ +SC with B = 100 bootstrap replicates and repeat on S = 10 test/train splits. +We exclude perfectly self-consistent instances to highlight differences (i.e., we plot (x, g) for which ˆ +SC < 1). +Table 21. Mean ± STD across S = 10 train/test splits × B = 100 bootstrap-replicate training runs on New Adult - CA - +Employment, g = race ∈ {NW, W}. These results correspond to Figure 20. +Decision trees +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.225 ± .001 .135 ± .001 +.09 ± .001 +.87 ± .001 +g = NW .221 ± .001 .135 ± .001 .086 ± .001 .871 ± .001 +g = W +.227 ± .001 .134 ± .001 .092 ± .001 +.87 ± .001 +Random forests +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.216 ± .001 .125 ± .001 .091 ± .001 .884 ± .001 +g = NW .213 ± .001 .127 ± .002 .086 ± .001 .883 ± .001 +g = W +.218 ± .001 .124 ± .001 .094 ± .001 .884 ± .001 +Logistic regression +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.233 ± .001 .161 ± .001 .072 ± .001 .996 ± 0 +g = NW +.233 ± .001 .166 ± .001 .066 ± .001 .996 ± 0 +g = W +.234 ± .001 .158 ± .001 .076 ± .001 .995 ± 0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Public Coverage. +By sex. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female +Male +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female +Male +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.0025 +0.0075 +0.0125 +0.0175 +0.0225 +0.0275 +0.0325 +0.0375 +Cumulative Proportion of Test Set +Female +Male +(c) Logistic regression +Figure 21. Cumulative proportion of test instances that attain the given level of self-consistency for New Adult - CA - Public +Coverage (g = sex). For all, we estimate ˆ +SC with B = 100 bootstrap replicates and repeat on S = 10 test/train splits. We exclude +perfectly self-consistent instances to highlight differences (i.e., we plot (x, g) for which ˆ +SC < 1). +Table 22. Mean ± STD across S = 10 train/test splits × B = 100 bootstrap-replicate training runs on New Adult - CA - Public +Coverage, g = sex ∈ {F, M}. These results correspond to Figure 21. +Decision trees +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.351 ± .001 .175 ± .001 .177 ± .002 +.72 ± .001 +g = F +.351 ± .002 .176 ± .001 .174 ± .002 +.72 ± .001 +g = M .352 ± .001 .172 ± .002 +.18 ± .002 +.721 ± .002 +Random forests +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.325 ± .001 .143 ± .001 .182 ± .002 .775 ± .001 +g = F +.323 ± .002 .144 ± .001 .179 ± .002 .775 ± .001 +g = M .327 ± .001 .142 ± .002 .186 ± .003 .776 ± .002 +Logistic regression +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.316 ± .002 .055 ± .001 .261 ± .002 .993 ± .0 +g = F +.312 ± .003 .055 ± .001 .257 ± .003 .993 ± .0 +g = M .321 ± .002 .055 ± .001 .266 ± .002 .993 ± .0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +By race. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Non-white +White +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Non-white +White +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.0025 +0.0075 +0.0125 +0.0175 +0.0225 +0.0275 +0.0325 +0.0375 +Cumulative Proportion of Test Set +Non-white +White +(c) Logistic regression +Figure 22. Cumulative proportion of test instances that attain the given level of self-consistency for New Adult - CA - Public +Coverage (g = race). For all, we estimate ˆ +SC with B = 100 bootstrap replicates and repeat on S = 10 test/train splits. We exclude +perfectly self-consistent instances to highlight differences (i.e., we plot (x, g) for which ˆ +SC < 1). +Table 23. Mean ± STD across S = 10 train/test splits × B = 100 bootstrap-replicate training runs on New Adult - CA - Public +Coverage, g = race ∈ {NW, W}. These results correspond to Figure 22. +Decision trees +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.351 ± .001 .175 ± .001 .177 ± .002 +.72 ± .001 +g = NW .364 ± .001 .177 ± .001 .187 ± .002 .713 ± .001 +g = W +.342 ± .001 .173 ± .002 .169 ± .002 .725 ± .001 +Random forests +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.325 ± .001 .143 ± .001 .182 ± .002 .775 ± .001 +g = NW +.34 ± .002 +.146 ± .002 .194 ± .002 .766 ± .001 +g = W +.313 ± .001 .141 ± .002 .172 ± .002 .782 ± .001 +Logistic regression +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.316 ± .002 .055 ± .001 .261 ± .002 .993 ± .0 +g = NW .334 ± .003 .056 ± .001 .278 ± .003 .993 ± .0 +g = W +.302 ± .003 .054 ± .001 .248 ± .002 .994 ± .0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +E.4.5. HMDA +We present the results for binarized sex, race, and ethnicity separately, in three panels each with the 3 model types +we examine for this dataset – decision tree classifiers, random forest classifiers, and logistic regression. As above, we defer +detailed discussion to Appendix E.4.6. +NY - 2017. +By sex. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +Cumulative Proportion of Test Set +Female (F) +Male (M) +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +Cumulative Proportion of Test Set +Female (F) +Male (M) +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +Cumulative Proportion of Test Set +Female (F) +Male (M) +(c) Logistic regression +Figure 23. Cumulative proportion of test instances that attain the given level of self-consistency for HMDA - NY - 2017 (g = sex). +For all, we estimate ˆ +SC with B = 100 bootstrap replicates and repeat on S = 10 test/train splits. We exclude perfectly self-consistent +instances to highlight differences (i.e., we plot (x, g) for which ˆ +SC < 1). +Table 24. Mean ± STD across S = 10 train/test splits × B = 100 bootstrap-replicate training runs on HMDA - NY - 2017, +g = sex ∈ {F, M}. These results correspond to Figure 23. +Decision trees +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.202 ± .001 +.1 ± .001 +.102 ± .001 .807 ± .001 +g = F +.205 ± .001 .101 ± .002 .104 ± .001 .803 ± .001 +g = M .201 ± .001 .099 ± .001 .101 ± .001 .809 ± .001 +Random forests +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.181 ± .001 .094 ± .001 .087 ± .001 .856 ± .001 +g = F +.184 ± .002 .094 ± .002 .089 ± .001 .853 ± .001 +g = M +.18 ± .001 +.094 ± .001 .085 ± .001 .858 ± .001 +Logistic regression +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.175 ± .001 .115 ± .001 +.06 ± .001 +.991 ± .001 +g = F +.178 ± .002 .116 ± .002 .062 ± .001 .991 ± .001 +g = M .173 ± .001 .114 ± .001 .059 ± .001 .991 ± .001 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +By race. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 +Cumulative Proportion of Test Set +Non-white (NW) +White (W) +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 +Cumulative Proportion of Test Set +Non-white (NW) +White (W) +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +Cumulative Proportion of Test Set +Non-white (NW) +White (W) +(c) Logistic regression +Figure 24. Cumulative proportion of test instances that attain the given level of self-consistency for HMDA - NY - 2017 (g = race). +For all, we estimate ˆ +SC with B = 100 bootstrap replicates and repeat on S = 10 test/train splits. We exclude perfectly self-consistent +instances to highlight differences (i.e., we plot (x, g) for which ˆ +SC < 1). +Table 25. Mean ± STD across S = 10 train/test splits × B = 100 bootstrap-replicate training runs on HMDA - NY - 2017, +g = race ∈ {NW, W}. These results correspond to Figure 24. +Decision trees +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.202 ± .001 +.1 ± .001 +.102 ± .001 .807 ± .001 +g = NW .227 ± .001 +.12 ± .002 +.107 ± .001 .786 ± .002 +g = W +.196 ± .001 .095 ± .001 .101 ± .001 .813 ± .001 +Random forests +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.181 ± .001 .094 ± .001 .087 ± .001 .856 ± .001 +g = NW .207 ± .001 .109 ± .002 .098 ± .001 .834 ± .002 +g = W +.174 ± .001 .091 ± .001 .084 ± .001 .862 ± .001 +Logistic regression +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.175 ± .001 .115 ± .001 +.06 ± .001 +.991 ± .001 +g = NW .197 ± .002 .113 ± .002 .084 ± .002 +.99 ± .001 +g = W +.169 ± .001 .115 ± .001 .054 ± .001 .991 ± .001 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +By ethnicity. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 +Cumulative Proportion of Test Set +Hispanic or Latino (HL) +Not Hispanic or Latino (NHL) +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +Cumulative Proportion of Test Set +Hispanic or Latino (HL) +Not Hispanic or Latino (NHL) +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +Cumulative Proportion of Test Set +Hispanic or Latino (HL) +Not Hispanic or Latino (NHL) +(c) Logistic regression +Figure 25. Cumulative proportion of test instances that attain the given level of self-consistency for HMDA - NY - 2017 (g = +ethnicity). For all, we estimate ˆ +SC with B = 100 bootstrap replicates and repeat on S = 10 test/train splits. We exclude perfectly +self-consistent instances to highlight differences (i.e., we plot (x, g) for which ˆ +SC < 1). +Table 26. Mean ± STD across S = 10 train/test splits × B = 100 bootstrap-replicate training runs on HMDA - NY - 2017, +g = ethnicity ∈ {HL, NHL}. These results correspond to Figure 25. +Decision trees +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.202 ± .001 +.1 ± .001 +.102 ± .001 .807 ± .001 +g = HL +.206 ± .002 .115 ± .002 .091 ± .002 .804 ± .002 +g = NHL .202 ± .001 .099 ± .001 .103 ± .001 .807 ± .001 +Random forests +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.181 ± .001 .094 ± .001 .087 ± .001 .856 ± .001 +g = HL +.191 ± .002 .101 ± .002 +.09 ± .003 +.843 ± .002 +g = NHL +.18 ± .001 +.094 ± .001 .086 ± .001 .857 ± .001 +Logistic regression +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.175 ± .001 .115 ± .001 +.06 ± .001 +.991 ± .001 +g = HL +.187 ± .003 .107 ± .002 +.08 ± .003 +.99 ± .001 +g = NHL .174 ± .001 .115 ± .001 .058 ± .001 .991 ± .001 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +TX - 2017. +We present the results for binarized sex, race, and ethnicity separately, in three panels each with the 3 model types +we examine for this dataset – decision tree classifiers, random forest classifiers, and logistic regression. As above, we defer +detailed discussion to Appendix E.4.6. +By sex. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +Cumulative Proportion of Test Set +Female (F) +Male (M) +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +Cumulative Proportion of Test Set +Female (F) +Male (M) +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +0.012 +0.014 +0.016 +0.018 +0.020 +0.022 +Cumulative Proportion of Test Set +Female (F) +Male (M) +(c) Logistic regression +Figure 26. Cumulative proportion of test instances that attain the given level of self-consistency for HMDA - TX - 2017 (g = sex). +For all, we estimate ˆ +SC with B = 100 bootstrap replicates and repeat on S = 10 test/train splits. We exclude perfectly self-consistent +instances to highlight differences (i.e., we plot (x, g) for which ˆ +SC < 1). +Table 27. Mean ± STD across S = 10 train/test splits × B = 100 bootstrap-replicate training runs on HMDA - TX - 2017, +g = sex ∈ {F, M}. These results correspond to Figure 26. +Decision trees +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.164 ± .0 +.083 ± .0 +.082 ± .001 +.843 ± .0 +g = F +.172 ± .001 .088 ± .001 .084 ± .001 .837 ± .001 +g = M +.161 ± .0 +.08 ± .0 +.081 ± .0 +.846 ± .0 +Random forests +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.148 ± .0 +.072 ± .0 +.076 ± .001 .883 ± .001 +g = F +.155 ± .001 .074 ± .001 .081 ± .001 .877 ± .001 +g = M +.145 ± .0 +.071 ± .0 +.074 ± .001 .885 ± .001 +Logistic regression +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.149 ± .001 .061 ± .001 .089 ± .001 .996 ± .0 +g = F +.157 ± .002 .058 ± .001 .098 ± .002 .996 ± .0 +g = M +.146 ± .0 +.062 ± .001 .084 ± .001 .996 ± .0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +By race. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +Cumulative Proportion of Test Set +Non-white (NW) +White (W) +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +Cumulative Proportion of Test Set +Non-white (NW) +White (W) +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +0.012 +0.014 +0.016 +0.018 +0.020 +0.022 +Cumulative Proportion of Test Set +Non-white (NW) +White (W) +(c) Logistic regression +Figure 27. Cumulative proportion of test instances that attain the given level of self-consistency for HMDA - TX - 2017 (g = race). +For all, we estimate ˆ +SC with B = 100 bootstrap replicates and repeat on S = 10 test/train splits. We exclude perfectly self-consistent +instances to highlight differences (i.e., we plot (x, g) for which ˆ +SC < 1). +Table 28. Mean ± STD across S = 10 train/test splits × B = 100 bootstrap-replicate training runs on HMDA - TX - 2017, +g = race ∈ {NW, W}. These results correspond to Figure 27. +Decision trees +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.164 ± .0 +.083 ± .0 +.082 ± .001 +.843 ± .0 +g = NW .164 ± .001 .092 ± .001 .072 ± .001 .843 ± .001 +g = W +.164 ± .0 +.08 ± .001 +.084 ± .001 .843 ± .001 +Random forests +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.148 ± .0 +.072 ± .0 +.076 ± .001 .883 ± .001 +g = NW .148 ± .001 .078 ± .001 .069 ± .001 .882 ± .001 +g = W +.148 ± .001 .071 ± .001 .077 ± .001 .883 ± .001 +Logistic regression +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.149 ± .001 .061 ± .001 .089 ± .001 .996 ± .0 +g = NW .149 ± .002 +.07 ± .002 +.079 ± .001 .996 ± .0 +g = W +.149 ± .001 .059 ± .001 .091 ± .001 .996 ± .0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +By ethnicity. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +Cumulative Proportion of Test Set +Hispanic or Latino (HL) +Not Hispanic or Latino (NHL) +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +Cumulative Proportion of Test Set +Hispanic or Latino (HL) +Not Hispanic or Latino (NHL) +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Self-Consistency +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +0.012 +0.014 +0.016 +0.018 +0.020 +0.022 +0.024 +Cumulative Proportion of Test Set +Hispanic or Latino (HL) +Not Hispanic or Latino (NHL) +(c) Logistic regression +Figure 28. Cumulative proportion of test instances that attain the given level of self-consistency for HMDA - TX - 2017 (g = +ethnicity). For all, we estimate ˆ +SC with B = 100 bootstrap replicates and repeat on S = 10 test/train splits. We exclude perfectly +self-consistent instances to highlight differences (i.e., we plot (x, g) for which ˆ +SC < 1). +Table 29. Mean ± STD across S = 10 train/test splits × B = 100 bootstrap-replicate training runs on HMDA - TX - 2017, +g = ethnicity ∈ {HL, NHL}. These results correspond to Figure 28. +Decision trees +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.164 ± .0 +.083 ± .0 +.082 ± .001 +.843 ± .0 +g = HL +.192 ± .001 .101 ± .001 .091 ± .001 .818 ± .001 +g = NHL .155 ± .001 +.077 ± .0 +.079 ± .0 +.852 ± .0 +Random forests +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.148 ± .0 +.072 ± .0 +.076 ± .001 .883 ± .001 +g = HL +.173 ± .001 .085 ± .001 .089 ± .001 .862 ± .001 +g = NHL +.14 ± .001 +.068 ± .0 +.072 ± .0 +.889 ± .001 +Logistic regression +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.149 ± .001 .061 ± .001 .089 ± .001 .996 ± .0 +g = HL +.175 ± .002 .056 ± .001 .119 ± .002 .995 ± .0 +g = NHL .141 ± .001 .062 ± .001 .079 ± .001 .996 ± .0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +E.4.6. EXTENDED DISCUSSION OF ILLUSTRATIVE EXAMPLES OF SELF-CONSISTENCY +With a few exceptions, such as Old Adult (Appendix E.4.1 and some HMDA by g = ethnicity results,, we were +surprised to find that our bootstrap estimations for expected error are almost all within a couple of percentage points of parity +across demographic groups. That is, we bootstrap with B = 100 replicates and use standard hyperparameter settings (e.g., +see Chen et al. (2018); Hardt et al. (2016); Barocas & Selbst (2016)) and computed both expected err +ˆ +Err, and underlying +ˆ +FPR and +ˆ +FNR and found greater parity than expected on almost all tasks. +This is not to say that all models performed equally well, in terms of their respective expected error results. It is common +in algorithmic fairness literature to see comments like those in Chen et al. (2018), which imply that results across logistic +regression, decision trees, and random forests are roughly similar. Perhaps this is true when sampling a few models and +doing cross validation (we investigate this more with respect to COMPAS in Section 5 and Appendix E.6), but our results +generally do not align with this finding (this is perhaps unsurprising, given the lower bias of logistic regression, but does not +cohere with the common wisdom we read in the literature). Overall, accuracy rates might be within a couple of percentage +points across models, but the underlying sources of error are different. Logistic regression tends to be quite low variance, +except on small datasets like COMPAS (Appendix E.4.1) and South German Credit (Appendix E.4.2). Generally +speaking random forests and decision trees exhibit much higher variance than logistic regression. This is clear in all of the +ˆ +SC plots above. SVMs and MLPs are significantly computationally more expensive, but do not yield significantly better +results than logistic regression or random forests, so we forgo running them on larger datasets (See Table 31 below). We did +not select for CPUs with any particular features, and thus the run times are quite variable. We therefore provide these results +just to give a general sense of run time. +Table 30. The run-times (hh:mm:ss) for our S = 10, B = 100 illustrative examples in Appendix E.4. These times are recorded for our +cluster environment, described in Appendix E.2. *Run in S parallel chunks and summed for each train/test split s. +Dataset +g +Logistic regression Decision trees Random forests MLPs +SVMs +South German Credit sex +00:00:32 +00:00:24 +00:00:31 +00:01:44 +00:03:16 +COMPAS +race +00:01:44 +00:01:13 +00:01:39 +07:51:21* 04:07:33* +Old Adult +sex +00:02:44 +00:02:33 +00:02:56 +14:27:36* 37:33:47* +Taiwan Credit +sex +00:02:12 +00:06:24 +00:05:15 +10:03:51* 27:57:58* +New Adult - CA +Income +sex, race +00:02:48 +00:08:45 +00:11:07 +Employment +sex, race +00:08:25 +00:12:37 +00:20:23 +Public Coverage +sex, race +00:03:48 +00:05:31 +00:09:49 +HMDA +NY +sex, race, ethnicity 00:08:39 +00:17:20 +00:17:14 +TX +sex, race, ethnicity 00:21:39 +00:38:26 +00:47:40 +In short, these experiments together indicate that similar error rates can conceal underlying patterns regarding sources of +error. While logistic regression is low variance, decision trees and random forests present a clear opportunity to investigate +variance reduction techniques. When we applied our algorithm to improve self-consistency, we did in fact see that variance +reduction can in general improve the performance of random forests and decision trees in comparison to logistic regression, +which it seems becomes limited by inherent bias-induced error (we hypothesize this, but do not try to estimate it directly). +Again, none of these results are perhaps surprising in the general case. Nevertheless, they diverge from current practices in +fair classification literature, which tend to train 3-5 of these hypothesis classes and conclude roughly similar performance. +After these initial results, we expected that our bagging with confidence approach could lead to significant improves in +accuracy. We did not expect major changes in fairness, given that, generally speaking, the expected subgroup-conditional +error rates were also closer together than we anticipated. +As we discuss in the main paper, these initial illustrations suggest normative concerns for algorithmic fairness that extend +beyond measuring and comparing false positive and false negative rates — whether those comparisons are done for one +model or across multiple possible models. Measuring self-consistency of a learning process provides more nuance about +what is happening under-the-hood of summary metrics like model misclassification rates. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +E.5. Validating our algorithm in practice +We provide comprehensive results to put in conversation with those in Appendix E.4 concerning ensembling with confidence +and super-ensembling with confidence. For all experiments, we set the ˆ +SC confidence level to be κ = .75. Before going into +the detailed results, we provide some context on the use of the Wasserstein-1 distance. +E.5.1. MEASURING THE DISTANCE BETWEEN EMPIRICAL SELF-CONSISTENCY CURVES +To accompany our visualizations of increased ˆ +SC, we can depict changes by measuring the Wasserstein-1 distance between +each set of curves (Ramdas et al., 2015), before and after, and computing the difference. This is the natural distance for +us to consider because it has a closed form when being applied to CDFs, which are effectively what we have plotted. For +each subgroup g, we can consider their ˆ +SC CDF curves to be A, B with probability measures a, b; we measure the change +between each set of curves as the distance over all possible self-consistency values κ with +W1(a, b) = +� +R +|A(κ) − B(κ)|dκ. +For self-consistency, which we have defined on [0.5, 1], this is just +W1(a, b) = +� 1 +0.5 +|A(κ) − B(κ)|dκ, +with a bit of a buffer to the left of the minimum to account for the fact that our empirical approximations ˆ +SC fall just shy of +SC. Practically, this just means we iterate over the y-values we have collected for our two CDF curves, sum up their absolute +differences, and then scale by 1 +z , where z is the number of empirical measurements that that we have for κ, at which both A +and B are defined. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +E.5.2. COMPAS +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +NW - before +W - before +NW - after +W - after +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +NW - before +W - before +NW - after +W - after +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +NW - before +W - before +NW - after +W - after +(c) Logistic regression +Figure 29. Cumulative proportion of test instances that attain the given level of self-consistency for COMPAS (g = race). For all, we +estimate ˆ +SC with 101 bags in the super-ensemble, and 51 underlying individual models in each bag. We repeat for 10 test/train splits. +Ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.235 ± .008 +.358 ± .027 +.091 ± .017 +.144 ± .014 +.629 ± .009 +g = F +.24 ± .009 +.434 ± .031 +.104 ± .018 +.136 ± .01 +.623 ± .018 +g = M +.223 ± .016 +.205 ± .014 +.064 ± .014 +.159 ± .023 +.642 ± .028 +Super-ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.324 ± .004 +.432 ± .013 +.153 ± .012 +.171 ± .012 +.131 ± .007 +g = F +.328 ± .004 +.485 ± .012 +.161 ± .009 +.166 ± .013 +.123 ± .008 +g = M +.316 ± .012 +.327 ± .017 +.136 ± .018 +.18 ± .012 +.146 ± .009 +Ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.236 ± .005 +.372 ± .017 +.089 ± .012 +.147 ± .016 +.538 ± .005 +g = F +.233 ± .008 +.456 ± .017 +.1 ± .013 +.134 ± .012 +.539 ± .016 +g = M +.239 ± .016 +.213 ± .013 +.068 ± .013 +.171 ± .024 +.536 ± .022 +Super-ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.315 ± .004 +.415 ± .009 +.138 ± .01 +.177 ± .014 +.107 ± .006 +g = F +.323 ± .004 +.478 ± .009 +.153 ± .012 +.169 ± .012 +.106 ± .005 +g = M +.302 ± .014 +.292 ± .014 +.108 ± .008 +.193 ± .02 +.108 ± .012 +Ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.283 ± .013 +.373 ± .012 +.104 ± .009 +.179 ± .022 +.228 ± .013 +g = F +.279 ± .009 +.438 ± .011 +.114 ± .01 +.165 ± .019 +.232 ± .013 +g = M +.29 ± .024 +.251 ± .016 +.084 ± .01 +.205 ± .03 +.221 ± .022 +Super-ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.313 ± .011 +.389 ± .006 +.123 ± .006 +.19 ± .015 +.041 ± .003 +g = F +.31 ± .01 +.442 ± .007 +.129 ± .008 +.18 ± .013 +.043 ± .005 +g = M +.319 ± .021 +.286 ± .006 +.111 ± .006 +.208 ± .021 +.038 ± .005 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +E.5.3. OL D AD U LT +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +(c) Logistic regression +Figure 30. Cumulative proportion of test instances that attain the given level of self-consistency for Old Adult (g = sex). For all, we +estimate ˆ +SC with 101 bags in the super-ensemble, and 51 underlying individual models in each bag. We repeat for 10 test/train splits. +Ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.067 ± .002 +.157 ± .002 +.019 ± .001 +.048 ± .002 +.313 ± .004 +g = F +.03 ± .002 +.045 ± .003 +.004 ± .0 +.027 ± .002 +.182 ± .004 +g = M +.09 ± .003 +.227 ± .002 +.029 ± .001 +.061 ± .002 +.377 ± .006 +Super-ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.138 ± .002 +.211 ± .003 +.055 ± .001 +.083 ± .001 +.058 ± .002 +g = F +.066 ± .004 +.078 ± .005 +.019 ± .003 +.047 ± .001 +.034 ± .002 +g = M +.175 ± .002 +.277 ± .004 +.073 ± .001 +.102 ± .002 +.069 ± .003 +Ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.083 ± .002 +.155 ± .004 +.023 ± .001 +.06 ± .002 +.228 ± .004 +g = F +.042 ± .002 +.048 ± .002 +.007 ± .001 +.035 ± .001 +.112 ± .003 +g = M +.107 ± .003 +.219 ± .005 +.032 ± .002 +.076 ± .003 +.284 ± .007 +Super-ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.138 ± .002 +.204 ± .004 +.05 ± .002 +.088 ± .001 +.043 ± .001 +g = F +.066 ± .002 +.077 ± .003 +.017 ± .002 +.049 ± .002 +.02 ± .003 +g = M +.173 ± .002 +.267 ± .005 +.067 ± .003 +.107 ± .003 +.054 ± .002 +Ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.142 ± .003 +.19 ± .005 +.045 ± .003 +.097 ± .001 +.036 ± .001 +g = F +.07 ± .003 +.071 ± .004 +.016 ± .001 +.054 ± .002 +.018 ± .002 +g = M +.178 ± .004 +.249 ± .005 +.06 ± .003 +.119 ± .001 +.044 ± .002 +Super-ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.152 ± .003 +.198 ± .004 +.05 ± .002 +.102 ± .001 +.006 ± .001 +g = F +.075 ± .002 +.076 ± .004 +.018 ± .001 +.057 ± .002 +.003 ± .001 +g = M +.189 ± .004 +.257 ± .005 +.066 ± .003 +.123 ± .001 +.008 ± .001 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +E.5.4. SO U T H GERMAN CREDIT +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +(c) Logistic regression +Figure 31. Cumulative proportion of test instances that attain the given level of self-consistency for South German Credit (g = +sex). For all, we estimate ˆ +SC with 101 bags in the super-ensemble, and 51 underlying individual models in each bag. We repeat for 50 +test/train splits. +Ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.095 ± .027 +.98 ± .01 +.089 ± .027 +.006 ± .006 +.652 ± .031 +g = F +.098 ± .08 +.996 ± .008 +.096 ± .079 +.002 ± .007 +.652 ± .06 +g = M +.095 ± .028 +.977 ± .012 +.088 ± .028 +.006 ± .007 +.652 ± .034 +Super-ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.203 ± .032 +.847 ± .024 +.148 ± .03 +.054 ± .017 +.175 ± .03 +g = F +.203 ± .082 +.879 ± .06 +.152 ± .07 +.052 ± .051 +.199 ± .059 +g = M +.202 ± .033 +.842 ± .023 +.148 ± .031 +.054 ± .015 +.171 ± .033 +Ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.119 ± .031 +.983 ± .009 +.115 ± .031 +.004 ± .004 +.474 ± .037 +g = F +.119 ± .078 +.999 ± .004 +.118 ± .078 +.001 ± .003 +.475 ± .067 +g = M +.119 ± .031 +.98 ± .01 +.115 ± .031 +.004 ± .004 +.474 ± .039 +Super-ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.215 ± .029 +.903 ± .019 +.186 ± .029 +.029 ± .013 +.089 ± .023 +g = F +.235 ± .096 +.945 ± .04 +.205 ± .091 +.03 ± .035 +.099 ± .057 +g = M +.211 ± .029 +.896 ± .019 +.183 ± .029 +.029 ± .012 +.087 ± .024 +Ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.188 ± .025 +.881 ± .028 +.149 ± .026 +.039 ± .015 +.212 ± .026 +g = F +.228 ± .087 +.96 ± .041 +.196 ± .084 +.032 ± .035 +.208 ± .066 +g = M +.182 ± .026 +.869 ± .029 +.142 ± .026 +.04 ± .016 +.212 ± .027 +Super-ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.226 ± .025 +.833 ± .031 +.164 ± .025 +.062 ± .02 +.04 ± .012 +g = F +.255 ± .074 +.917 ± .05 +.211 ± .078 +.044 ± .038 +.041 ± .031 +g = M +.221 ± .027 +.82 ± .033 +.156 ± .024 +.065 ± .02 +.04 ± .013 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +E.5.5. TA I W A N CREDIT +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +(c) Logistic regression +Figure 32. Cumulative proportion of test instances that attain the given level of self-consistency for Taiwan Credit (g = sex). For +all, we estimate ˆ +SC with 101 bags in the super-ensemble, and 41 underlying individual models in each bag. We repeat for 10 test/train +splits. +Ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.102 ± .003 +.035 ± .003 +.007 ± .002 +.095 ± .003 +.58 ± .005 +g = F +.096 ± .004 +.03 ± .004 +.006 ± .001 +.09 ± .004 +.567 ± .006 +g = M +.112 ± .004 +.042 ± .004 +.008 ± .002 +.103 ± .005 +.599 ± .005 +Super-ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.163 ± .002 +.107 ± .004 +.033 ± .002 +.13 ± .002 +.072 ± .001 +g = F +.153 ± .003 +.099 ± .003 +.03 ± .001 +.123 ± .003 +.067 ± .002 +g = M +.178 ± .004 +.119 ± .005 +.037 ± .003 +.142 ± .004 +.08 ± .001 +Ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.127 ± .002 +.045 ± .002 +.01 ± .001 +.117 ± .002 +.256 ± .008 +g = F +.12 ± .004 +.041 ± .003 +.009 ± .001 +.11 ± .003 +.24 ± .008 +g = M +.139 ± .004 +.051 ± .002 +.012 ± .002 +.126 ± .004 +.281 ± .007 +Super-ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.168 ± .003 +.111 ± .003 +.035 ± .001 +.133 ± .003 +.042 ± .002 +g = F +.158 ± .004 +.103 ± .002 +.033 ± .001 +.126 ± .004 +.039 ± .003 +g = M +.183 ± .005 +.124 ± .004 +.039 ± .002 +.144 ± .004 +.047 ± .003 +Ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.182 ± .002 +.066 ± .001 +.018 ± .0 +.164 ± .002 +.023 ± .002 +g = F +.17 ± .002 +.062 ± .001 +.017 ± .001 +.153 ± .002 +.021 ± .001 +g = M +.201 ± .005 +.072 ± .002 +.02 ± .001 +.18 ± .006 +.025 ± .003 +Super-ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.188 ± .002 +.073 ± .001 +.021 ± .001 +.167 ± .002 +.004 ± .001 +g = F +.175 ± .003 +.069 ± .001 +.02 ± .001 +.156 ± .003 +.004 ± .0 +g = M +.207 ± .006 +.079 ± .002 +.023 ± .002 +.184 ± .006 +.004 ± .002 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +E.5.6. NE W AD U LT - CA +As in Appendix E.4.4, We present the results for binarized sex and race separately, in two panels each with the 3 model +types we examine for this dataset – decision tree classifiers, random forest classifiers, and logistic regression. +Income. +By sex. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +Female +Male +F - after +M - after +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +Female +Male +F - after +M - after +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +Female +Male +F - after +M - after +(c) Logistic regression +Figure 33. Cumulative proportion of test instances that attain the given level of self-consistency for New Adult - CA - Income +(g = sex). For all, we estimate ˆ +SC with 101 bags in the super-ensemble, and 21 underlying individual models in each bag. We repeat for +5 test/train splits. +Ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.074 ± .001 +.333 ± .002 +.035 ± .001 +.039 ± .001 +.503 ± .002 +g = F +.069 ± .001 +.268 ± .004 +.036 ± .001 +.033 ± .001 +.492 ± .003 +g = M +.079 ± .001 +.393 ± .003 +.035 ± .002 +.044 ± .001 +.513 ± .001 +Super-ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.152 ± .002 +.392 ± .002 +.073 ± .001 +.079 ± .001 +.137 ± .001 +g = F +.145 ± .002 +.341 ± .003 +.076 ± .001 +.069 ± .001 +.133 ± .002 +g = M +.159 ± .002 +.438 ± .002 +.071 ± .002 +.088 ± .002 +.141 ± .002 +Ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.099 ± .001 +.36 ± .002 +.046 ± .001 +.053 ± .001 +.336 ± .001 +g = F +.093 ± .002 +.306 ± .004 +.049 ± .001 +.044 ± .001 +.328 ± .002 +g = M +.105 ± .002 +.409 ± .002 +.044 ± .001 +.062 ± .001 +.342 ± .001 +Super-ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.16 ± .001 +.394 ± .002 +.076 ± .001 +.084 ± .001 +.087 ± .0 +g = F +.153 ± .001 +.349 ± .003 +.081 ± .001 +.072 ± .001 +.086 ± .001 +g = M +.167 ± .002 +.435 ± .002 +.072 ± .002 +.095 ± .001 +.088 ± .001 +Ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.216 ± .001 +.396 ± .001 +.101 ± .001 +.115 ± .001 +.01 ± .0 +g = F +.211 ± .003 +.381 ± .002 +.122 ± .001 +.089 ± .002 +.01 ± .0 +g = M +.221 ± .001 +.41 ± .001 +.083 ± .001 +.138 ± .001 +.009 ± .0 +Super-ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.218 ± .001 +.397 ± .001 +.102 ± .001 +.116 ± .001 +.002 ± .0 +g = F +.213 ± .003 +.382 ± .002 +.123 ± .001 +.091 ± .002 +.003 ± .0 +g = M +.223 ± .001 +.41 ± .001 +.084 ± .001 +.139 ± .001 +.002 ± .0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +By race. +For these results we binarize the protected attribute into two groups: “Non-white” (0) and “White” (1). +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +Non-white +White +NW - after +W - after +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +Non-white +White +NW - after +W - after +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +Non-white +White +NW - after +W - after +(c) Logistic regression +Figure 34. Cumulative proportion of test instances that attain the given level of self-consistency for New Adult - CA - Income +(g = race). For all, we estimate ˆ +SC with 101 bags in the super-ensemble, and 21 underlying individual models in each bag. We repeat +for 5 test/train splits. +Ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.074 ± .001 +.333 ± .002 +.035 ± .001 +.039 ± .001 +.503 ± .002 +g = NW +.073 ± .0 +.27 ± .002 +.032 ± .001 +.041 ± .001 +.488 ± .003 +g = W +.075 ± .001 +.373 ± .002 +.037 ± .001 +.038 ± .001 +.513 ± .002 +Super-ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.152 ± .002 +.392 ± .002 +.073 ± .001 +.079 ± .001 +.137 ± .001 +g = NW +.148 ± .002 +.331 ± .003 +.069 ± .001 +.08 ± .001 +.131 ± .002 +g = W +.154 ± .002 +.43 ± .002 +.076 ± .001 +.078 ± .001 +.141 ± .001 +Ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.099 ± .001 +.36 ± .002 +.046 ± .001 +.053 ± .001 +.336 ± .001 +g = NW +.098 ± .001 +.293 ± .002 +.042 ± .001 +.056 ± .001 +.321 ± .003 +g = W +.1 ± .002 +.403 ± .002 +.049 ± .001 +.051 ± .001 +.345 ± .001 +Super-ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.16 ± .001 +.394 ± .002 +.076 ± .001 +.084 ± .001 +.087 ± .0 +g = NW +.157 ± .002 +.332 ± .003 +.071 ± .002 +.086 ± .002 +.082 ± .001 +g = W +.162 ± .002 +.433 ± .001 +.08 ± .001 +.083 ± .001 +.09 ± .001 +Ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.216 ± .001 +.396 ± .001 +.101 ± .001 +.115 ± .001 +.01 ± .0 +g = NW +.214 ± .002 +.339 ± .003 +.098 ± .001 +.116 ± .003 +.01 ± .0 +g = W +.218 ± .001 +.431 ± .0 +.103 ± .001 +.115 ± .001 +.01 ± .0 +Super-ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.218 ± .001 +.397 ± .001 +.102 ± .001 +.116 ± .001 +.002 ± .0 +g = NW +.216 ± .002 +.34 ± .003 +.099 ± .001 +.117 ± .003 +.002 ± .0 +g = W +.22 ± .001 +.432 ± .001 +.104 ± .001 +.116 ± .0 +.002 ± .0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Employment. +By sex. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +Female +Male +F - after +M - after +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +Female +Male +F - after +M - after +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +Female +Male +F - after +M - after +(c) Logistic regression +Figure 35. Cumulative proportion of test instances that attain the given level of self-consistency for New Adult - CA - +Employment (g = sex). For all, we estimate ˆ +SC with 101 bags in the super-ensemble, and 15 underlying individual models in +each bag. We repeat for 5 test/train splits. +Ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.149 ± .0 +.496 ± .001 +.108 ± .001 +.042 ± .001 +.224 ± .001 +g = F +.178 ± .001 +.5 ± .003 +.138 ± .001 +.04 ± .001 +.225 ± .001 +g = M +.12 ± .0 +.493 ± .001 +.076 ± .0 +.043 ± .001 +.223 ± .001 +Super-ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.18 ± .001 +.509 ± .001 +.121 ± .001 +.06 ± .001 +.082 ± .002 +g = F +.209 ± .002 +.512 ± .002 +.152 ± .001 +.056 ± .002 +.082 ± .002 +g = M +.151 ± .001 +.506 ± .001 +.088 ± .0 +.063 ± .001 +.081 ± .003 +Ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.148 ± .0 +.484 ± .0 +.103 ± .0 +.045 ± .001 +.199 ± .001 +g = F +.175 ± .001 +.487 ± .002 +.132 ± .001 +.043 ± .001 +.203 ± .002 +g = M +.121 ± .001 +.482 ± .001 +.073 ± .0 +.047 ± .001 +.196 ± .002 +Super-ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.178 ± .001 +.5 ± .001 +.116 ± .001 +.062 ± .001 +.075 ± .002 +g = F +.205 ± .002 +.503 ± .002 +.147 ± .001 +.058 ± .001 +.077 ± .002 +g = M +.15 ± .0 +.497 ± .001 +.085 ± .0 +.066 ± .001 +.073 ± .002 +Ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.231 ± .001 +.546 ± .001 +.161 ± .0 +.07 ± .0 +.008 ± .0 +g = F +.255 ± .001 +.568 ± .001 +.201 ± .0 +.054 ± .0 +.008 ± .0 +g = M +.206 ± .001 +.523 ± .001 +.12 ± .0 +.086 ± .001 +.007 ± .0 +Super-ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.232 ± .001 +.546 ± .001 +.161 ± .0 +.071 ± .001 +.003 ± .0 +g = F +.256 ± .001 +.567 ± .001 +.201 ± .0 +.055 ± .0 +.003 ± .0 +g = M +.207 ± .001 +.523 ± .001 +.12 ± .0 +.087 ± .001 +.003 ± .0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +By race. +For these results we binarize the protected attribute into two groups: “Non-white” (0) and “White” (1). +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +Non-white +White +NW - after +W - after +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +Non-white +White +NW - after +W - after +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +Non-white +White +NW - after +W - after +(c) Logistic regression +Figure 36. Cumulative proportion of test instances that attain the given level of self-consistency for New Adult - CA - +Employment (g = race). For all, we estimate ˆ +SC with 101 bags in the super-ensemble, and 15 underlying individual models +in each bag. We repeat for 5 test/train splits. +Ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.149 ± .0 +.496 ± .001 +.108 ± .001 +.042 ± .001 +.224 ± .001 +g = NW +.145 ± .001 +.5 ± .001 +.111 ± .0 +.034 ± .001 +.224 ± .002 +g = W +.152 ± .001 +.494 ± .001 +.105 ± .001 +.047 ± .001 +.224 ± .001 +Super-ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.18 ± .001 +.509 ± .001 +.121 ± .001 +.06 ± .001 +.082 ± .002 +g = NW +.177 ± .001 +.516 ± .0 +.125 ± .0 +.052 ± .001 +.082 ± .002 +g = W +.183 ± .001 +.504 ± .001 +.118 ± .001 +.065 ± .001 +.081 ± .003 +Ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.148 ± .0 +.484 ± .0 +.103 ± .0 +.045 ± .001 +.199 ± .001 +g = NW +.144 ± .001 +.491 ± .001 +.107 ± .0 +.037 ± .001 +.201 ± .001 +g = W +.151 ± .001 +.481 ± .001 +.101 ± .001 +.05 ± .001 +.198 ± .002 +Super-ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.178 ± .001 +.5 ± .001 +.116 ± .001 +.062 ± .001 +.075 ± .002 +g = NW +.175 ± .001 +.509 ± .0 +.121 ± .0 +.054 ± .001 +.076 ± .001 +g = W +.18 ± .001 +.495 ± .001 +.113 ± .001 +.067 ± .001 +.075 ± .002 +Ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.231 ± .001 +.546 ± .001 +.161 ± .0 +.07 ± .0 +.008 ± .0 +g = NW +.23 ± .0 +.553 ± .002 +.165 ± .0 +.064 ± .001 +.007 ± .0 +g = W +.232 ± .001 +.541 ± .001 +.158 ± .001 +.074 ± .001 +.008 ± .0 +Super-ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.232 ± .001 +.546 ± .001 +.161 ± .0 +.071 ± .001 +.003 ± .0 +g = NW +.231 ± .001 +.553 ± .002 +.166 ± .0 +.065 ± .001 +.003 ± .0 +g = W +.233 ± .001 +.541 ± .001 +.159 ± .001 +.074 ± .001 +.003 ± .0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Public Coverage. +By sex. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +(c) Logistic regression +Figure 37. Cumulative proportion of test instances that attain the given level of self-consistency for New Adult - CA - Public +Coverage (g = sex). For all, we estimate ˆ +SC with 101 bags in the super-ensemble, and 21 underlying individual models in each bag. +We repeat for 5 test/train splits. +Ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.19 ± .002 +.235 ± .002 +.046 ± .001 +.145 ± .002 +.607 ± .002 +g = F +.188 ± .003 +.205 ± .004 +.046 ± .002 +.142 ± .003 +.607 ± .004 +g = M +.193 ± .002 +.273 ± .003 +.045 ± .002 +.148 ± .001 +.608 ± .004 +Super-ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.268 ± .002 +.288 ± .003 +.099 ± .001 +.169 ± .002 +.185 ± .003 +g = F +.267 ± .003 +.275 ± .003 +.1 ± .003 +.167 ± .001 +.186 ± .004 +g = M +.269 ± .004 +.305 ± .003 +.097 ± .002 +.172 ± .004 +.183 ± .003 +Ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.197 ± .002 +.218 ± .002 +.042 ± .001 +.156 ± .002 +.48 ± .002 +g = F +.193 ± .003 +.195 ± .003 +.041 ± .002 +.152 ± .002 +.481 ± .003 +g = M +.203 ± .001 +.248 ± .003 +.043 ± .001 +.16 ± .002 +.479 ± .003 +Super-ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.264 ± .002 +.269 ± .003 +.086 ± .001 +.178 ± .003 +.132 ± .002 +g = F +.263 ± .002 +.257 ± .004 +.087 ± .003 +.175 ± .002 +.132 ± .001 +g = M +.266 ± .003 +.283 ± .003 +.084 ± .002 +.182 ± .003 +.13 ± .003 +Ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.31 ± .001 +.159 ± .001 +.051 ± .001 +.259 ± .002 +.014 ± .0 +g = F +.308 ± .002 +.147 ± .002 +.051 ± .001 +.257 ± .002 +.014 ± .001 +g = M +.313 ± .002 +.173 ± .001 +.051 ± .001 +.262 ± .002 +.014 ± .0 +Super-ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.312 ± .001 +.162 ± .001 +.053 ± .001 +.259 ± .002 +.004 ± .0 +g = F +.31 ± .002 +.151 ± .002 +.053 ± .001 +.257 ± .002 +.004 ± .0 +g = M +.315 ± .002 +.177 ± .001 +.053 ± .002 +.262 ± .002 +.004 ± .0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +By race. +For these results we binarize the protected attribute into two groups: “Non-white” (0) and “White” (1). +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +NW - before +W - before +NW - after +W - after +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +NW - before +W - before +NW - after +W - after +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +NW - before +W - before +NW - after +W - after +(c) Logistic regression +Figure 38. Cumulative proportion of test instances that attain the given level of self-consistency for New Adult - CA - Public +Coverage (g = race). For all, we estimate ˆ +SC with 101 bags in the super-ensemble, and 15 underlying individual models in each bag. +We repeat for 5 test/train splits. +Ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.19 ± .002 +.235 ± .002 +.046 ± .001 +.145 ± .002 +.607 ± .002 +g = NW +.208 ± .003 +.248 ± .003 +.05 ± .002 +.158 ± .002 +.623 ± .003 +g = W +.178 ± .004 +.226 ± .005 +.043 ± .002 +.135 ± .003 +.595 ± .003 +Super-ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.268 ± .002 +.288 ± .003 +.099 ± .001 +.169 ± .002 +.185 ± .003 +g = NW +.283 ± .002 +.296 ± .004 +.102 ± .002 +.181 ± .003 +.192 ± .003 +g = W +.256 ± .003 +.282 ± .003 +.096 ± .002 +.16 ± .002 +.179 ± .004 +Ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.197 ± .002 +.218 ± .002 +.042 ± .001 +.156 ± .002 +.48 ± .002 +g = NW +.214 ± .003 +.224 ± .004 +.044 ± .002 +.17 ± .002 +.5 ± .002 +g = W +.185 ± .003 +.214 ± .003 +.04 ± .002 +.145 ± .002 +.464 ± .003 +Super-ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.264 ± .002 +.269 ± .003 +.086 ± .001 +.178 ± .003 +.132 ± .002 +g = NW +.279 ± .003 +.273 ± .004 +.089 ± .002 +.191 ± .003 +.138 ± .002 +g = W +.253 ± .002 +.266 ± .004 +.084 ± .002 +.168 ± .002 +.126 ± .002 +Ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.31 ± .001 +.159 ± .001 +.051 ± .001 +.259 ± .002 +.014 ± .0 +g = NW +.326 ± .003 +.158 ± .003 +.052 ± .001 +.274 ± .003 +.015 ± .001 +g = W +.299 ± .002 +.159 ± .002 +.051 ± .001 +.248 ± .002 +.014 ± .001 +Super-ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.312 ± .001 +.162 ± .001 +.053 ± .001 +.259 ± .002 +.004 ± .0 +g = NW +.328 ± .003 +.162 ± .003 +.054 ± .002 +.274 ± .003 +.004 ± .0 +g = W +.3 ± .002 +.163 ± .002 +.053 ± .001 +.247 ± .002 +.003 ± .0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +E.5.7. HMDA +As in Section E.4.5, we present the results for binarized sex, race, and ethnicity separately, in three panels each with +the 3 model types we examine for this dataset – decision tree classifiers, random forest classifiers, and logistic regression. +NY - 2017. +By sex. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +(c) Logistic regression +Figure 39. Cumulative proportion of test instances that attain the given level of self-consistency for HMDA - NY - 2017 (g = sex). +For all, we estimate ˆ +SC with 101 bags in the super-ensemble, and 21 underlying individual models in each bag. We repeat for 5 test/train +splits. +Ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.033 ± .001 +.954 ± .001 +.024 ± .001 +.01 ± .001 +.422 ± .002 +g = F +.033 ± .001 +.948 ± .001 +.022 ± .001 +.011 ± .001 +.431 ± .002 +g = M +.034 ± .001 +.957 ± .001 +.024 ± .001 +.009 ± .001 +.417 ± .002 +Super-ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.116 ± .002 +.817 ± .001 +.061 ± .001 +.055 ± .001 +.129 ± .002 +g = F +.118 ± .002 +.805 ± .002 +.061 ± .001 +.057 ± .002 +.132 ± .002 +g = M +.115 ± .002 +.824 ± .001 +.062 ± .001 +.053 ± .001 +.127 ± .002 +Ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.063 ± .001 +.905 ± .001 +.04 ± .0 +.023 ± .001 +.306 ± .001 +g = F +.063 ± .002 +.894 ± .002 +.038 ± .001 +.025 ± .001 +.314 ± .001 +g = M +.062 ± .001 +.911 ± .001 +.041 ± .001 +.021 ± .0 +.302 ± .001 +Super-ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.128 ± .001 +.803 ± .001 +.068 ± .001 +.06 ± .001 +.082 ± .001 +g = F +.13 ± .002 +.79 ± .002 +.067 ± .002 +.063 ± .002 +.084 ± .001 +g = M +.126 ± .002 +.81 ± .001 +.068 ± .001 +.058 ± .001 +.081 ± .001 +Ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.168 ± .001 +.832 ± .001 +.111 ± .001 +.057 ± .001 +.019 ± .001 +g = F +.172 ± .002 +.822 ± .003 +.113 ± .002 +.059 ± .001 +.02 ± .001 +g = M +.166 ± .001 +.837 ± .001 +.11 ± .001 +.056 ± .001 +.019 ± .001 +Super-ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.172 ± .001 +.827 ± .001 +.114 ± .001 +.059 ± .001 +.005 ± .0 +g = F +.176 ± .002 +.817 ± .003 +.115 ± .002 +.061 ± .002 +.005 ± .0 +g = M +.17 ± .001 +.832 ± .001 +.113 ± .001 +.058 ± .001 +.005 ± .0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +By race. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +NW - before +W - before +NW - after +W - after +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +NW - before +W - before +NW - after +W - after +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +NW - before +W - before +NW - after +W - after +(c) Logistic regression +Figure 40. Cumulative proportion of test instances that attain the given level of self-consistency for HMDA - NY - 2017 (g = race). +For all, we estimate ˆ +SC with 101 bags in the super-ensemble, and 21 underlying individual models in each bag. We repeat for 5 test/train +splits. +Ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.033 ± .001 +.954 ± .001 +.024 ± .001 +.01 ± .001 +.422 ± .002 +g = NW +.039 ± .002 +.93 ± .002 +.025 ± .001 +.014 ± .001 +.471 ± .004 +g = W +.032 ± .001 +.96 ± .001 +.023 ± .001 +.009 ± .0 +.409 ± .002 +Super-ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.116 ± .002 +.817 ± .001 +.061 ± .001 +.055 ± .001 +.129 ± .002 +g = NW +.135 ± .002 +.744 ± .003 +.066 ± .002 +.069 ± .002 +.152 ± .002 +g = W +.112 ± .002 +.835 ± .002 +.06 ± .001 +.051 ± .001 +.123 ± .002 +Ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.063 ± .001 +.905 ± .001 +.04 ± .0 +.023 ± .001 +.306 ± .001 +g = NW +.073 ± .001 +.853 ± .002 +.04 ± .001 +.033 ± .001 +.356 ± .004 +g = W +.06 ± .001 +.918 ± .001 +.04 ± .001 +.02 ± .001 +.293 ± .001 +Super-ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.128 ± .001 +.803 ± .001 +.068 ± .001 +.06 ± .001 +.082 ± .001 +g = NW +.149 ± .001 +.724 ± .003 +.072 ± .001 +.077 ± .001 +.096 ± .003 +g = W +.122 ± .002 +.822 ± .002 +.067 ± .001 +.056 ± .002 +.078 ± .001 +Ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.168 ± .001 +.832 ± .001 +.111 ± .001 +.057 ± .001 +.019 ± .001 +g = NW +.191 ± .002 +.739 ± .004 +.11 ± .002 +.081 ± .002 +.02 ± .001 +g = W +.162 ± .001 +.855 ± .001 +.112 ± .001 +.051 ± .001 +.019 ± .001 +Super-ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.172 ± .001 +.827 ± .001 +.114 ± .001 +.059 ± .001 +.005 ± .0 +g = NW +.196 ± .001 +.735 ± .004 +.113 ± .002 +.083 ± .002 +.005 ± .001 +g = W +.166 ± .001 +.85 ± .001 +.114 ± .001 +.053 ± .001 +.005 ± .0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +By ethnicity. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +HL - before +NHL - before +HL - after +NHL - after +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +HL - before +NHL - before +HL - after +NHL - after +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +HL - before +NHL - before +HL - after +NHL - after +(c) Logistic regression +Figure 41. Cumulative proportion of test instances that attain the given level of self-consistency for HMDA - NY - 2017 (g = +ethnicity). For all, we estimate ˆ +SC with 101 bags in the super-ensemble, and 21 underlying individual models in each bag. We repeat +for 5 test/train splits. +Ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.033 ± .001 +.954 ± .001 +.024 ± .001 +.01 ± .001 +.422 ± .002 +g = HL +.027 ± .002 +.94 ± .003 +.014 ± .001 +.013 ± .002 +.433 ± .004 +g = NHL +.034 ± .001 +.955 ± .001 +.025 ± .001 +.009 ± .001 +.421 ± .002 +Super-ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.116 ± .002 +.817 ± .001 +.061 ± .001 +.055 ± .001 +.129 ± .002 +g = HL +.122 ± .004 +.741 ± .006 +.051 ± .002 +.07 ± .004 +.15 ± .003 +g = NHL +.116 ± .002 +.823 ± .001 +.062 ± .001 +.054 ± .001 +.127 ± .002 +Ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.063 ± .001 +.905 ± .001 +.04 ± .0 +.023 ± .001 +.306 ± .001 +g = HL +.058 ± .004 +.858 ± .007 +.026 ± .001 +.033 ± .003 +.339 ± .003 +g = NHL +.063 ± .001 +.909 ± .001 +.041 ± .0 +.022 ± .001 +.303 ± .001 +Super-ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.128 ± .001 +.803 ± .001 +.068 ± .001 +.06 ± .001 +.082 ± .001 +g = HL +.137 ± .004 +.716 ± .005 +.057 ± .002 +.08 ± .003 +.095 ± .004 +g = NHL +.127 ± .002 +.81 ± .001 +.069 ± .001 +.058 ± .001 +.081 ± .001 +Ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.168 ± .001 +.832 ± .001 +.111 ± .001 +.057 ± .001 +.019 ± .001 +g = HL +.184 ± .004 +.735 ± .006 +.102 ± .002 +.082 ± .004 +.019 ± .002 +g = NHL +.167 ± .001 +.84 ± .001 +.112 ± .001 +.055 ± .001 +.019 ± .001 +Super-ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.172 ± .001 +.827 ± .001 +.114 ± .001 +.059 ± .001 +.005 ± .0 +g = HL +.189 ± .005 +.731 ± .006 +.105 ± .002 +.084 ± .005 +.004 ± .001 +g = NHL +.171 ± .001 +.835 ± .001 +.114 ± .001 +.057 ± .001 +.005 ± .0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +TX - 2017. +By sex. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +F - before +M - before +F - after +M - after +(c) Logistic regression +Figure 42. Cumulative proportion of test instances that attain the given level of self-consistency for HMDA - TX - 2017 (g = sex). +For all, we estimate ˆ +SC with 101 bags in the super-ensemble, and 7 underlying individual models in each bag. We repeat for 3 test/train +splits. +Ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.024 ± .0 +.934 ± .0 +.011 ± .0 +.013 ± .0 +.348 ± .001 +g = F +.026 ± .0 +.918 ± .0 +.01 ± .0 +.016 ± .0 +.362 ± .002 +g = M +.024 ± .0 +.94 ± .001 +.012 ± .0 +.012 ± .0 +.342 ± .001 +Super-ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.069 ± .0 +.832 ± .001 +.03 ± .001 +.039 ± .0 +.18 ± .001 +g = F +.072 ± .001 +.798 ± .001 +.028 ± .0 +.045 ± .001 +.19 ± .001 +g = M +.067 ± .001 +.846 ± .001 +.031 ± .001 +.036 ± .0 +.176 ± .001 +Ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.044 ± .0 +.883 ± .001 +.019 ± .0 +.025 ± .0 +.272 ± .0 +g = F +.046 ± .0 +.855 ± .001 +.017 ± .0 +.03 ± .0 +.284 ± .001 +g = M +.042 ± .0 +.895 ± .001 +.02 ± .0 +.022 ± .0 +.267 ± .001 +Super-ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.088 ± .001 +.8 ± .001 +.038 ± .0 +.051 ± .0 +.113 ± .0 +g = F +.093 ± .0 +.763 ± .002 +.035 ± .001 +.058 ± .001 +.119 ± .002 +g = M +.086 ± .001 +.815 ± .001 +.039 ± .0 +.047 ± .0 +.111 ± .0 +Ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.146 ± .001 +.75 ± .001 +.059 ± .0 +.087 ± .001 +.01 ± .0 +g = F +.153 ± .0 +.711 ± .002 +.056 ± .001 +.096 ± .0 +.01 ± .0 +g = M +.143 ± .001 +.767 ± .002 +.06 ± .0 +.083 ± .001 +.01 ± .0 +Super-ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.148 ± .001 +.748 ± .001 +.06 ± .0 +.088 ± .001 +.004 ± .0 +g = F +.155 ± .0 +.709 ± .002 +.057 ± .0 +.097 ± .0 +.004 ± .0 +g = M +.145 ± .001 +.765 ± .002 +.061 ± .0 +.084 ± .001 +.004 ± .0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +By race. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +NW - before +W - before +NW - after +W - after +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +NW - before +W - before +NW - after +W - after +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +NW - before +W - before +NW - after +W - after +(c) Logistic regression +Figure 43. Cumulative proportion of test instances that attain the given level of self-consistency for HMDA - TX - 2017 (g = race). +For all, we estimate ˆ +SC with 101 bags in the super-ensemble, and 7 underlying individual models in each bag. We repeat for 3 test/train +splits. +Ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.024 ± .0 +.934 ± .0 +.011 ± .0 +.013 ± .0 +.348 ± .001 +g = NW +.024 ± .0 +.919 ± .001 +.011 ± .0 +.014 ± .0 +.347 ± .002 +g = W +.024 ± .0 +.937 ± .001 +.011 ± .0 +.013 ± .0 +.348 ± .001 +Super-ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.069 ± .0 +.832 ± .001 +.03 ± .001 +.039 ± .0 +.18 ± .001 +g = NW +.068 ± .001 +.798 ± .003 +.029 ± .001 +.039 ± .001 +.18 ± .003 +g = W +.069 ± .001 +.839 ± .001 +.03 ± .001 +.038 ± .0 +.18 ± .0 +Ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.044 ± .0 +.883 ± .001 +.019 ± .0 +.025 ± .0 +.272 ± .0 +g = NW +.043 ± .0 +.856 ± .001 +.018 ± .001 +.026 ± .001 +.273 ± .001 +g = W +.044 ± .0 +.889 ± .001 +.019 ± .0 +.024 ± .0 +.272 ± .0 +Super-ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.088 ± .001 +.8 ± .001 +.038 ± .0 +.051 ± .0 +.113 ± .0 +g = NW +.087 ± .001 +.763 ± .001 +.037 ± .001 +.051 ± .0 +.113 ± .001 +g = W +.088 ± .001 +.808 ± .001 +.038 ± .0 +.05 ± .0 +.113 ± .001 +Ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.146 ± .001 +.75 ± .001 +.059 ± .0 +.087 ± .001 +.01 ± .0 +g = NW +.144 ± .001 +.728 ± .0 +.066 ± .001 +.078 ± .0 +.01 ± .0 +g = W +.146 ± .001 +.755 ± .001 +.057 ± .0 +.089 ± .001 +.01 ± .0 +Super-ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.148 ± .001 +.748 ± .001 +.06 ± .0 +.088 ± .001 +.004 ± .0 +g = NW +.147 ± .001 +.726 ± .0 +.068 ± .001 +.079 ± .001 +.004 ± .0 +g = W +.148 ± .001 +.753 ± .001 +.058 ± .0 +.09 ± .001 +.004 ± .0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +By ethnicity. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +HL - before +NHL - before +HL - after +NHL - after +(a) Decision tree classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +HL - before +NHL - before +HL - after +NHL - after +(b) Random forest classifiers +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set += . 75 +HL - before +NHL - before +HL - after +NHL - after +(c) Logistic regression +Figure 44. Cumulative proportion of test instances that attain the given level of self-consistency for HMDA - TX - 2017 (g = +ethnicity). For all, we estimate ˆ +SC with 101 bags in the super-ensemble, and 7 underlying individual models in each bag. We repeat +for 3 test/train splits. +Ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.024 ± .0 +.934 ± .0 +.011 ± .0 +.013 ± .0 +.348 ± .001 +g = HL +.03 ± .001 +.902 ± .001 +.012 ± .0 +.018 ± .0 +.403 ± .001 +g = NHL +.023 ± .0 +.943 ± .0 +.011 ± .0 +.012 ± .0 +.33 ± .001 +Super-ensembling with confidence: Decision trees +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.069 ± .0 +.832 ± .001 +.03 ± .001 +.039 ± .0 +.18 ± .001 +g = HL +.084 ± .001 +.767 ± .001 +.033 ± .001 +.05 ± .0 +.211 ± .0 +g = NHL +.064 ± .0 +.852 ± .001 +.029 ± .001 +.035 ± .0 +.17 ± .001 +Ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.044 ± .0 +.883 ± .001 +.019 ± .0 +.025 ± .0 +.272 ± .0 +g = HL +.053 ± .001 +.83 ± .002 +.02 ± .0 +.033 ± .001 +.319 ± .0 +g = NHL +.041 ± .0 +.899 ± .001 +.019 ± .0 +.022 ± .0 +.257 ± .001 +Super-ensembling with confidence: Random forest classifiers +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.088 ± .001 +.8 ± .001 +.038 ± .0 +.051 ± .0 +.113 ± .0 +g = HL +.105 ± .001 +.727 ± .002 +.041 ± .0 +.064 ± .001 +.132 ± .001 +g = NHL +.083 ± .0 +.823 ± .001 +.037 ± .001 +.046 ± .0 +.107 ± .0 +Ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.146 ± .001 +.75 ± .001 +.059 ± .0 +.087 ± .001 +.01 ± .0 +g = HL +.171 ± .002 +.651 ± .001 +.054 ± .0 +.116 ± .002 +.011 ± .0 +g = NHL +.138 ± .001 +.782 ± .001 +.06 ± .0 +.077 ± .001 +.01 ± .0 +Super-ensembling with confidence: Logistic regression +ˆ +Err +ˆ +PR +ˆ +FPR +ˆ +FNR +ˆ +AR +Total +.148 ± .001 +.748 ± .001 +.06 ± .0 +.088 ± .001 +.004 ± .0 +g = HL +.173 ± .002 +.65 ± .001 +.056 ± .0 +.117 ± .002 +.004 ± .0 +g = NHL +.14 ± .0 +.78 ± .001 +.062 ± .0 +.078 ± .0 +.004 ± .0 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +E.5.8. DISCUSSION OF EXTENDED RESULTS FOR ALGORITHM 1 +Overall, our results support that examining self-consistency and error together provide a much richer picture of model +behavior, both with respect to arbitrariness and fairness metric disparities. Particularly in smaller datasets, the learning +process produces models with a large degree of variance. As a result, ensembling with confidence can lead to huge abstention +rates. +Improving self-consistency by doing a round of variance reduction first and then ensembling with confidence (i.e., super- +ensembling) can lead to improvements in error over baselines while having a lower abstention rate. These improvements +are typically shared across subgroups, but may not be symmetric; some subgroups may benefit more than others. As a +result, even though accuracy increases absolutely for both groups, relative fairness metrics can decrease. This is a different +instantiation of the fairness-accuracy trade-off than is often written about, which posits a necessary decrease in accuracy for +one subgroup to improve fairness between binarized groups. Our results suggest that it is worth first tuning for accuracy, and +then seeing how fairness interventions can balance the benefits across subgroups. Of course, it is possible that doing this +could lead to injecting variance back into the model outputs, thereby reducing self-consistency and inducing arbitrariness. +We leave this investigation to future work. +Additionally, our results reify that choice of model matters a lot. While overall error rates across model types may be similar, +the sources of that error are not necessarily the same. This is an obvious point, relating to bias and variance (Appendix B.1. +However, a lot of fair classification work describes similar performance across logistic regression, decision trees, random +forests, SVMs, and MLPs (e.g., Chen et al. (2018)). Looking at self-consistency confirms that this is not the case, with +decision trees and random forests in particular exhibiting higher variance, and thus being more amenable to variance +reduction and improvements in overall error. As fair classification research transitions to larger benchmarks, it will likely be +fruitful to investigate more complex model classes. +We provide run times on our cluster environment in Table 31. We did not select for CPUs with any particular features, and +thus the run times are quite variable. +Table 31. These times are recorded for our cluster environment (hh:mm:ss), described in Appendix E.2 for our Algorithm 1 experiments. +At the time of running, due to time constraints, the authors had not yet parallelized this part of the code. +Dataset +g +Logistic regression Decision trees Random forests +South German Credit sex +00:42:50 +00:25:28 +00:34:42 +COMPAS +race +00:57:05 +00:39:24 +00:31:47 +Old Adult +sex +01:08:37 +01:23:39 +00:57:11 +Taiwan Credit +sex +00:31:35 +01:34:57 +01:53:33 +New Adult - CA +Income +sex, race +01:39:53 +02:51:13 +04:59:07 +Employment +sex, race +02:20:15 +02:18:16 +03:00:15 +Public Coverage +sex, race +01:13:33 +02:02:57 +02:24:08 +HMDA - 2017 +NY +sex, race, ethnicity 03:50:52 +05:00:19 +05:39:44 +TX +sex, race, ethnicity 05:18:59 +04:10:34 +04:18:59 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +0.04 +0.02 0.00 +0.02 +0.04 +0.06 +ErrorNW - ErrorW +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative prop. of models +(a) +ˆ +Err disparity +0.050 0.025 0.000 0.025 0.050 0.075 +FPNW -FPW +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative prop. of models +(b) +ˆ +FPR disparity +0.06 +0.04 +0.02 +0.00 +0.02 +FNNW - FNW +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative prop. of models +(c) +ˆ +FNR disparity +Figure 45. Cumulative distribution of error disparity across 1000 logistic regression models trained on COMPAS. +E.6. Reliability and fairness metrics in COMPAS and South German Credit +Even before we apply our intervention to improve self-consistency, our results in Section 3 and Appendix E.3 show close-to- +parity +ˆ +Err, +ˆ +FPR, and +ˆ +FNR across subgroups in COMPAS, and similarly for South German Credit in Appendix E.4. +These results are surprising. We run B = 100 models to produce estimates of variance and self-consistency, but of course +doing this also has the effect of estimating the expected error more generally (with variance representing a portion of +that error). Our estimates of expected error for these tasks indicate that the average model produced training on COMPAS +and South German Credit, with respect to popular fairness definitions like Equality of Opportunity and Equalized +Odds (Barocas et al., 2019; Hardt et al., 2016) are in fact baseline close to parity, with no fairness intervention applied. We +found this across model types for both datasets, though the story becomes more complicated when we apply techniques to +improve self-consistency (discussed in Appendix E.5.8). +Of course, we did not expect this result, as these are two of the de facto standard benchmark datasets in algorithmic fairness. +They are used in countless other studies to probe and verify algorithmic fairness interventions (Fabris et al., 2022). As a +result, we initially thought that our results must be incorrect. We therefore lookedthe underlying models in our bootstrap +runs to see the error of the underlying models. In Figure 45, we plot the bootstrap models that went into the baseline result +for logistic regression, presented in Figure 5. For another view on analogous information, in Table 32, we provide an excerpt +of the results for COMPAS regarding the underlying 1000 random forest classifiers used to produce Figure 2a. +Overall, we can see that there is a wide range of error disparities that trend in both directions, with a skew toward higher +ˆ +FPR for g = NW. These results support our claim that training many models is necessary to get an accurate picture of error. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +Table 32. Comparing subgroup error rates in COMPAS for different random forest classifiers trained to produce Figure 2a. Each table +looks at the top-3 highest differences between subgroups for the specified metric: (a) +ˆ +ErrNW − +ˆ +ErrW, when +ˆ +ErrNW > +ˆ +ErrW; (b) +ˆ +ErrW − +ˆ +ErrNW, when +ˆ +ErrW > +ˆ +ErrNW; (c) +ˆ +FPRNW − +ˆ +FPRW, when +ˆ +FPRNW > +ˆ +FPRW; (d) +ˆ +FPRW − +ˆ +FPRNW, when +ˆ +FPRW > +ˆ +FPRNW; (e) +ˆ +FNRNW − +ˆ +FNRW, when +ˆ +FNRNW > +ˆ +FNRW; and, (f) (e) +ˆ +FNRW − +ˆ +FNRNW, when +ˆ +FNRW > +ˆ +FNRNW. We highlight the overall error metric +in gray, the larger metric (being subtracted from) in blue, the smaller metric (being subtracted) in red, and the difference in the metric +between subgroups in purple. Note that run 757 appears twice, which we mark in orange. +(a) The top-3 most unfair models according to subgroup-specific +ˆ +Err, when +ˆ +ErrNW > +ˆ +ErrW (i.e., unfair toward NW). +Run # s b +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +ErrNW +ˆ +FPRNW +ˆ +FNRNW +ˆ +ErrW +ˆ +FPRW +ˆ +FNRW +ˆ +ErrNW − +ˆ +ErrW +762 +8 504 0.374 0.179 0.196 +0.405 +0.204 +0.201 +0.315 +0.13 +0.186 +0.09 +757 +8 464 0.369 0.167 0.202 +0.395 +0.201 +0.193 +0.318 +0.101 +0.218 +0.077 +328 +4 116 0.371 0.165 0.206 +0.395 +0.181 +0.214 +0.323 +0.134 +0.189 +0.072 +(b) The top-3 most unfair models according to subgroup-specific +ˆ +Err, when +ˆ +ErrW > +ˆ +ErrNW (i.e., unfair toward W). +Run # s b +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +ErrNW +ˆ +FPRNW +ˆ +FNRNW +ˆ +ErrW +ˆ +FPRW +ˆ +FNRW +ˆ +ErrW − +ˆ +ErrNW +414 +5 75 +0.376 0.167 0.209 +0.352 +0.158 +0.194 +0.422 +0.186 +0.236 +0.07 +435 +5 180 0.376 0.199 0.177 +0.355 +0.189 +0.166 +0.416 +0.217 +0.198 +0.061 +413 +5 70 +0.378 0.189 0.189 +0.359 +0.188 +0.171 +0.413 +0.191 +0.222 +0.054 +(c) The top-3 most unfair models according to subgroup-specific +ˆ +FPR, when +ˆ +FPRNW > +ˆ +FPRW (i.e., unfair toward NW). +Run # s b +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +ErrNW +ˆ +FPRNW +ˆ +FNRNW +ˆ +ErrW +ˆ +FPRW +ˆ +FNRW +ˆ +FPRNW − +ˆ +FPRW +757 +8 464 0.369 +0.167 0.202 +0.395 +0.201 +0.193 +0.318 +0.101 +0.218 +0.1 +729 +8 240 0.358 0.162 0.197 +0.376 +0.189 +0.187 +0.323 +0.107 +0.216 +0.082 +791 +8 736 0.377 +0.171 0.205 +0.395 +0.198 +0.197 +0.341 +0.118 +0.222 +0.08 +(d) The top-3 most unfair models according to subgroup-specific +ˆ +FPR, when +ˆ +FPRW > +ˆ +FPRNW (i.e., unfair toward W). +Run # s b +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +ErrNW +ˆ +FPRNW +ˆ +FNRNW +ˆ +ErrW +ˆ +FPRW +ˆ +FNRW +ˆ +FPRW − +ˆ +FPRNW +639 +7 280 +0.36 +0.187 0.173 +0.352 +0.174 +0.178 +0.376 +0.212 +0.164 +0.038 +807 +9 72 +0.381 0.191 +0.19 +0.372 +0.179 +0.192 +0.398 +0.214 +0.184 +0.035 +543 +6 264 0.358 0.155 0.203 +0.351 +0.144 +0.206 +0.37 +0.175 +0.196 +0.031 +(e) The top-3 most unfair models according to subgroup-specific +ˆ +FNR, when +ˆ +FNRNW > +ˆ +FNRW (i.e., unfair toward NW). +Run # s b +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +ErrNW +ˆ +FPRNW +ˆ +FNRNW +ˆ +ErrW +ˆ +FPRW +ˆ +FNRW +ˆ +FNRNW − +ˆ +FNRW +246 +3 141 0.379 0.166 +0.213 +0.398 +0.169 +0.229 +0.345 +0.161 +0.184 +0.045 +506 +6 42 +0.367 +0.17 +0.197 +0.386 +0.175 +0.211 +0.332 +0.161 +0.171 +0.04 +204 +3 15 +0.384 0.185 +0.199 +0.394 +0.181 +0.213 +0.365 +0.192 +0.173 +0.04 +(f) The top-3 most unfair models according to subgroup-specific +ˆ +FNR, when +ˆ +FNRW > +ˆ +FNRNW (i.e., unfair toward W). +Run # s b +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +ErrNW +ˆ +FPRNW +ˆ +FNRNW +ˆ +ErrW +ˆ +FPRW +ˆ +FNRW +ˆ +FNRW − +ˆ +FNRNW +474 +5 375 0.373 0.175 0.199 +0.356 +0.183 +0.174 +0.406 +0.159 +0.247 +0.073 +401 +5 10 +0.378 0.189 +0.19 +0.363 +0.197 +0.167 +0.406 +0.173 +0.233 +0.066 +52 +1 53 +0.367 0.172 0.196 +0.351 +0.178 +0.173 +0.397 +0.16 +0.238 +0.065 + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +This detailed view provides insight into how such a result is possible. Broadly speaking, individual runs have roughly similar +error;18 yet, the subgroup-specific error rates that compose the overall error can nevertheless vary widely depending on +the underlying training data. This observation aligns with current interest in model multiplicity in the algorithmic fairness +community (Black et al., 2022b; Watson-Daniels et al., 2022), which imports the idea from Breiman (2001). In this case, as +suggested by Table 32, there are models that demonstrate unfairness toward both subgroups with respect to each error rate +metric +ˆ +Err, +ˆ +FPR, and +ˆ +FNR. When we move away from attempting to find a single model that performs well (accurately or +fairly) on COMPAS, and instead consider the information contained across different possible models, we yield the result that +the average, expected behavior smooths over the variance in underlying models such that the result is close to fair: The +average of unfair models with high variance in subgroup error rates is essentially fair. +Verifying the stability of our illustrative results in Appendix E.4. +To verify the stability of this result, we re-execute our experiments for increasing numbers of train/test splits S and replicates +B. While our results for COMPAS are generally tight for small S (e.g., Figure 2a, this was not the case for South German +Credit, for which it was difficult to estimate self-consistency consistently (Appendix E.4). As a result, for COMPAS, we +did not expect markedly different results for increased S, given that the confidence intervals in Figure 2a are so tight using +S = 10, B = 100. Our results for larger B also follow a similar trend. Our results for S = 100, B = 100 (Figure 46, +Table 33) and S = 100, B = 1000 (Figure 47, Table 34) confirm this intuition. Overall, these results provide the same +conclusion as those with B = 100, S = 10. The run-time for the S = 100, B = 100 and S = 100, B = 1000 experiments +on our cluster were 00:12:58 and 02:21:53, respectively. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Non-white +White +Figure 46. COMPAS split by g = race, B = 100, S = 100 +Table 33. Mean ± STD across S = 100 train/test splits +× B = 100 bootstrap-replicate training runs for random +forest classifiers on COMPAS. +COMPAS +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.366 ± .006 +.173 ± .007 +.193 ± .008 +.729 ± .005 +g = NW +.37 ± .007 +.179 ± .009 +.19 ± .009 +.73 ± .005 +g = W +.36 ± .011 +.162 ± .01 +.198 ± .014 +.728 ± .008 +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Non-white (NW) +White (W) +Figure 47. COMPAS split by g = race, S = 100, B = 1000 +Table 34. Mean ± STD across S = 100 train/test splits +× B = 1000 bootstrap-replicate training runs for random +forest classifiers on COMPAS. +COMPAS +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.366 ± .006 +.173 ± .007 +.193 ± .008 +.729 ± .005 +g = NW +.37 ± .007 +.179 ± .009 +.19 ± .009 +.73 ± .006 +g = W +.359 ± .01 +.162 ± .01 +.197 ± .013 +.728 ± .008 +18This should be taken relatively. In general, COMPAS demonstrates high error; the error is relatively tight given just how much error +there is. The error fluctuates depending on the training data, but the average error rate across S is rather tight, despite the fluctuations in +error within the B runs of each train/test split run s. + +Variance, Self-Consistency, and Arbitrariness in Fair Classification +We provide analogous results for South German Credit, with S = 1000, B = 100 (Figure 48, Table 35) and +S = 2000, B = 200 (Figure 49, Table 36). The run-times for these experiments on our cluster were 00:51:09 and 03:56:36, +respectively. +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female (F) +Male (M) +Figure 48. South German Credit split by g = sex, S = 1000, B = 100 +Table 35. Mean ± STD across S = 1000 train/test splits +× B = 100 bootstrap-replicate training runs for random +forest classifier. +South German Credit +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.28 ± .021 +.172 ± .028 +.107 ± .017 +.769 ± .015 +g = F +.288 ± .065 +.183 ± .073 +.105 ± .038 +.766 ± .041 +g = M +.279 ± .023 +.171 ± .029 +.108 ± .018 +.769 ± .016 +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Self-Consistency +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Cumulative Proportion of Test Set +Female (F) +Male (M) +Figure 49. South German Credit split by g = sex, S = 2000, B = 200 +Table 36. Mean ± STD across S = 2000 train/test splits +× B = 200 bootstrap-replicate training runs for random +forest classifiers. +South German Credit +ˆ +Err +ˆ +FPR +ˆ +FNR +ˆ +SC +Total +.28 ± .021 +.173 ± .027 +.107 ± .017 +.769 ± .015 +g = F +.287 ± .065 +.183 ± .072 +.105 ± .038 +.768 ± .04 +g = M +.279 ± .023 +.171 ± .029 +.108 ± .018 +.769 ± .016 + diff --git a/xNFJT4oBgHgl3EQfgiz6/content/tmp_files/load_file.txt b/xNFJT4oBgHgl3EQfgiz6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d24f60c4669754e060d198efade78197bb9e823b --- /dev/null +++ b/xNFJT4oBgHgl3EQfgiz6/content/tmp_files/load_file.txt @@ -0,0 +1,10717 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNFJT4oBgHgl3EQfgiz6/content/2301.11562v1.pdf,len=10716 +page_content='Variance, Self-Consistency, and Arbitrariness in Fair Classification A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNFJT4oBgHgl3EQfgiz6/content/2301.11562v1.pdf'} +page_content=' Feder Cooper 1 Solon Barocas 2 3 Christopher De Sa 1 Siddhartha Sen 2 Abstract In fair classification, it is common to train a model, and to compare and correct subgroup-specific error rates for disparities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNFJT4oBgHgl3EQfgiz6/content/2301.11562v1.pdf'} +page_content=' However, even if a model’s classification decisions satisfy a fairness metric, it is not necessarily the case that these deci- sions are equally confident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNFJT4oBgHgl3EQfgiz6/content/2301.11562v1.pdf'} +page_content=' This becomes clear if we measure variance: We can fix everything in the learning process except the subset of training data, train multiple models, measure (dis)agreement in predictions for each test example, and interpret disagreement to mean that the learning process is more unstable with respect to its classification decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNFJT4oBgHgl3EQfgiz6/content/2301.11562v1.pdf'} +page_content=' Empirically, some decisions can in fact be so unstable that they are effectively arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNFJT4oBgHgl3EQfgiz6/content/2301.11562v1.pdf'} +page_content=' To reduce this arbitrariness, we formalize a notion of self-consistency of a learning process, develop an ensembling algorithm that provably increases self-consistency, and empirically demonstrate its utility to often improve both fairness and accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNFJT4oBgHgl3EQfgiz6/content/2301.11562v1.pdf'} +page_content=' Further, our evaluation reveals a startling observation: Applying ensembling to common fair classification benchmarks can significantly reduce subgroup error rate disparities, without employing common pre-, in-, or post-processing fairness interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNFJT4oBgHgl3EQfgiz6/content/2301.11562v1.pdf'} +page_content=' Taken together, our results indicate that variance, particularly on small datasets, can muddle the reliability of conclusions about fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNFJT4oBgHgl3EQfgiz6/content/2301.11562v1.pdf'} +page_content=' One solution is to develop larger benchmark tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNFJT4oBgHgl3EQfgiz6/content/2301.11562v1.pdf'} +page_content=' To this end, we release a toolkit that makes the Home Mortgage Disclosure Act datasets easily usable for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNFJT4oBgHgl3EQfgiz6/content/2301.11562v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNFJT4oBgHgl3EQfgiz6/content/2301.11562v1.pdf'} +page_content=' Introduction Consider the following experiment: We fit 10 logistic regression models on different training sets from the COMPAS benchmark (Larson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNFJT4oBgHgl3EQfgiz6/content/2301.11562v1.pdf'} +page_content=', 2016), and compare the resulting classifications for two individuals reserved in a test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNFJT4oBgHgl3EQfgiz6/content/2301.11562v1.pdf'} +page_content=' As shown in Figure 1, while the 10 models 1Department of Computer Science, Cornell University, Ithaca, NY, USA 2Microsoft Research, New York, NY, USA 3Department of Information Science, Cornell University, Ithaca, NY, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNFJT4oBgHgl3EQfgiz6/content/2301.11562v1.pdf'} +page_content=' Cor- respondence to: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNFJT4oBgHgl3EQfgiz6/content/2301.11562v1.pdf'} +page_content=' Feder Cooper